# Structural engineering from an inverse problems perspective

A. Gallet1, S. Rigby1, T. N. Tallman2, X. Kong3,
I. Hajirasouliha1, A. Liew1, D. Liu4, L. Chen1,
A. Hauptmann5,6, and D. Smyl7

1Department of Civil and Structural Engineering, University of Sheffield
2School of Aeronautics and Astronautics, Purdue University
3Department of Physics and Engineering Science, Coastal Carolina University
4School of Physical Sciences, University of Science and Technology of China
5Research Unit of Mathematical Sciences, University of Oulu
6Department of Computer Science, University College London
6Department of Civil, Coastal, and Environmental Engineering, University of
South Alabama

###### Abstract

The field of structural engineering is vast, spanning areas from the design of
new infrastructure to the assessment of existing infrastructure. From the
onset, traditional entry-level university courses teach students to analyse
structural response given data including external forces, geometry, member
sizes, restraint, etc. – characterising a _forward_ problem (structural
causalities $\to$ structural response). Shortly thereafter, junior engineers
are introduced to structural design where they aim to, for example, select an
appropriate structural form for members based on design criteria, which is the
inverse of what they previously learned. Similar inverse realisations also
hold true in structural health monitoring and a number of structural
engineering sub-fields (response $\to$ structural causalities). In this light,
we aim to demonstrate that many structural engineering sub-fields may be
fundamentally or partially viewed as inverse problems and thus benefit via the
rich and established methodologies from the inverse problems community. To
this end, we conclude that the future of inverse problems in structural
engineering is inexorably linked to engineering education and machine learning
developments.

###### Contents

  1. 1 Introduction
  2. 2 Inverse problems, methods, and contemporary use
  3. 3 Structural design as an inverse problem
    1. 3.1 Demarcating structural analysis and design
    2. 3.2 The link to structural optimisation
    3. 3.3 Implications of treating structural designs as inverse problems
  4. 4 Extreme loads on structures
    1. 4.1 Blast loading and inverse analysis
    2. 4.2 Proof-of-concept experimental studies
    3. 4.3 Outlook
  5. 5 Structural health monitoring
    1. 5.1 Background on inverse methods in SHM
    2. 5.2 Static inverse problems in SHM
    3. 5.3 Dynamical inverse problems in SHM
    4. 5.4 Computer vision inverse problems in SHM
    5. 5.5 Digital twins and outlook
  6. 6 Smart materials and structures
    1. 6.1 Piezoresistive nanocomposites
    2. 6.2 Conductivity imaging via EIT/ERT
    3. 6.3 Piezoresistive inversion
  7. 7 A look forward: machine learning and education
    1. 7.1 Machine learned inversion
    2. 7.2 Inverse methodology in structural engineering education
  8. 8 Conclusions

##  1 Introduction

The estimation of structural response from loading and boundary conditions is
a fundamental concept in structural analysis, from elementary Euler-Bernoulli
beam theory to non-linear simulations involving complex structures subjected
to extreme earthquake excitation. In fact, numerical computation of structural
response from known causalities characterises a forward problem (cause $\to$
effect) and has rightly been the source of significant research since the
advent of modern computing. Amongst the myriad of computational frameworks,
the finite element method (FEM) [1, 2, 3, 4], finite difference [5, 6],
spectral element [7, 8], and hybridisations [9] have proven both widely
applicable and successful over the years. The implementation of such forward
models have aided engineers in their ability to model, analyse, and design
structures with arbitrary geometry and precision, contributing greatly to the
presence of skyscrapers, supersonic aircraft, large cruise ships, and many
more engineering examples. In the near future, the pervasiveness of, for
example, the FEM appears inevitable while its usefulness is unquestionable in
structural engineering applications.

Pragmatically however, the final configuration of structural members is not
known at the beginning of the design process; i.e. one iteration of a
structural simulation is not generally sufficient in a real project. This
reality implies that the design of structures is an iterative process – for
example the identification of appropriately sized structural members,
connections, and restraints (causalities) from design constraints, building
codes, and environmental considerations (data). As is often the case, the
iterative design process is carried out initially using design tables, rules
of thumb, handcrafted protocols, optimisation regimes, etc. Nonetheless, this
process is emphatically an inverse problem, where an engineer is given data
alongside design objectives and challenged to determine the appropriate
structural configuration (causalities).

Of course, the field of structural engineering is diverse, in which structural
design is one of many sub-fields where inverse problems are applicable.
Perhaps a more straightforward implementation of inverse problems is
structural health monitoring (SHM), where real-time (or near real-time) data
is used in the prognosis of structural condition. Indeed, the detection,
localisation, prediction, and prognosis of potential damage processes is
enabled in one of two ways, (a) via pattern recognition or (b) solving an
inverse problem (or series of inverse problems) [10]. Moreover, certain non-
destructive evaluation (NDE) approaches are also known to employ inverse
problems (e.g. X-ray CT and emerging NDE approaches in academia) to asses the
damage state of structural elements measured offline [11]. For
contextualisation, a schematic example contrasting the differences between
forward and inverse models is provided in Fig. 1; this well known problem is
referred to as an inverse elasticity problem.

Figure 1: Schematic illustration depicting the forward and inverse problem
relationship for a stretched elastic plate with randomised stiffness
properties. The forward finite element model inputs (causalities) are shown as
the non-homogeneous stiffness properties while the model output is the
displacement field. In contrast, the inverse model aims to estimate the
stiffness properties given the displacement field.

These, and many more sub-fields of structural engineering and structural
mechanics [12], can not only be (fundamentally or partially) viewed as inverse
problems, but, as we aim to illustrate herein, are benefited by the systematic
approaches comprising the rich area of inverse problems. Too often overlooked
by structural engineers and structural researchers, the mathematics of inverse
problems is an established field, ranging from classical statistical/Bayesian
methodology [13] to cutting-edge implementation of deep neural networks [14].
Moreover, while the present use of inverse problem methodologies in structural
engineering is limited, its potential is immense across the expanse of the
structural engineering sub-fields.

In this paper, this potential will be discussed in detail and contextualised
among a broad suite of existing inversion-based applications. To begin, a
clear description of inverse problems and methods will be detailed. Following
this, a review and discussion of inverse methodologies in modern structural
engineering applications will be provided. Inasmuch, the intent of this
manuscript is to examine the following topics in structural engineering
through the lens of inverse problems. We remark, however, that the forthcoming
topical sections are _not_ intended to be exhaustive reviews, but rather, to
provide substantiating evidence for the pervasiveness of inverse problems in
structural engineering. Lastly, realisations, overview, paths forward, and
conclusions will be presented.

##  2 Inverse problems, methods, and contemporary use

Traditionally, the field of inverse problems is concerned with the
mathematical question of if and how one can determine the cause for certain
measurements. Despite being primarily mathematically oriented, the underlying
questions always stem from relevant physics and engineering applications. This
is especially true for one of the most prototypical inverse problems, the so-
called Calderón problem [15] that asks: can one determine the conductivity of
a body from electrical measurements at the boundary. In fact, this question
arose during Calderón’s time as a civil engineer, before he pursued an
academic career in mathematics. In the following, we want to close the loop
back to civil and structural engineering application that once motivated an
entire field of mathematical studies, by utilising the insights gained in the
last decades.

More generally speaking, inverse problems consist of finding the unknown
characteristics of a structural system from some of the outputs, or
measurements of that system. Most notably, this includes the above mentioned
inverse conductivity problem in geophysics [16] and engineering, but also
includes a large field with applications in medical image reconstruction [17,
18]. Mathematically, such problems are ill-posed, broadly meaning that the
parameters to be estimated $\theta$ are highly sensitive to changes in the
measurement data $d$. The solution to the inverse problem involves estimating
the parameter $\theta$ from a fixed set of measured data $d$, in contrast to
the forward problem of computing $d$ from knowledge of the system parameter
$\theta$. Specifically, this means given the forward model $U$, that models
the system equations, we first formulate the underlying observation model

$d=U(\theta)+\delta d.$ (1)

Here, $\delta d$ denotes an error term, modelling several sources of possible
errors, such as inaccurate measurements or even inaccuracies in the model
simulation. The question that remains is: ”how can we obtain $\theta$ from $d$
given the above relationship?” Which we will call the reconstruction problem
in the following. It is important to note that we can not simply invert the
forward problem (1), as the ill-posed nature implies that there can be either
no or multiple solutions and additionally under inevitable measurement noise
these solutions are not stable to compute by direct inversion. This ill-
posedness of the inversion procedure constitutes the underlying paradigm of an
_inverse problem_.

In order to obtain stable reconstructions, we make use of a concept known as
regularisation [19], which aims to assign a unique solution to each set of
measurements in a stable manner, that means if the noise in the measurement
vanishes, we would obtain the original system parameter. We can separate such
stable reconstruction procedures into two primary classes: ones that compute a
solution $\theta^{*}$ directly from measured data and those that iteratively
aim to fit a solution by minimising a suitable cost functional. In the first
case we aim to formulate an inverse mapping $U^{\dagger}$, such that

$U^{\dagger}(d)\approx\theta.$ (2)

The primary problem in obtaining such direct inversion algorithms is that they
can be highly dependent on the problem under consideration. Especially so,
when the relationship between $d$ and $\theta$ is non-linear. Thus, obtaining
such a mapping is a highly non-trivial task, but reveals much about the
underlying problem characteristics and hence is a primary interest of
mathematical research [17, 20, 21].

The second case, is a more principled approach that can be formulated for a
large class of problems. The underlying premise is to reformulate the
reconstruction problem as an optimisation problem. That is, we formulate a
cost functional that measures how good our reconstruction fits the data while
simultaneously enforcing some additional characteristics and acting as
regularisation for the reconstruction process. Specifically, the
reconstruction problem then writes as finding a minimiser of

$\theta^{*}=\operatorname*{arg\,min}_{\theta}\frac{1}{2}\|U(\theta)-d\|_{2}^{2}+\alpha
R(\theta).$ (3)

Here, the first term enforces that reconstructions fit the data, whereas the
second term is the so-called regularisation term. As discussed previously,
this regularisation term is necessary when dealing with inverse problems, as
it prevents a solution from over-fitting the measurement noise. Importantly,
by incorporating prior knowledge in the design of $R$ [22, 23], we effectively
choose preferred solutions and overcome the problem of non-uniqueness.
Finally, the parameter $\alpha>0$ balances both terms and depends on the noise
amplitude. Solutions to (3) are computed by suitable optimisation schemes, for
which repeated evaluation of the forward model $U$ will be necessary.
Consequently, computing solutions to (3) can be computationally highly
expensive, if the evaluation of $U$ is expensive. Thus, for nonlinear problems
fast converging algorithms, such as Gauss-Newton or related methods [24], are
preferred.

Lastly, with the recent rising popularity of data-driven methods, researchers
have designed computationally more efficient ways to address the
reconstruction problem [25]. Such data-driven approaches are often inspired by
classical reconstruction algorithms discussed above. For instance, one can
replace the direct reconstruction operator or parts of it, with a data-driven
component, typically consisting of a neural network [26, 27, 28]. Given a set
of informative reference data, one can then learn a suitable mapping mimicking
(2). Alternatively, researchers have investigated the possibility to improve
the iterative process to compute solutions to (3), by replacing parts in the
optimisation algorithm with learned components [14, 29, 30, 31, 32], or
entirely building network architectures motivated by the iterative solution
process [33, 34]. The use of data-driven methods is expanded upon in §7.

##  3 Structural design as an inverse problem

###  3.1 Demarcating structural analysis and design

Prior to World War II, engineering higher education was originally focused on
the art and practice of engineering design [35]. By the 1960s however, due to
the success of science-based ventures such as the Manhattan Project and the
rise of government-sponsored research grants that severed the link between
academia and industry, engineering science became the main field of research
and teaching at universities [36]. This research led to the development of
powerful structural analysis techniques, yet also left engineering graduates
with a noticeable loss of practical engineering skills [37]. In the 1990s and
early 2000s, there was a push to introduce capstone design projects in
university engineering degrees to address this shortcoming, the success of
which is disputed [38].

There is a broad agreement within the literature that analysis and design are
two distinct activities. Structural analysis, which falls into the field of
engineering science, is primarily concerned with establishing knowledge-that
explains the world, whereas design is concerned with knowledge-how something
works [39]. Design is often characterised as being ill-structured [40], open-
ended [41] or even “wicked” [42], qualities which do not necessarily lend it
to science-based research and helps motivate the search for a different
paradigm which more adequately addresses its true nature.

One perspective to account for these differences is to recognise structural
analysis and structural design as being two different types of problems.
Although structural analysis is typically seen as a sub-field of design to
validate and justify the adequacy of structural elements [43], we like to
advocate the view that structural design is in fact an inverse-problem, with
structural analysis forming the related forward-problem.

Unlike typical inverse problems such as the one shown in Fig. 1, where
physically measured data (the displacement field) is used to identify the
causalities (stiffness properties), in structural design we are dealing with a
theoretical construct, or “theoretically” measured data. This data contains
the set of complex design constraints which need to be adhered to, such as
ultimate (ULS), serviceability limit states (SLS), sustainability and
constructability. From this perspective, structural design can be seen as the
process of arriving from a specific set of constraints to a viable structural
solution, with analysis being the process of checking if the proposed
structural solution adheres to those constraints as shown in Fig. 2.

Figure 2: Relationship between structural analysis and design, in which design
is the inverse problem of evaluating a suitable structural solution given a
set of constraints. Some constraints (such as stability constraints) render
themselves more easily to quantitative treatments.

This perspective is evidenced by realising that the key features of inverse
problems, specifically their ill-posed nature as explained in §2, manifest
themselves in design. Their shared characteristics include being unstable,
non-unique and sometimes un-solvable problems [44]. Notice how these qualities
were alluded to as “ill-structured”, “open-ended” and “wicked” by previous
researchers, yet to the best of our knowledge, this is the first time that
literature has attempted to view structural design as an inverse problem.

We suggest that the application of the inverse problem perspective in design
gives rise to the idea of a “design model”. Similar to how an “analysis model”
allows us to evaluate the action effects of a given structural solution,
solving the forward problem shown in Fig. 2, a “design model” would generate
structural solutions which adhere to the given set of user defined
constraints, solving the inverse problem. This perspective sheds light on the
characteristics which design problems directly share with other inverse
problems, such as the non-uniqueness property. Multiple viable solutions often
exist to any given design brief, one example of which are the various truss-
archetypes for bridge designs to span a similar distance. In order to create a
viable design model, it would be necessary to provide some form of
regularisation, examples of which are provided in §5 in the context of
structural health monitoring, which effectively encourage design models to
choose preferred solutions based on prior knowledge. This could be achieved by
constraining the solution space to a sub-set of structural systems, cross-
sections or materials based on the specific constraints provided. Other
properties of inverse problems which arise in design are shown in Table 1 and
a comprehensive understanding of the application of the inverse problem
perspective warrants further research using specific design examples.

Ill-posed inverse problem characteristics |  General description |  Examples in structural design
---|---|---
Unstable111In reference to the solution space, not structural instability (buckling), see [44]. |  Small changes in constraints can lead to large changes in the solution space |  Impact of grid-spacing on the selection of appropriate floor options
Non-unique |  For a given set of constraints, multiple solutions exist |  Various truss archetypes which can span the same distance
Unsolvable |  Lack of realistic constraining data impedes the search for an adequate solution |  Current lack of appropriate structural materials for space elevator designs
|  |
Table 1: Overview of ill-posed inverse problem features in the context of
structural design.

We note that the design problem can be also formulated as an optimal control
problem [45], where the optimal design parameters are thought to be found as
minimiser of a penalty function while satisfying the system equations.
Whereas, the optimal control approach can provide an effective way to solve
complicated design problems, it falls short in accounting for uncertainties or
inaccuracies in the forward model, and especially the link between measurement
and system parameters. We believe that the here lies the strength of the
inverse problems viewpoint, that offers a rich interpretation and link between
system parameters and measurements given as the forward model. This is not
only promising for the optimisation task in the design process, but also for
new ways to approach the modelling of the forward problem.

###  3.2 The link to structural optimisation

As explained in §2, inverse problems can effectively be solved iteratively
[46]. Effectively, this process involves making an estimate of the structural
solution based on experience and design heuristics (such as simple rules of
thumbs), checking those estimates through an analysis (forward) model and
updating the model if necessary; in other words, this approach is
characterised by creating an optimisation problem. Some examples of forward-
driven optimisation models used in structural design include: optimising the
deformed shape of flexible formwork structures to predefined target geometries
[47, 48], best-fit geometry optimisation of thrust networks in the design of
shell structures [49], and finding the optimal structural forms for long-span
bridges [50], gridshells [51], trusses [52], portal frames [53] and structural
sections [54].

A key theme in these research works is that the structural geometry or member
proportions are not initially defined, but rather are form-found or discovered
in the process, based on the defined loading, boundary conditions and
objectives. Often these discovered structural forms may lead to step change
benefits in terms of performance or reduced material usage, as they are
unbiased by our preconceived perception of what a ”good” structure is, or by
what we currently design and build in the construction industry with standard
template solutions. It is also true for many cases, that a solution may not
even exist, forcing us to accept a closest best fit solution, or it can be
that an optimal solution may have multiple candidates by virtue of the
structure’s static indeterminacy. A common problem with a forward (sometimes
brute-force) optimisation approach, can be lengthy computational times for
structural analyses in the objective functions, and the sheer size of the
design (and hence optimisation) search space, stemming from many wide ranging
input design variables. While fast and globally convergent convex optimisation
programs can be set-up, many practical structural engineering design problems
are inherently non-linear in nature, forcing a slower approach that uses local
searches or heuristic methods with no guarantee of a global optimum. This is a
current challenge faced in solving inverse problems iteratively with forward
models.

###  3.3 Implications of treating structural designs as inverse problems

This inverse problem perspective has various implications. Firstly, it should
emphasise that design problems ask a different question than those related to
analysis. An analysis model solves the forward-problem, and answers questions
such as “what is the ULS utilisation ratio of a particular beam system for a
specific load, with the following specific cross-sections and support
conditions?”. An appropriate design model would ask the reverse: “what is the
group of cross-sections and support-conditions which ensures a utilisation
ratio of less than 1.0 for this particular beam system to carry this
particular load?”. In this formulation, the magnitude of loading and the
utilisation ratio serve as the design constraints (both are known “data”).
Whilst engineering science has produced sophisticated analysis models, the
research of such “design models” is lacking in academia.

Secondly, the inverse problem perspective sheds insight on the possibility of
using data-driven approaches as opposed to more typical optimisation
techniques. The rise of machine learning and deep neural networks could be
used for the development of such “design models”, which focus on directly
identifying a set of structural solutions from a given set of constraints
using learned data. Such models could address some of the challenges faced in
design due to tight deadlines that force early design decisions, whose full
implications might only be realised at the detailed design stage where changes
become cost and time prohibitive. This can lead to structural solutions that
are difficult to build, have poor sustainability metrics, and be costly to
engineer and fabricate. If one instead considers the design process in an
inverse manner, rooting firmly first at the end goal, it could be possible to
reduce project risk and pick more effective structures by appreciating many
solutions to the brief from the onset. This was already alluded to in §2 and
is expanded upon in section §7.

Lastly, the ultimate benefit of an inverse problem perspective is that it
helps to clearly distinguish between analysis and design procedures and
provides academia with an adequate framework to contextualise design model
research. Engineering academia has been dominated by the engineering science
perspective [55], predominantly choosing to research and teach forward
problems. One of the uncomfortable implications of this view is that engineers
from over 150 years ago, which trained primarily in the art and practice of
engineering design, may in fact have been better “inverse-problem solvers”
than academics and graduating engineers of today (who are stronger in solving
the well-structured forward problems) [56]. This might help swing the pendulum
away from focusing exclusively on forward models (analysis) towards a more
stable equilibrium with inverse models (design) by acknowledging the existence
of these two related, albeit distinctively different types of problems. The
use of inverse problem and inverse analysis in a related field to structural
design, notably blast engineering, is discussed next.

##  4 Extreme loads on structures

###  4.1 Blast loading and inverse analysis

High-rate dynamic loads can arise from events such as earthquakes, wind, tidal
waves, impacts, and accidental or malicious explosions. Here, the imparted
load may be comparable or several times larger than the strength of the
material it is acting on, it can be applied and removed in sub-second
durations, and is often highly localised. Accordingly, the notion of static
structural design according to a pre-determined distribution of stresses and
strains may not be appropriate, and instead the designer must consider energy
balance, non-linear analyses, and deformation modes for which there is no
equivalent static counterpart.

Blast loading is undoubtedly one of the most aggressive forms of dynamic
loading. When an explosive detonates it undergoes a violent and self-
perpetuating exothermic chemical reaction, releasing energy through the
breaking of inter-molecular bonds during oxidation [57]. The explosive
material is converted into a high pressure (10–30 GPa), high temperature
(3000–4000∘C) gas which violently expands, displacing the surrounding air at
supersonic velocities (6–8 km/s). This displacement causes a shock wave to
form in the air, termed a blast wave, which eventually detaches from the
expanding detonation product ‘fireball’ and continues to propagate into the
surrounding air, decreasing in pressure and density as it expands.

When a blast wave encounters an obstacle some distance from the source it will
impart momentum as the air is momentarily (either fully or partially,
depending on the compliance of the obstacle) brought to rest at the
air/obstacle interface. Prediction of blast even in the most simple settings
is a considerable challenge to the scientific community. This becomes an
increasingly complex and multi-faceted problem when considering issues such
as: obstacle orientation; proximity of the obstacle to the source and
additional momentum transfer from fireball impingement; secondary combustion
effects either at the air/obstacle interface or in late-time due to partial or
full confinement of the explosive products; the presence of mitigating or
blast-enhancing materials (soil, reactive munitions, etc.).

Real-world blast events are highly uncertain, and the need for inverse
analysis is clear: it is very rare that the exact size, shape, composition,
and location of an explosive device is known _a priori_. Instead, information
relating to the _cause_ of an explosion should be estimated, within reason,
from the more readily observed _effects_ , i.e. the magnitude and severity of
structural damage to surrounding buildings, and cratering of the ground
surface. Whilst inverse analysis is well-established for practical post-event
assessment of explosions—and has been used to determine the size/location of
blast events through forensic investigation of social media videos [58] or
numerical modelling correlating structural damage [59, 60]—the use of inverse
modelling in an academic context is yet to be exploited fully. In the former,
order-of-magnitude estimates are typically deemed sufficient, whereas the
latter requires repeatable, precise measurements and high levels of
experimental control.

The lack of robust yet high-fidelity experimental techniques has stifled
academic research into close-in blast for some time. Close-in blast is
typically defined as the region within approximately 20 radii from the charge
centre, where loading in this region is characterised by a near-instantaneous
rise to peak pressure in the order of 100–1000 MPa, followed by a rapid decay
to ambient conditions typically occurring in sub-ms. Subsequent structural
response may reach a peak value in the order of 10-100 mm and, whilst this may
occur orders of magnitude slower than load application, deformation cycles are
still generally within ms durations.

Recently, two notable advancements have been made in experimental
characterisation of close-in blast and structural response. In the first of
these, researchers at the University of Sheffield (UoS), UK, developed a large
scale apparatus for the spatial and temporal measurement of blast pressures
from close-in explosions [61, 62]. In the second, researchers at the
University of Cape Town (UCT), South Africa, adapted the well-known digital
image correlation (DIC) technique to measure the transient response of the
rear-face of blast loaded plates [63]. These two techniques were combined in a
recent study [64, 65] where, for the first time, detailed loading maps _and_
temporal structural response profiles were developed independently, in a
single-blind study, for identical (scaled) experimental set ups.

###  4.2 Proof-of-concept experimental studies

Of the experiments performed in [64], 12 are relevant to the notion of inverse
analysis of blast loading and structural response, and will be discussed here.
Six tests were performed with spherical explosive charges; three at UoS
measuring blast loading and three at UCT measuring structural response. In the
UoS spherical tests, 100 g PE4 charges were located at 55.4 mm clear distance
from the centre of a nominally rigid target, on which the reflected blast
pressures were measured. In the UCT spherical charge tests, 50 g PE4 charges
were located at 44.0 mm clear stand-off distance from the centre of the
flexible target plates: 300 mm diameter, 3 mm thick, Domex 355MC steel plates,
fully clamped around the periphery. The plate response was filmed using a pair
of stereo high speed video cameras and DIC was used to determine transient
plate response. The two test series can be expressed at the same scale using
well-known geometric/cube-root scaling laws. Here, it is assumed that the
flexible targets deform on timescales orders of magnitude longer than loading
application, and therefore differences between the loaded imparted to the
rigid and flexible plates are negligible.

Specific impulse is given as the integral of pressure with respect to time.
Numerical integration of the UoS pressure histories (at various distances from
the centre of the plate) yields _directly-measured_ specific impulse
distributions. The first few frames of the UCT tests were used to determine
initial velocity distributions of the plate, from which imparted specific
impulse could be _inferred_ through localised conservation of momentum:
$i(x)=v(x)\rho t$, where $i$ is specific impulse, $x$ is distance from the
plate centre, $v$ is out-of-plane velocity of the plate, $\rho$ is density
(7830 kg/m3 for Domex 355MC), and $t$ is plate thickness.

The results for the spherical tests are shown in Fig. 3. The full-field
inferred specific impulse distributions are in close agreement with the
discreet, directly measured values, and both measurements form a tight banding
in an approximate Gaussian distribution [66]. Not only does this indicate a
high level of test-to-test repeatability for each method, but demonstrates
that the two methods are measuring the same underlying phenomena, albeit in
entirely different ways. Thus, it can be said that an imparted impulse will
result in an initial velocity uptake which is directly proportional, and
therefore measurement of one allows for the other to be determined. This
proves the concept of using plate deformation under blast loads in an inverse
approach—namely that from knowledge of plate deformation one may be able to
determine the imparted load—and provides physical verification of the inverse
approach developed by [67, 68].

In addition to the spherical charge tests, six tests were performed in [64]
using squat cylinders (height:diameter of 1:3). Such charges are known to
produce a more concentrated load, with the fireball propagating at higher
velocities along the axis of the charge [69]. This accelerates the growth and
emergence of surface instabilities [70], which gives rise to a more variable
loading distribution [71]. A key research question in this study was: ”will a
more variable loading result in more variable structural response?” In the UoS
cylindrical tests, 78 g PE4 charges were located at 168.0 mm clear distance
from the centre of the target, and in the UCT cylindrical tests, 50 g PE4
charges were located at 145.0 mm clear distance from the centre of the target.
Again, specific impulse distributions were both directly-measured and inferred
from plate response respectively, and the experiments were expressed at the
same scale using common scaling laws.

The results for the cylindrical tests are shown in Fig. 4. Whilst the two
methods again show good agreement, the results can be seen to form a much
wider spread. In contrast to the spherical tests, where peak specific impulse
was seen to consistently act in the plate centre for all tests, here the peak
value is often up to 25 mm from the plate centre (approximately equal to the
charge radius), in both the directly-measured and inferred values. The
inferred values are generally bounded by the directly-measured values, which
suggests that this spread is indeed a genuine feature caused by application of
more variable load.

Figure 3: Directly-measured (UoS) and inferred (UCT) specific impulse
distributions from studies of blast loading and plate deformation following
detonation of spherical explosives, expressed at 100 g (UoS) scale, after [64]
Figure 4: Directly-measured (UoS) and inferred (UCT) specific impulse
distributions from studies of blast loading and plate deformation following
detonation of cylindrical explosives, expressed at 78 g (UoS) scale, after
[64]

###  4.3 Outlook

The aforementioned studies [64, 65] have provided a firm physical basis for
inverse analysis in the context of extreme loading and structural response.
The results have clear implications for the future of research in this area.
Namely, it has been demonstrated that not only can inverse analysis provide
excellent predictions of blast loading in repeatable, well-controlled
situations (as with the spherical tests in [64]), but structural response
measurements are potentially sensitive enough to detect localised variations
in loading (as with the cylindrical tests in [64]). This is particular
important in situations where a highly variable loading might be expected
(e.g. from explosives buried in well-graded soils [72]), but statistical
variations cannot be determined in a robust sense when using direct
measurements (note the discrete nature of the direct measurements in this
study, compared to the effectively continuous nature of the inferred
measurements). This technique may permit, through inverse methods, fundamental
scientific studies of complex mechanisms governing blast loading following
close-in detonation of explosive charges in situations where previous research
has not yet been possible.

##  5 Structural health monitoring

###  5.1 Background on inverse methods in SHM

Unlike the fields of structural design or blast engineering, it is well known
that inverse problems are deeply connected to SHM. In fact, computerised
damage detection can generally be recognised as either a pattern recognition
or inverse problem [10, 73, 74] where unknown or uncertain parameters
(causalities) are estimated via quasi-static or dynamic structural response
data. Among the numerous inversion-based approaches, FEM updating
methodologies are among the most pervasive [75, 76, 77, 78]. Much of the noted
popularity is owed to the flexibility of the FEM for comparing modal
parameters to an undamaged state and in compensating model errors.

Meanwhile, analytical [79], wavelet [80], fractal [81], fuzzy system [82],
Kalman filter [83], chaotic interrogation [84], shape function [85], and
particle filtering [86] approaches, among many others, have proven successful
in uncertain inverse parameter identification applications within SHM. Broadly
speaking, inverse SHM approaches can be grouped as either deterministic or
probabilistic [87] – which is also generally the case in classical inverse
problems [13]. In the latter case, estimation of uncertain SHM parameters
takes the form of a probabilistic term, for example a value with an associated
certainty, a probability itself, etc.

Irrespective of the computational approach used in damage detection, two key
realisations affect the efficacy of inversion methodologies: (a) a baseline is
generally needed to detect/quantify damage [88] and (b) the presence of damage
inherently influences the linearity of structural behaviour [89]. In
addressing (a), reference-free or baseline-free frameworks have been
introduced [90, 91, 92, 93] via the introduction of either assumptions on the
reference state, implementation of prior physics knowledge, or probabilistic
regimes. On the other hand, non-linearity in the structural response can
either act as a corrupting entity when linear forward models are used (e.g.
unacceptable forward model error) or used as an advantage when properly
leveraged. Regarding the latter, as noted in [89], methods based on non-linear
indicators, dynamical systems theories, and non-linear systems identification
approaches can be used to aid or enrich the damage identification process;
such a conclusion can also be extended to the pure usage of inverse approaches
in damage detection.

In the past thirty years, implementation of inversion-based damage detection
methods in SHM has steadily increased. This is due to both the increase in
inverse problems know-how and computational resources. Yet, since the
emergence of contemporary machine learning, the ability to solve problems
deemed previously intractable has exponentially increased opportunities in
this area. For example, in many cases, forward models may not be available or
are too computationally expensive, sufficient non-linearity may exist to
effectively model the desired physics, errors in highly reduced models may be
excessive, the ability to compute model gradients may be overly expensive,
etc. Moreover, the ability of classification networks to readily classify
important variables such as the probability and/or severity of damage from
structural data is intuitively appealing and pragmatically useful. In the
following, we will provide contemporary examples highlighting the use of both
classical and machine learning based SHM inverse approaches.

###  5.2 Static inverse problems in SHM

Incorporation of discrete static (or quasi-static) data measured from
structures into SHM frameworks is well established. For example, a number of
sources, including corrosion, relative humidity, fibre-optic, topography,
laser, potentiometer, strain gauge, electrical, and thermal sources, have been
successfully integrated into long-term condition monitoring protocols [94].
The richness of spatial-temporal data obtained from these sources lends itself
well for use in inversion-based SHM, i.e. given a set or sets of static SHM
data, use an inverse methodology in capturing (potential) damage. This is true
in cases where numerical models are available for the problem physics and when
they are not (e.g. learned models can act as surrogates when physics-based
models are unavailable).

The sheer volume of literature available reporting the successful use of
static inversion methods in SHM is formidable. However, roughly speaking,
static inverse methodologies have been implemented within three areas, (a)
point sensing, (b) area sensing, and (c) volumetric sensing. Holistically, it
can be difficult to distinguish between each of these areas; for example, when
lower dimensional measurements are extrapolated to characterise damage in area
or volume targets [95, 96]. One such example is Digital Image Correlation
(DIC), where the displacement field at discrete points on a structure is
inversely computed via pixel movement and then extrapolated (interpolated) to
a full-field, whereby the quality of the computed field is highly dependent on
the quality of the contrasting area speckle pattern [97]. Similarly, one may
consider the distinguishing of static measurement dimensionality as a local-
global phenomenon where discrete local changes contribute to the analysis of
the global structural system [95]. Lastly, to complicate matters even more,
the use of discrete measurements can yield 2- or 3D information – as in the
case of penetrative electrical measurements, where currents diffuse through
the entirety of a body [98].

Fortunately these, perhaps philosophical realisations, are often washed out
via the nature of inversion methodologies themselves. Pragmatically, at least
in the context of SHM, the solution to static inverse problems generally
requires a model, either physics based or learned. As such, the amalgamation
or assimilation of data and solution dimensionality is often simply a matter
of discretisation or model generation. In a similar vane, when static inverse
problems are ill-posed, solutions generated using lower dimensional data are
regularised/biased using prior models consistent with the solution dimension.

Many examples are available in the literature illustrating the efficacy of
static inversion methodologies for applications in a suite of SHM
implementations. For example, the use of displacement measurements for
capturing SHM causalities in various structural geometries was reported in
[99, 100, 101]. Of note, the specific applications using displacement fields
to reconstruction elastic ans elasto-plastic properties (and corresponding
damage characteristics) has been the source of significant research [102, 103,
104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117]. In the
pervasive case where displacement/strain measurements are discretely measured
from strain gauges/fibre optics, inverse methodologies have also been
fruitfully employed for damage characterisation, pressure and strain mapping,
and shape sensing [118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128,
129]. Perhaps one measure illustrating the success of such inverse approaches
is highlighted by the recent interest in optimising the related sensing
schemes [130, 131, 118, 132].

In the past decades, the emergence of electrical inverse methods has also
proven a viable approach to static condition monitoring [133, 134]. This
family of inversion-based modalities generally utilises three different data
sources including capacitative, direct current, and alternating current based
measurements. Capacitative sensing is perhaps the newest of these approaches
to sensing, where SHM causalities can be inversely computed using smart bricks
[135], area sensors [136, 137, 138, 139], and electrical capacitance
tomography [140, 141]. On the other hand, owing to its established history in
medical and geophysical applications, electrical impedance tomography (EIT, or
electrical resistance tomography, ERT) is becoming a well established approach
to damage detection, reconstruction, and localisation especially in concrete
applications. For these, EIT presently manifests via two approaches,
reconstructing conductivity maps using boundary voltages measured from area
sensing skins [142, 143] or directly imaging a 3D cementitous body [144].
Representative 2D EIT reconstructions are provided in Fig. 5 using a machine
learned approach (direct reconstruction) for imaging flexural and shear cracks
in concrete elements. To this end, EIT has also been used for characterising
area corrosion [143] and localising area temperature variations [145].

|
---|---
a) Simulated shear crack patterns | b) Reconstruction of shear crack patterns
|
c) Experimental image of a flexural crack | d) Reconstruction of a flexural crack
Figure 5: Reconstructions (right column) report probabilistic predictions of
local flexural and shear cracking in a concrete elements. The colour bars
represent the probability of cracks at nodal locations: (a) simulated shear
cracking pattern, (b) probabilistic prediction of the shear crack pattern
using a convolutional Neural Network, (c) Photo of a flexural crack in an area
sensing skin, and (d) probabilistic prediction of the flexural crack using a
feedforward neural network.

In summary of this subsection, it is clear that the use of inverse
methodologies in static SHM applications is pervasive. Meanwhile, a number of
inversion-based modalities are still emerging – or are yet to emerge. Indeed,
given the number of potential static data sources available at present, there
exist substantial opportunities to investigate or formulate new inversion
based modalities. In the light that some physical models for various
underlying physics remain unavailable (either in open source or in general),
the use of learned models may bridge this gap. Lastly, there currently exists
tremendous opportunities in the areas of data fusion and joint imaging, which
remain predominately unexplored in the area of static inversion based SHM
[146].

###  5.3 Dynamical inverse problems in SHM

The use of dynamical data for monitoring the health and condition of
structures is well established [147]. For this, a number of data sources are
available, for example discrete acceleration, strain, displacement
measurements, and recently, coupled electromechanical impedance via
piezoelectric transducers [148, 149]. In the case of typical civil
infrastructure [150], the ability to actively excite monitored structures is
pragmatically challenging due to the extreme magnitude of the excitation
required to attain a distinguishable response. For this reason, ambient
monitoring methodologies have gained significant popularity in recent years
[151, 152, 153]. Irrespective of the dynamical monitoring approach used,
extracting dynamical structural properties of interest can be viewed as an
ill-posed inverse problem [154]. The ill-posedness of such problems results
from a number of actualities, not limited to uncertainties in environmental
conditions (wind, temperature, ground conditions, humidity, etc.), traffic,
measurement noise, the discrete nature of measurements, material
characteristics and numerical modelling error.

One of the most pervasive frameworks used in solving dynamical inverse
problems is model updating, which generally aims to match a physics-based
model (such as a representative finite element model) to measured dynamical
data [89, 73], commonly using a form of modal analysis [155, 156, 157]. The
physics-based techniques are particularly efficient in providing higher
accuracy when testing is restricted. It is often the case that reconstructing
the dynamical SHM properties of interest proves difficult, requiring an
innovative approach; some proposed frameworks have included advanced
optimisation protocols [158] and mode decomposition/superposition [159, 159].
Alternatively, the use of phase space [160], state space [161], singular value
decomposition, $\lambda$-curves method [162][163], auto-regressive [164, 165],
Gaussian process [166], and Bayesian/stochastic approaches [167, 168, 169,
170] have proven successful. As noted in the previous subsection, one metric
for assessing the progress in this field is the number of works aiming to
optimise sensing information, for example in [171, 172, 132, 173, 174, 175,
176, 177, 178].

In the past twenty years, guided wave based modalities have emerged as a
viable approach to dynamical inversion based SHM [179, 180, 181, 182, 183,
184, 185, 186]. Common physical manifestations of guided waves include Lamb
waves (propagating through thin shell and plate structures) [187], Rayleigh
waves (surface waves) [188], and shear waves. Generally speaking, SHM systems
consist of transducer systems used for actuation and measurement accompanied
with an inversion algorithm aiming to reconstruct SHM causalities of interest.
Owing to a number of numerical challenges, conventional solutions to related
inverse guided waves problems are generally not feasible [183].
Consequentially, alternative methodologies have been proposed including, for
example, inverse filtering [189], reverse time migration [190, 191],
Bayesian/probabilistic [192, 193], amongst emerging inversion approaches.

While the use of dynamical inversion based techniques is well established in
conventional monitoring,in some areas (such as guided wave monitoring) it
remains in the early stages of development and affords numerous research
opportunities. It is worth mentioning that with the advent of modern machine
learning methods, we can only anticipate significant advances in forthcoming
years as trained networks are now capable of addressing key SHM challenges
related to, for example, model error estimation/correction [194] and reducing
computational demands associated with many SHM facets [195, 196].

###  5.4 Computer vision inverse problems in SHM

In addition to the discussed static and dynamic inverse problems in SHM,
computer vision-based SHM methods have become an emerging field in inverse
engineering problems in approximately the past decade. Relying on digital
images and videos, vision-based SHM techniques enable affordable and rapid
structural prognosis. The concept of vision-based inverse problems is
straightforward: visual information from the external surfaces of structures
is captured through digital cameras, serving as input data for computer vision
algorithms in detecting, localising, and quantifying structural damage in a
variety of contexts.

Computer vision-based inverse problems can be either static or dynamic in
nature. In dynamical environments, a digital camera is treated as a vision
sensor measuring dynamic structural responses. Instead of directly capturing
the structural vibration through contact-based sensors (e.g. accelerometers),
vision-based algorithms can track structural responses through a video stream.
For example, in [197], researchers applied a video feature tracking technique
to measure the pixel movements of a steel girder in a football stadium under a
service load using a consumer-grade digital camera. These movements were then
converted into displacements using a scaling factor. Similar efforts have been
reported in [198, 199]. Furthermore, through the usage of cameras as
displacement sensors, other key structural features, such as natural
frequencies/mode shapes [200, 201], beam influence lines [202], and bridge
cable loads [203] have been be estimated.

In addition to tracking the surface motion, computer vision algorithms can
offer rapid and reliable inspections against structural damage such as cracks
[204], concrete spalling [205], steel corrosion [206], and other structural
deterioration [207]. To make it is viable, researchers develop vision-based
algorithms to scan and extract damage-induced visual features either across
the entire image scene or within a small predefined image patch (e.g. region
of interest) that is prone to structural deterioration. In general, the image-
based damage extraction techniques can be categorised as: (a) machine or deep
learning-based methods; and (b) non-learning based methods.

The idea of machine or deep learning-based (computer vision) methods is to
train a damage detection classifier through an image dataset with pre-labeled
structural damage. Then, the classifier is applied in characterising
structural damage from newly captured images. Some of the successful
applications include detection of concrete cracks [208, 209] and spalling
[205], steel cracks [210], bolt loosening [211, 212], steel surface defects
[205], pavement cracks [207], and complex situations where multiple damage
types exist [213]. In contrast, non-learning based methods can directly pick
up image features caused by structural damage, and hence do not require any
prior knowledge in training the classifier. For instance, fatigue cracks in
steel bridges can be identified through crack breathing behaviour [214]. Also,
loosened bolts in steel connections can be quantified by extracting the
differential features provoked by bolt head rotations [215]. Fig. 6
illustrates an example by comparing two images of a steel connection at
different inspection periods to extract the differential features provoked by
the loosened bolts.

Figure 6: An example of vision-based bolt loosening detection where image (a)
and (b) are images of a bolted connection taken at different inspection
periods. Two loosened steel bolts are shown in the blow-up figures with
counterclockwise rotations in their bolt heads. Using a series of image
processing techniques, the differential features caused by the bolt loosening
can be identified in c). Detailed discussion can be found in [215].

Computer vision-based methods can be extended for damage detection and pattern
recognition in full scale civil structures. Utilising the platform of unmanned
aerial vehicles (UAVs), the on-board UAV camera can rapidly scan the structure
including the locations that are challenging to be accessed by traditional
contact-based sensors. For example, researchers in [216, 217] applied UAVs and
vision algorithms to leverage effective approaches for post-earthquake
building safety inspections. Similar efforts have been reported for inspecting
dams [218, 219], tunnels [220], and railways [221]. Other researchers adopted
satellite images to examine damage status over a larger scope of work (i.e.,
multiple buildings at the community level) after natural disasters such as
flood, earthquake, volcanic eruption, hurricane, and wildfire [222, 223].

UAV platforms are also capable of collecting a large volume of images of civil
structures under different camera angles through automated route planning.
Such an advantage can be further enhanced by a computer vision workflow,
termed photogrammetry, for the purpose of reconstructing the 3D model of the
structure. Relying on structure-from-motion with multi-view stereo (SfM-MVS)
algorithms [224], photogrammetry technique can create a 3D structure model
(e.g. a 3D point cloud) based on 2D images. Photogrammetry leverages several
potentials in inverse SHM problems. For instance, in recent work, [225]
created a dense point cloud of a building in the construction site based on
UAV images. Then the point cloud was integrated with a building information
modelling (BIM) model for labelling the structural components in original UAV
images. In [226], the researchers utilised the dense point cloud to assist the
creation of a finite element model of a masonry bridge. The authors argued the
benefit of using the point cloud was twofold: the point cloud depicted the
accurate geometric information of the bridge and offered the results of bridge
crack distribution. In [227], researchers leveraged the photogrammetry
workflow to find concrete cracks and spalling of a highway bridge. Lastly, in
the context of the 2021 Hernando de Soto Bridge incident [228], where a large
crack was discovered in a ”fracture critical” element by a private engineering
firm – yet, previously identified approximately two years earlier by a local
operating a commercially available drone: the use of coupled UAV/computer
vision approaches to SHM may be more valuable than ever.

###  5.5 Digital twins and outlook

As this section has illustrated, the use of inverse methodologies in SHM is
both well established and an area of active development. With the rapid
digital transformation of structural assessment and infrastructure asset
management, the emergence of numerous digitally-inspired technologies will
play a key role in the future trajectory of inverse problems in SHM. At the
forefront, digital twins have been the source of increasing research and
industrial interest in recent years [229]. While the scope of digital twins’
applications spans beyond SHM alone, its basic aim is to provide information
on the current or future state of an asset by combining real-time data and a
physical/data-driven model offers many potential avenues for engagement with
the inverse problems community. Nonetheless, in specifically considering a
classical SHM application, such as damage localisation [230]: developments
stemming from the inverse community including, for example, state estimation
[231, 232, 233], uncertainty/model error approximation/compensation [234, 194,
235], regularisation [236], and model reduction [237], have excellent
potential for enriching or enhancing digital twin frameworks.

As a whole, the future outlook for the integration and advancement of inverse
methodologies in SHM is very bright. Indeed in the past 20 years, we have seen
an exponential increase in high-performance computing and graphical processing
units development and assimilation into modern civil and mechanical
engineering applications [238, 239]. Coupled with powerful inverse frameworks
for large-scale problems (e.g. Krylov solvers [240] and distributed computing
[241]) and machine learning [242], we can only expect a steady increase in (a)
the breadth of inverse problems the SHM community is able to address and (b)
and an evolution in innovative inversion-based approaches to solving
increasingly challenging SHM problems.

##  6 Smart materials and structures

_Smart_ , _self-sensing_ materials have received immense attention in recent
decades [243, 244, 245]. A material is said to be self-sensing if it exhibits
a property change in response to external stimuli. These materials are able to
intrinsically report on their health or condition in a spatially continuous
manner and with less hardware/instrumentation burden than traditional point-
based sensing technologies (e.g. strain gauges, piezoelectric patches,
accelerometers, etc.). In structural engineering, external stimuli are often
mechanical effects such as deformation, damage, or loads. Hence, integrating
smart materials into next-generation structures may allow for unprecedented
health monitoring and diagnostics. A discussion on smart materials in
structural contexts may therefore also be considered as a subset of the
preceding discussion on SHM (see §5). Nonetheless, we will treat it as a
distinct topic in the forthcoming due to its unique inverse problems.

Although self-sensing is an umbrella term encompassing many different physical
effects, the _piezoresistive effect_ has perhaps received the most attention
to date (see recent reviews [243, 246, 247]). Piezoresistive materials are so-
named because they exhibit a change in electrical conductivity (or its
inverse, resistivity) upon deformation. This means that every point of the
material is capable of relaying information on its mechanical state. Damages
such as voids, ruptures, or fractures can also be detected since the removal
of material represents a conductivity loss. But spatially-continuous
piezoresistive sensing presents two challenges of relevance to inverse
problems: (a) It is not practical to instrument electrodes at every point on a
structure, which means that it is necessary to deduce conductivity
distributions from a finite set of measurements. And (b), even if we could
recover a spatially continuous mapping of the conductivity, electrical
properties are of little consequence to the structural engineer. We would much
rather know the underlying mechanics that give rise to an observed
conductivity distribution. We will address both of these inverse problems.
First however, a brief summary of the physics and piezoresistive modelling
approaches is given. It will be seen later that modelling techniques are
essential to solving inverse problem (b) above.

###  6.1 Piezoresistive nanocomposites

Many materials intrinsically exhibit piezoresistive properties. For example,
carbon fibre-reinforced polymers (CFRPs) are well known to change conductivity
when loaded elastically [248, 249]. Here however, we will instead focus on
materials that have been _engineered_ to be self-sensing; that is, an
additional constituent has been added to the material system without which it
does not exhibit piezoresistivity. This is most commonly done by adding a
conductive phase to a non-conductive matrix such as polymers (including
structural polymers like epoxy vinyl ester [250], polymeric thin films for use
as sensing skins [251], laser-induced graphene inter-layers in continuous
fibre composites [252, 253], and even polymer binders in energetic materials
[254, 255]), cements [256], or ceramics [257]. Electrical transport is then a
consequence of percolation – the composite conducts electricity when enough
fillers have been added to form an electrically connected network. Because the
percolation threshold decreases with aspect ratio, fillers with ultra-high
aspect ratios like carbon nanotubes (CNTs) are popular. There are considerable
challenges associated with manufacturing CNT-based nanocomposites such as
achieving good dispersion – ultra-small fillers such as CNTs have a tendency
to agglomerate which can degrade the mechanical properties of the composite.
But manufacturing is outside of the scope of this manuscript and is well-
covered elsewhere [258, 259].

Considerable effort has also been devoted to the development of
piezoresistivity models – computational and/or analytical means of predicting
how conductivity changes for a prescribed strain. These efforts can be broadly
categorised as (a) equivalent resistor network models [260, 261, 262], (b)
computational micro-mechanics models [254, 263, 264, 265], or (c) homogenised
macroscale models [266, 267, 268]. In (a) equivalent resistor network models,
high aspect ratio fillers like CNTs are represented as either straight or
wavy/curved sticks in a micro-domain [260, 261, 262]. These sticks are
discretised into resistors based on the length, diameter, and conductivity of
the fillers, and junctions between nearby fillers are discretised into
resistor elements based on the equivalent resistance felt by an inter-filler
tunnelling electron [269]. The conductivity of the nanocomposite can then be
calculated from the overall resistance of the discretised nanofiller network
and the dimensions of the micro-domain. For a given deformation of the micro-
domain, the translation and rotation of the fillers can be calculated by
treating them as rigid inclusions [270]. Post-deformation, the conductivity of
the micro-domain is recalculated thereby allowing piezoresistive properties to
be predicted.

Second, (b) computational micro-mechanics models use computational means to
simulate both phases of the composite – the non-conductive matrix and the
conductive fillers [254, 263, 264, 265]. Because of this, and unlike
equivalent resistor network models, computational micro-mechanics models can
incorporate nuanced effects such as nanofiller-to-matrix debonding, nanofiller
deformation, etc. A limitation of this approach is computational expense due
to individual nanofillers and the enveloping matrix being explicitly
simulated. It is therefore difficult to scale these models to structural
levels.

And third, (c) homogenised macroscale models describe the conductivity of the
nanocomposite as an analytical function of the strain state without simulating
individual fillers. Rather conductivity/resistivity-strain are coupled through
analytical functions based on excluded volume theory [267] or via
‘constitutive’ tensors (note that these are not proper constitutive relations
because conductivity and strain are not energy complements) [268, 271, 272].
This approach is therefore less computationally expensive than equivalent
resistor network and computational micro-mechanics models. Importantly,
homogenised approaches can be readily integrated with structural analysis
tools such as the finite element method, thereby allowing for macroscale
piezoresistive analyses. As will be discussed in §66.3, this allows for strain
recovery via piezoresistive inversion. Despite these advantages, analytical
models suffer from having to make assumptions regarding average inter-filler
spacing, average orientation of nanofillers, and the need for calibration
data.

###  6.2 Conductivity imaging via EIT/ERT

As discussed in §55.2, conductivity imaging modalities such as EIT (or DC
resistivity imaging via ERT) have been explored for health monitoring in
structural materials [247]. EIT is a natural complement for piezoresistive
materials because it allows for the spatial localisation of damage and the
_mapping of deformation and strain._ There are several factors that make the
EIT interesting to pair with self-sensing materials: (a) This combination
allows for sub-surface strain imaging. That is, a myriad of techniques exist
for monitoring surface strains such as strain gauges, DIC, holographic
methods, etc. Tools for studying sub-surface strains, however, are much more
limited, often require ionising radiation, and can be costly (e.g. volumetric
strains via X-ray digital volume correlation [273]). (b) Traditional
‘limitations’ of EIT can be strengths in structural applications. For example,
the simplest implementation of EIT favours spatially smooth solutions. This is
obviously undesirable when imaging discontinuous features with distinct
boundaries such as organs in a living body. Strain fields, however, are often
smoothly varying thereby playing into EIT’s strengths. And (c), because
piezoresistive materials are engineered, we can leverage our knowledge of
their piezoresistive response to build bounds into the EIT inverse problem.
Two examples of strain imaging via EIT in self-sensing polymeric composites
are shown in Fig. 7. Both examples leverage knowledge such knowledge of
conductivity change bounds to build constraints into the solution space.
Beyond these advantages for structural imaging, the pairing of EIT/ERT with
self-sensing materials may also have keen, as-of-yet unrealized potential for
extreme loading (particularly self-sensing energetic materials [254, 255]).
That is, it may be illuminating to image energetic materials as they detonate.
However, most prevailing imaging modalities have not the temporal resolution
to capture these fleeting moments and require hardware that is too expensive
to risk damaging. EIT, on the other hand, can have ultra-high temporal
resolution (on the order of hundreds of microseconds for optimized systems
[274]) and uses only low-cost hardware (i.e. sacrificing the hardware during a
detonation is of virtually no financial consequence). Thus, there may be much
exciting potential for overlap between smart materials + EIT and extreme
loading as described in §4.

There have been many studies on the topic of EIT and piezoresistive materials.
A few representative examples are summarised in this manuscript, but
interested readers are directed to a recent review for a more in-depth
discussion [247]. Some of the first work in this area made use of self-sensing
nanocomposite _sensing skins_ produced by a layer-by-layer fabrication
technique [251, 275]. These skins were applied to substrates, and EIT was then
used to identify and spatially localise deleterious effects including
mechanical etching, pH exposure, impact damage, and residual strains from
impacts. Further on the topic of polymer-based composites, thin pressure
sensors were produced by embedding a non-woven textile modified with CNTs into
a soft elastomer. EIT was then employed to visualise various pressure
distributions including non-uniform distributions [276]. EIT was also recently
applied for damage detection in a ceramic-based composite that was modified
with micron-sized waste-iron particles [277]. And as a final representative
example, self-sensing cement composites have been produced by spray-depositing
a CNT-modified latex on the aggregate phase. EIT was then successfully used to
localise various damages in the material [278].

###  6.3 Piezoresistive inversion

Even though solving the EIT inverse problem allows for the spatially
continuous visualisation of mechanical effects in these materials, this poses
an obvious problem – structural engineers are generally not interested in
conductivity. They would much rather know the spatially-varying components of
the strain tensor, stress tensor, or precise damage characteristics since
these factors drive structural analyses and health assessments. Recalling also
that various macroscale piezoresistivity models exist as described in §66.1,
we can formulate another inverse problem as follows: “for a given EIT
conductivity distribution (or, more directly, for a given set of EIT boundary
data) and with an accurate model of conductivity-strain or conductivity-damage
coupling for a particular material, can the precise mechanics of the system be
recovered?”

Figure 7: Examples of EIT and piezoresistive inversion applied to self-sensing
nanocomposites. a) A soft CNF/PU is deformed by rigid, non-conductive
indentors [279]. EIT is then used to image the deformation-induced
conductivity changes, and piezoresistive inversion is used to recover the
displacement field (multiplied by a factor of five for ease of visibility). b)
A hard CNF/epoxy is loaded in tension with a stress raiser at its centre
[280]. EIT is again used to image the conductivity change. Lastly,
piezoresistive inversion is used to recover the underlying displacement field.
With knowledge of the material’s elastic properties, strains and stresses can
be spatially mapped. The first principal stress of the guage section is shown
here along with comparison to a traditional FEM solution for validation.

Both of these piezoresistive inversion problems—strain recovery and damage
recovery—are challenging. The former is challenging because, under ideal
circumstances, we seek six components (in 3D) of a strain tensor from a scalar
conductivity field. Prospects can be improved somewhat by instead seeking the
displacement field (i.e. three unknowns in 3D) from the conductivity data and
making use of reasonable assumptions (e.g. plane strain and plane stress)
where applicable, but the displacements-from-conductivity inverse problem is
nonetheless under-determined. The challenge is exacerbated by the fact that
circumstances are never truly ideal; conductivity and conductivity changes are
not exactly isotropic, and EIT cannot image individual components of a
conductivity tensor. Even for the case of damage imaging, an accurate model of
material breakage-induced conductivity change is needed. For simple
nanofiller/matrix phase nanocomposites (i.e. without reinforcing fibre),
material breakage can be treated as a cessation of conductivity. For more
complicated material systems such as nanofiller-modified continuous fibre
composites, however, material breakage-induced conductivity changes must
account for factors such as anisotropy and residual post-damage conductivity
due to inter-laminae contact. And even if a suitable damage model is
developed, the inverse solver needs to be capable of reproducing potentially
complex damage shapes that are not readily amenable to parameterisation. Both
strain and damage recovery are additionally hampered by the fact that EIT does
not produce accurate conductivity distributions in an absolute sense or a
spatial sense.

Despite these challenges, the piezoresistive inversion problem has received
some attention. An initial effort used an analytical inversion framework
predicated on iteratively minimising the $l_{2}$-norm of an error vector
between a predicted and observed conductivity distribution [281]. Although
this work was entirely computational and limited to simple deformations and
infinitesimal strains, it nonetheless demonstrated that piezoresistive
inversion was possible. The next work in this area used EIT to image strain-
induced conductivity changes in a carbon nanofiber (CNF)-modified polyurethane
(PU) composite [279]. Three marbles (i.e. comparatively rigid non-conductive
indentors) were pushed into the CNF/PU as EIT measurements were taken. A
similar analtyical approach was used to reproduce the displacement field. Two
important factors differentiated this study – experimental validation of
piezoresistive inversion and successful application to materials undergoing
finite strains. Later works looked at utilising metaheuristic algorithms for
solving the strains-from-conductivity inverse problem in a CNF-modified epoxy
[280, 282]. The CNF/epoxy was moulded in the shape of a plate with a hole and
loaded in tension, causing strain concentrations in the vicinity of the hole.
Genetic algorithms, simulated annealing, and particle swarm optimisation were
explored because it was observed that the analytical formulation failed to
converge to the physically correct solution for this more complex loading
state. It was found that genetic algorithms performed the best for this
inverse problem, but all methods compared favourably to DIC experimental
validation. Because epoxy is relatively brittle, these studies were
necessarily limited to infinitesimal strains. Despite the successes of the
preceding studies, they were all limited to electro-mechanically isotropic
materials. Translating these capabilities to electro-mechanically anisotropic
materials remains a daunting challenge. Fig. 7 summarises results for strain
recovery via piezoresistive inversion.

Some work has also been done for damage recovery via piezoresistive inversion.
Recall that a key challenge with the damage recovery inverse problem is shape
parameterisation. To that end, these preliminary studies have considered
relatively simple damage cases. For example, various machine learning
algorithms were used to categorise three damage conditions in a self-sensing
bone cement directly from EIT boundary voltage data [283]. Thus, self-sensing
materials also have overlap with machine learning methods (see §7). Bone
cement is poly(methyl methacrylate) (PMMA) used to facilitate robust contact
between an othopaedic implant (e.g. a total joint replacement) and hard bone.
Failure of the PMMA interface is often difficult to detect via radiographic
imaging, hence the motivation for alternative diagnostic tools. This
clarification aside, the parameterisation was relatively simple in this case –
only four distinct states were possible (three damaged plus one healthy). The
combination of EIT, self-sensing PMMA bone cement, and machine learning
allowed for correct damage classification with over 90% accuracy. In another
study, image recognition-based machine learning was used to identify, size,
and localise through-hole damage to a self-sensing composite plate [284]. The
image recognition algorithm was trained using computationally generated EIT
images on a simple square domain punctured by a random number of randomly
sized circular holes. The trained network was able to adeptly predict through-
hole size, location, and number from EIT conductivity maps with good accuracy
– likely better accuracy than human interpretation of EIT images. But this
again utilised a very simple damage parameterisation (i.e. needing only to
predict hole number, radius, and in-plane coordinates). As a final example, a
recent study looked at delamination shaping from EIT images in CNF-modified
glass fibre/epoxy laminates [285]. In this study, delaminations induced by
low-velocity impacts were parameterised as ellipses of unknown major and minor
axes and centred at unknown in-plane coordinates. A genetic algorithm was used
to inversely determine these parameters by minimising the $l_{1}$-norm of the
difference between experimentally collected EIT boundary voltage data and
boundary voltage data predicted by a computational model of the damaged
domain. Destructive analyses of the post-impacted laminates revealed that the
GA-predicted damage state much more closely matched the actual delamination
size and shape than the EIT conductivity images. This third example of damage
recovery is particularly noteworthy because it represents a much more
realistic damage state.

In summary, this section has looked at smart, self-sensing materials from the
perspective of structural inverse problems. Two noteworthy inverse problems
were discussed – the EIT inverse problem and the strain/damage recovery
problem. The former has been extensively researched in other fields (e.g.
[286]). The latter, however, is much more recent and has only been the subject
of a few precursory studies. Much work remains to be done regarding the
inversion of electrical data to obtain underlying mechanical effects.
Nonetheless, it can be seen that the field of smart materials in structural
applications has much potential and cross-cutting overlap with other topics of
this article including extreme loading, SHM, and machine learning.

##  7 A look forward: machine learning and education

This paper has evidenced the pervasiveness of inverse problems and
methodologies used in the field of structural engineering. Yet, the following
remains to question: ”what is guiding the future trajectory of inversion in
structural engineering?” In this section, we examine what we foresee as the
two most influential future areas in addressing this question: machine
learning and education.

###  7.1 Machine learned inversion

Many areas of structural engineering rely heavily on applied mathematics and
science – from the integration of material models within finite element
frameworks to experimental measurement of structural response excitation. From
a broad scientific perspective, there are two paradigms to research, either
(a) the Keplerian paradigm (data driven, obtaining discoveries via data
analysis) or (b) the Newtonian paradigm (first principles, discovery through
fundamental principles) [287]. Without question, structural engineering
research uses both principles.

Often, first principles approaches are manifested via partial differential
equations (PDEs) and their analytical or numerical solutions. However, both in
research and practice, obtaining solutions to PDEs can be infeasible or
intractable, for example owing to computational demand, a dearth in available
numerical regimes and/or the “curse of dimensionality” [288]. In such cases
where engineering problems are governed by such PDEs, solving a related
inverse problem using a conventional methodology would also be a dubious task.
When faced with this situation, we are most likely constrained to adopting a
Keplerian approach.

Unmistakably, machine learning has provided the science and engineering
communities with a powerful tool for data-driven analysis, prediction,
assessment, and significantly more. Structural engineering research has also
greatly benefited from machine learning, especially in the areas of
performance assessment [289], SHM [10] and analysis of various structural
phenomena [290]. Yet, significantly less attention has been paid to the use of
machine learning for solving inverse problems in structural engineering
outside of areas such as SHM and NDE (as identified in §5).

Exemplifying this reality, structural design highly under utilises machine
learning and data science. Design is traditionally associated with an
iterative nature, in which various structural concepts are tested conceptually
until a prevailing option is identified which adheres to constraints initially
identified. The nature of this iterative design process has been understood in
the past using both positivist, pragmatic and post-modernist epistemologies
ranging from Simon’s “science of the artificial” [291], Schön’s “reflexive
practice” [292] and Buchanan’s “placements for contextualisation” [293]
respectively. The complexities involved in design from satisfying conflicting
demands to exercising appropriate judgements is hence often attributed as
being an innate human skill [56].

However, recent developments successfully solving inverse problems using data-
driven approaches suggest that such methodologies could also be incorporated
within structural design [25]. This challenges the notion that design is an
exclusive human ability, a development which mimics the success of self-
driving cars through data-driven approaches [294]. What is typically
considered “intuition” or “engineering judgement” may in fact be recalling
“data-points”, committed to long-term memory through the process of
experience, that are suitable solutions based on the unique set of constraints
one is presented with [291]. Indeed, supervised machine learning was
originally understood to create mapping functions that correlate a set of
inputs (a specific set of constraints for a design situation) to associated
outputs (viable structural solutions) by learning from a given data set
(experience) [295]. We note that this view has significantly evolved with the
onset of deep-learning, as well as the use of unsupervised and reinforcement
learning regimes.

We therefore believe that this inverse problem perspective evidences the need
for research on the development of machine learning tools for structural
design. Data-driven approaches are often dismissed because they are ”black-
boxes” which lack a scientific rationale for their outputs. However for
design, this criticism might be unwarranted, and instead highlights the
dogmatic concentration within academia on exclusively solving forward
problems, which due to their well-posed nature, lend themselves to engineering
science thinking. Recently, researchers have started investigating data-driven
models. In one example, neural networks and clustering were applied to form a
bridge and navigate the design space and shortlist viable and fitting
solutions [296, 297]. Other examples include researchers applying machine-
learning models to build suitable structural predictors for conceptual design
related to building massing [298].

The former developments are, in our view, only a starting point for the
employment of machine learned models used in solving inverse problems in
structural engineering as a new horizon of data-driven approaches emerge.
Especially, for the cases involving intractable forward problems, model
reduction techniques have been promising [299, 13, 300, 301], but these are
either difficult to design by hand or are restricted by overly simplistic
assumptions. Here, data driven approaches are a powerful alternative to
compensate for modelling errors [194, 234, 302] or reducing computational cost
of iterative optimisation schemes by model approximations [32, 303]. Finally,
we note that recent developments in geometric learning extend deep networks on
Euclidean meshes to general meshes, such as finite elements, by a embedding
them into graph structures essentially utilising the underlying geometry [304,
305]. This opens the possibility to extend many data driven approaches to
complex structural problems.

###  7.2 Inverse methodology in structural engineering education

Over the last 100 years, engineering education has experienced a number of
fundamental shifts: a shift in focus away from design to engineering science
in the 1960s, the rise of outcome-based accreditation in the UK and USA in the
1990s, along with a re-emphasis of teaching design through capstone projects
in the 2000s [306]. There also is the continued tension between teaching
graduates both the technical knowledge as well as the interpersonal skills
demanded from industry deemed necessary to become effective designers [35]. To
this day, there exists the debate on how to find the required balance between
knowledge-that and knowledge-how as identified in §3 [307]. For civil and
structural engineering disciplines, we believe that some of these challenges
might be addressed by communicating the existence of inverse problems, their
pervasive occurrences as shown by the sections above, and teaching the various
methods and techniques for solving such problems.

The dominance of engineering science within university curricula today, which
primarily focuses on identifying and solving forward problems, might
unintentionally generate the wrong supposition that all problems in
engineering are well-posed with idealised assumptions. Without adequately
addressing the existence of inverse-problems, and their distinctively
qualitative differences with forward problems, it is easy to mistakenly assume
that, for example, structural design is the application of such “forward
problems”. However, the idea that engineering is simply “putting theory intro
practice” [56] or “applied science” [308] has been strongly argued against by
numerous engineers and philosophers [55, 309, 310, 311]. The challenge which
students face when dealing with real-world design problems might be accounted
for by the fact that during the majority of their engineering education, they
might lack a conceptual framework to adequately demarcate design from
analysis. Similarly, in order to engage with other promising fields of
structural engineering, such as structural health monitoring, blast
engineering and smart materials, educating students on inverse problems is
crucial.

Hence, a possible improvement for current civil and structural engineering
curricula is introducing students to the existence of forward and inverse
problems, how they relate to one another and provide examples where each type
of problem arises and how to solve them. As identified previously, this will
also potentially require teaching students a host of new skills, especially if
data-driven models continue to be effective tools for solving inverse
problems, as is the case in structural health monitoring and increasingly
likely in design. More importantly, especially when taking into consideration
the recent developments in data science ranging from CNNs [312], transformers
[313] and graph neural networks [314], there exists a vast spectrum of
knowledge and applications we may not even be aware of.

As a matter of fact, in terms of research, we are perhaps faced with a unique
situation in academia today. Although only time will tell, it could be argued
that similar to how the “invention” (or discovery) of calculus in the 18th
century was instrumental in providing us necessary tools for solving forward
problems, resulting in material models and PDEs which allow the creation of
complex finite-element methods, so too might the rise of machine-learning and
data-science, which is only now starting to gain serious attention in
mathematics [287], allow a more rigorous treatment of solving inverse
problems. By realising the pervasiveness of inverse problems in structural
engineering, but also the fundamental differences with forward problems, there
is potentially a vast, untouched and exciting realm of research which awaits.

##  8 Conclusions

This article aimed to demonstrate that numerous structural engineering sub-
fields may be either fundamentally or partially viewed as inverse problems. It
was shown that this concept is well accepted in, for example structural health
monitoring; however, sub-fields such as structural design are not commonly
(formally) defined as inverse problems. We argue that, by shifting this
paradigm in structural engineering academia and industry, we may collectively
capitalise from the rich methodologies and approaches already established in
the inverse problems community. This beneficial relationship between
structural and inverse communities is expected to pay exponential dividends as
new tools, such as machine learned models, emerge and develop – offering new
opportunities for solving previously inaccessible, intractable, and/or
unforeseen structural challenges.

## Acknowledgements

AH is funded by Academy of Finland Project 338408 and Project 336796 (Finnish
Centre of Excellence in Inverse Modelling and Imaging, 2018–2025). DS was
supported by Engineering and Physical Sciences Research Council Project
EP/V007025/1. DL was supported by the National Natural Science Foundation of
China under Grant 61871356.

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