# Optimized Dual-Volumes for Tetrahedral Meshes

Alec Jacobson
University of Toronto and Adobe Research Toronto

###### Abstract

Constructing well-behaved Laplacian and mass matrices is essential for
tetrahedral mesh processing. Unfortunately, the _de facto_ standard linear
finite elements exhibit bias on tetrahedralized regular grids, motivating the
development of finite-volume methods. In this paper, we place existing methods
into a common construction, showing how their differences amount to the choice
of simplex centers. These choices lead to satisfaction or breakdown of
important properties: continuity with respect to vertex positions, positive
semi-definiteness of the implied Dirichlet energy, positivity of the mass
matrix, and unbiased-ness on regular grids. Based on this analysis, we propose
a new method for constructing dual-volumes which explicitly satisfy all of
these properties via convex optimization.

††volume: 43††issue: 5 Figure 1:  Linear FEM corresponds to dual-volumes formed by connecting simplex barycenters (top, volumes scaled by 50%), but exhibit bias on tetrahedralized grids, e.g., a generalized eigenvector $\mathbf{L}\mathbf{v}_{5}=\lambda_{5}\mathbf{M}\mathbf{v}_{5}$ with a pinned base (bottom). Our optimized dual-volumes coincide with circumcentric dual-volumes on a tetrahedralized regular grid (right), avoiding bias.  | $C^{0}$ | $-(\mathbf{L}+\mathbf{L}^{T})\succeq 0$ | $\ \mathbf{M}>0\ $ | Grid
---|---|---|---|---
Barycentric |  |  |  |
Circumcentric |  |  |  |
[AHKS20] |  |  |  |
[MMdGD11] |  |  |  |
Ours, snapping |  |  |  |
Ours, optimized |  |  |  |
Table 1:  Given a mesh with prescribed positions and connectivity we consider
existing finite-volume methods. Are their resulting dual volumes continuous
functions of vertex positions ($C^{0}$)? Do they yield a positive semi-
definite Dirichlet energy ($-(\mathbf{L}+\mathbf{L}^{T})\succeq 0$) and
generate a positive mass matrix ($\mathbf{M}>0$)? Do they maintain unbiased
performance on a tetrahedralized regular grid (Grid)?

##  1 Introduction

While the finite-element method with piecewise-linear shape functions may be
the _de facto_ standard for tetrahedral mesh processing of solids, finite-
volume methods are useful for a variety of reasons. In particular, unlike
linear finite elements they can result in unbiased operators for
tetrahedralized regular grids (see Figure 1). At the heart of the finite-
volume method is the definition of local dual-volumes around each vertex of a
given tetrahedral mesh. Existing methods attempt to generalize orthogonal
Voronoi dual-volumes, which behave well for Delaunay meshes. Unfortunately,
when considering general (e.g., non-Delaunay) meshes, each approach gives up
one or more desirable properties of its resulting Laplacian and mass matrices:
continuity with respect to vertex positions, positive semi-definiteness of its
implied Dirichlet energy, positivity of the mass matrix, unbiasedness on
regular grids (see Table 1).

In this paper, we categorize existing methods according to these criteria.
Placing methods into a common construction, their differences amount to the
choice of simplex centers, most importantly triangle centers. Each choice
drastically affects the discretization of the divergence operator in a finite-
volume construction. From their constructions we show how properties of each
method arise or break down algebraically.

Figure 2:  Previous methods such as [AHKS20] reproduce circumcentric duals for
perfect grids, where tetrahedra are on the cusp of becoming non-acute. A
gentle nudge and their dual volumes jump to totally different shapes and
sizes. Our optimized dual volumes are continuous functions of mesh vertex
positions.

Based on this analysis, we propose a new method for constructing dual-volumes
which explicitly satisfies all of these properties via convex optimization. In
contrast to previous methods, our dual-volumes are continuous functions of
vertex positions (see Figure 2). As a warm-up, we consider modifying the
positive-volume filtering procedure of Alexa et al. [AHKS20] to ensure
continuity as tetrahedra morph from acute to non-acute shapes (“Ours snapping”
in Table 1). Unfortunately, this — like [AHKS20] — does not ensure that the
Laplacian matrix defines a positive semi-definite Dirichlet energy. Instead,
we propose directly optimizing the centers associated with each simplex to
explicitly ensure symmetry and negative semi-definiteness of the Laplacian.
All properties are satisfied by construction, including continuity as the
optimization problem is strictly convex.

Let’s plunge into our problem’s details. We will consider prevalent prior
methods after introducing the details of our desired properties and postpone a
broader discussion of related work until our final discussion in Section 6.

##  2 Finite Volume Operators

Given a tetrahedral mesh of a solid region $\Omega\subset\mathbb{R}^{3}$ with
$n$ vertices, we will follow a general finite-volume construction to build
matrices $\mathbf{M}\in\mathbb{R}^{n\times n}$ and
$\mathbf{L}\in\mathbb{R}^{n\times n}$.

###  2.1 Mass-matrix

Our goal is to compute a diagonal or _lumped_ mass matrix
$\mathbf{M}\in\mathbb{R}^{n\times n}$, which as a quadratic form computes the
inner product of functions defined by per-vertex scalar values
$\mathbf{f}\in\mathbb{R}^{n}$.

$\displaystyle\mathbf{f}^{T}\mathbf{M}\mathbf{f}\approx\int_{\Omega}f^{2}\,dV$
(1)

We can also consider $\mathbf{M}$ as a linear operator acting on a _yet to be
determined_ local dual-volume $\star_{i}\subseteq\Omega$ around each vertex
$i$:

$\displaystyle(\mathbf{M}\mathbf{f})_{i}\approx\int_{\star_{i}}f\,dV$ (2)

To ensure the diagonally of $\mathbf{M}$ we now assume that $f$ is constant
over each local volume $\star_{i}$. This local dual volume $\star_{i}$ may
further be broken into contributions within the primal volume $\bullet_{t}$ of
each tetrahedron $t$ (usually only those incident on $i$):

$\displaystyle(\mathbf{M}\mathbf{f})_{i}=\left(\sum_{t}\int_{\bullet_{t}\cap\star_{i}}\,dV\right)\mathbf{f}_{i},\quad\mathbf{M}_{ii}:=\sum_{t}\int_{\bullet_{t}\cap\star_{i}}\,dV,$
(3)

with $\mathbf{M}_{ij}=0$ for $i\neq j$.

###  2.2 Dirichlet Energy / Laplacian Matrix

Our goal is also to compute a sparse Laplacian matrix
$\mathbf{L}\in\mathbb{R}^{n\times n}$ which approximates the continuous
Laplacian operator acting on scalar functions or equivalently acting as a
quadratic form computing the integral of squared gradients:

$\displaystyle-\mathbf{f}^{T}\mathbf{L}\mathbf{f}\approx\int_{\Omega}\nabla
f\cdot\nabla f\,dV=-\int_{\Omega}f\Delta
f\,dV\cancelto{0}{+\int_{\partial\Omega}f\nabla f\cdot\hat{n}\,dS}$ (4)

where the boundary term vanishes because we assume zero Neumann boundary
conditions ($\nabla f\cdot\hat{n}=0$), which are this energy’s natural
boundary conditions.

Again, we consider the matrix as computing the local integral at each vertex
$i$:

$\displaystyle(\mathbf{L}\mathbf{f})_{i}\approx\int_{\star_{i}}\Delta f\,dV$
(5)

This time, we assume that gradient $\nabla f$ is constant over each local dual
volume $\star_{i}$, which will result in a sparse (but not diagonal) matrix
$\mathbf{L}$. Again the local dual volume $\star_{i}$ may be broken into
contributions within each primal tetrahedron volume $\bullet_{t}$, upon which
divergence theorem may be applied:

$\displaystyle(\mathbf{L}\mathbf{f})_{i}=\sum_{t}\int_{\bullet_{t}\cap\star_{i}}\Delta
f\,dV=\sum_{t}\left(\int_{\partial(\bullet_{t}\cap\star_{i})\cap\,\Omega}\hat{n}\,dS\right)\cdot\left(\nabla
f\right)_{t},$ (6)

where the “$\cap\,\Omega$” enforces the zero Neumann boundary conditions by
_not_ including any part of the boundary of the overall solid volume $\Omega$.
By now $\mathbf{L}_{ij}$ are fully determined by the choice of local dual
volume $\star_{i}$, but we momentarily postpone its explicit formula until
after defining our general dual volume construction.

##  3 Dual Volumes

Figure 3:  In our general construction, each tetraheron $t$ contributes a
portion of its volume $\bullet_{t}\cap\star_{i}$ to the local finite dual-
volume of each incident vertex $i$. This subvolume is determined by centers
defined for each simplex. Different methods choose different centers leading
to different properties of constructed operators.

In this paper, we provide a general construction of the local volume
contribution $\bullet_{t}\cap\star_{i}$ from each tetrahedron $t$ to the dual
volume of each mesh vertex $i$ as a hexahedron with vertices at “centers” of
all shared simplices (see Figure 3):

  * 1 at the vertex $i$,

  * 3 at edges of $t$ incident on $i$,

  * 3 at triangle facets of $t$ incident on $i$, and

  * 1 at the tetrahedron $t$.

For each tetrahedron $t$ incident on a vertex $i$, these centers define the
eight corners of a hexahedron $\bullet_{t}\cap\star_{i}$. To compute the
entries of $\mathbf{M}$ and $\mathbf{L}$, we may explicitly create a mesh of
this local contribution and compute its volume (for $\mathbf{M}$) or integral
of surface normals (for $\mathbf{L}$). For simple choices of centers, we can
cook up closed-form expressions for the entries of $\mathbf{M}$ and
$\mathbf{L}$ without explicitly realizing a mesh of
$\bullet_{t}\cap\star_{i}$.

Nevertheless, the choice of how each center is defined will affect the
following criteria we may care about in the mass and Laplacian matrix
definitions above:

###  $C^{0}$ — continuous function of vertex positions

The local volume $\star_{i}$ should be a continuous function of tetrahedral
mesh vertex positions. This ensures that integrated quantities stored on mesh
vertices don’t jump in value as the underlying mesh changes continuously, for
example, in Lagrangian simulation or mesh deformation methods (see Figure 2).

###  $-(\mathbf{L}+\mathbf{L}^{T})\succeq 0$ — positive semi-definite
Dirichlet energy

Treating the Laplacian matrix $\mathbf{L}$ as a quadratic form should result
in a semi-definite111We use $\mathbf{X}\succeq 0$ to write that the matrix
$\mathbf{X}$ is positive semi-definite and $\mathbf{z}\geq 0$ to write that
every element of $\mathbf{z}$ is non-negative. Dirichlet energy. That is, for
any input scalar function $\mathbf{f}\in\mathbb{R}^{n}$ we expect a non-
negative Dirichlet energy:

$\displaystyle-\mathbf{f}^{T}\mathbf{L}\mathbf{f}\geq
0\quad\forall\,\mathbf{f}\in\mathbb{R}^{n},$ (7)

with equality only for constant functions (see Figure 4). Various methods
(e.g., [VL08, Rus07, SOG09, OBS*12, SACO22, BZC*23]) rely on explicit spectral
decomposition and Dirichelt energy indefiniteness presents a major problem.

Dirichlet energy positive semi-definiteness requires that the sum of
$\mathbf{L}$ and its transpose is negative semi-definite or equivalently that
all eigenvalues of the Laplacian plus its transpose are real and non-positive:

$\displaystyle\lambda_{\mathbf{L}+\mathbf{L}^{T}}\leq 0,$ (8)

with a single zero eigenvalue corresponding to the constant function. While
symmetry and negative semi-definiteness together are _sufficient_ to ensure
this property, they are not _necessary_.

$\displaystyle\mathbf{L}=\mathbf{L}^{T}\quad\text{and}\quad\lambda_{\mathbf{L}}\leq
0\quad\begin{subarray}{c}\Longrightarrow\\\\[5.0pt]
\centernot\Longleftarrow\end{subarray}\quad-(\mathbf{L}+\mathbf{L}^{T})\succeq
0.$ (9)

Indeed, symmetry alone or non-positive eigenvalues alone does not appear to
ensure any practical value of $\mathbf{L}$. Meanwhile, an indefinite
$\mathbf{L}$ which does define a positive semi-definite Dirichlet energy is
effective for many or most applications.

Figure 4:  In this numerical example, there exists a non-trivial unit-norm
function $\hat{\mathbf{f}}$ (left) such that the implied Dirichlet energy of
[AHKS20] is negative (and the energy is nonconvex):
$\hat{\mathbf{f}}^{T}(-\mathbf{L})\hat{\mathbf{f}}\approx-2.96$. For the
barycentric and our optimized centers, $\mathbf{L}$ is symmetric negative
semi-definite and that the smallest energy value is zero from a constant
function (right).

The risk of losing Dirichlet energy positive semi-definiteness is an immediate
consequence of the choice to discretize at the operator level rather than the
quadratic form level. Alternatively, linear FEM or discrete calculus of
variations would alternatively discretize the function space and operators
present in the energy functional itself, where all terms are squared and thus
non-negative.

Positive semi-definite or not, we can consider the Dirichlet “energy” implied
by $\mathbf{L}$ as a summation of contributions per-tetrahedron:

$\displaystyle\mathbf{f}^{T}\mathbf{L}\mathbf{f}=\sum_{t}\mathbf{f}^{T}\mathbf{S}_{t}^{T}\mathbf{L}_{t}\mathbf{S}_{t}\mathbf{f},$
(10)

where $\mathbf{S}_{t}\in\mathbb{R}^{4\times n}$ selects the entries of
$\mathbf{f}$ corresponding to the vertices of tetrahedron $t$, and
$\mathbf{L}_{t}\in\mathbb{R}^{4\times 4}$ is the contribution of tetrahedron
$t$ to the Laplacian matrix. By assuming that $\nabla f$ is constant in each
tetrahedron $t$, we have reduced the discretization of the integrated
Laplacian to that of the per-vertex divergence operator, which in turn can be
decomposed into a sum of contributions per-tetrahedron, namely:

$\displaystyle\mathbf{L}_{t}$
$\displaystyle=\mathbf{D}_{t}\underbrace{\mathbf{G}_{t}}_{\mathrlap{\text{fixed
by assumption}}}$ (11)

where $\mathbf{D}_{t}\in\mathbb{R}^{4\times 3}$ and
$\mathbf{G}_{t}\in\mathbb{R}^{3\times 4}$ and the entries are determined
according to Eq. (6):

$\displaystyle\left(\mathbf{S}_{t}^{T}\mathbf{D}_{t}\right)_{i}=\int_{\partial(t\cap\star_{i})\cap\,\Omega}\hat{n}\,dS\,$
(12)

If we require symmetry $\mathbf{L}=\mathbf{L}^{T}$ for all input meshes, we
are — by reduction — requiring that:

$\displaystyle\mathbf{D}_{t}\mathbf{G}_{t}=\mathbf{G}_{t}^{T}\mathbf{D}_{t}^{T}=-\mathbf{G}_{t}\mathbf{A}\mathbf{G}_{t}^{T}$
(13)

for all tetrahedra $t$, where $\mathbf{A}_{t}\in\mathbb{R}^{3\times 3}$ is
some symmetric matrix and minus sign appears to foreshadow the interpretation
of $\mathbf{A}_{t}$ as a piecewise-constant metric or diffusion tensor.
Letting $\mathds{1}$ be the vector of ones, then because
$\mathbf{G}_{t}\mathds{1}=0$, we immediately have
$\mathbf{L}_{t}\mathds{1}=0$. If we further require that
$\mathbf{L}_{t}\preceq 0$, then we are requiring that $\mathbf{A}_{t}$ is
positive definite:

$\displaystyle\mathbf{A}_{t}\succ 0.$ (14)

###  $\mathbf{M}>0$ — positive mass matrix

The mass matrix $\mathbf{M}$ should be positive for a non-degenerate input
mesh. As $\mathbf{M}_{ii}/\sum_{j}\mathbf{M}_{jj}$ represents the percentage
of the total volume of $\Omega$, we expect each local volume $\star_{i}$ to be
positive. Since $\mathbf{M}$ is diagonal by construction, this property is
true if and only if the mass-matrix $\mathbf{M}$ is strictly positive
definite. Indefinite or worse singular $\mathbf{M}$ causes all sorts of
problems in numerical methods. A sufficient condition for this property is
that the local volume contribution in each tetrahedron is positive:
$\int_{\bullet_{t}\cap\star_{i}}\,dV>0$ for $t\ni i$, or more strictly that
all centers lie within their respective simplices.

###  Grid — unbiased volumes on regular grids

As a unit-test for “connectivity bias,” the dual volume definition should
result in unbiased local volumes on a regular grid where each cube cell is
subdivided into six tetrahedra around an arbitrary diagonal of symmetry (see
Figure 1).

With conditions in hand, let’s immediately consider some obvious center
choices and then those of prevalent prior works (summarized in Table 1).

###  3.1 Barycentric dual-volumes

The barycentric dual-volumes are defined by using the barycenter (a.k.a.,
centroid or center of mass) of each simplex. For the mass-matrix, this results
in an equal division of each tetrahedron volume to its incident vertices:

$\displaystyle\mathbf{M}^{\text{bary}}_{ii}=\frac{1}{4}\sum_{t\ni
i}\int_{\bullet_{t}}dV$ (15)

Using barycenters results in the same Laplacian matrix as direct application
of the linear finite element method: the ubiquitous cotangent Laplacian.

Figure 5:  In our construction, any choice of dual-volume implies a $3\times
3$ matrix $\mathbf{A}_{t}$ which can be interpreted as a piecewise-constant
diffusion tensor. Barycentric dual-volumes, corresponding to linear FEM, imply
tensors proportional to the identity: in a way, _trusting_ the connectivity
too much. Circumcentric dual-volumes imply tensors which counteract
connectivity bias on tetrahedralized grids.

Barycenters are clearly continuous with respect to vertex positions and the
cotagent Laplacian is negative semi-definite. In particular,
$\mathbf{D}_{t}^{\text{bary}}=\mathbf{G}_{t}^{T}\mathbf{A}^{\text{bary}}_{t}$
where $\mathbf{A}_{t}^{\text{bary}}=(\int_{\bullet_{t}}dV)\,\mathbf{I}$ (see
Figure 5). Barycenters always lie inside each respective simplex. This
immediately implies that the mass matrix is positive. However, because
tetrahedra volumes are distributed to each incident vertex based on
connectivity not geometry, the baricentric dual-volumes suffer strong bias for
a tetrahedralized regular grid (see Figure 1).

###  3.2 Circumcentric dual-volumes

Circumcentric dual-volumes are defined by using the circumcenter of each
simplex. For interior vertices of a Delaunay mesh, this choice results in
local dual volumes matching the Voronoi diagram of vertices. Circumcenters are
a continuous function of vertex positions for non-denegerate meshes.

While not obvious _a priori_ , circumcentric dual-volumes result in a
symmetric Laplacian matrix. Unlike their 2D analogs for triangle meshes,
circumcentric dual-volumes do not result, in general, in the same Laplacian
matrix as the linear-FEM or barycentric dual-volume Laplacian:

$\mathbf{L}^{\text{circum}}\neq\mathbf{L}^{\text{bary}}.$ (16)

That is, in general,
$\mathbf{A}^{\text{circum}}_{t}\neq\mathbf{A}^{\text{bary}}_{t}$ for a given
tetrahedron $t$ as $\mathbf{A}^{\text{circum}}_{t}$ is, in general, not
diagonal. We could alternatively interpret a Laplacian built with
circumcentric dual-volumes as a piecewise-linear finite-element method applied
to a mesh with non-trivial piecewise-constant metric diffusion tensors
$\mathbf{A}^{\text{circum}}_{t}\in\mathbb{R}^{3\times 3}$; in other words,
$\mathbf{A}^{\text{circum}}_{t}$ determines a per-tetrahedron transformation
$\mathbf{J}_{t}$ applied to each tetrahedron independently before applying
linear-FEM construction (see Figure 5). There is no guarantee that the per-
tetrahedron transformations could be realized by a vertex deformation of the
whole mesh in $\mathbb{R}^{3}$.

Figure 6:  For non-Delaunay tetrahedral meshes such as this heart,
circumcentric dual-volumes can result in negative entries in the mass matrix,
due to self-intersecting, flipped-inside-out contributions from each element
(left). Barycentric (center) and our optimized dual-volumes (right) are always
positive.

For a regular grid, circumcentric dual-volumes are unbiased, coinciding with a
finite-differences. Unfortunately, if the mesh is non-Delaunay the mass-matrix
is not guaranteed to be positive. Circumcenters may slip outside simplices,
causing the boundary of the dual volume to self-intersect. This may eventually
cause the total signed volume of $\star_{i}\cap t$ and even the total signed
volume of $\star_{i}$ to be negative (see Figure 6).

Figure 7:  As a tetrahedron continuously deforms from equilateral to non-
acute, the circumcenters of the tetrahedron and its triangles pass through the
triangle and edge circumcenters, respectively. This inspires our circumcentric
snapping generalization of [MDSB02], which ensures continuity of dual-volumes
but not symmetry of the Laplacian $\mathbf{L}$.

###  3.3 [AHKS20] dual-volumes: barycentric snapping

To combat negative volumes in 2D triangle meshes, Meyer et al. [MDSB02]
proposed using circumcentric triangle centers _except_ when the triangle is
obtuse. In this case, Meyer et al. would snap the center to the midpoint of
the edge opposite the obtuse angle. Citing inspiration from the idea of
center-filtering at a high-level, Alexa et al. [AHKS20] snap any centers lying
outside its simplex to that simplex’s _barycenter_. By construction the mass-
matrix is positive, but dual-volumes on a $\epsilon$-perturbed regular grid
are biased as every entry suddenly jumps in shape and size to the biased
barycentric dual-volume mass matrix (see Figure 2). Consider a simplex as it
morphs from non-obtuse to obtuse: its circumcenter and barycenter will not
coincide during the transition. Therefore, the dual volumes of [AHKS20] are
not a continuous function of vertex positions.

For _triangle_ meshes, the choice of triangle center does not affect the
definition of the Laplacian matrix (by Stokes Theorem and that the gradient is
constant in each triangle. See, e.g., Sec. 3.3.4 in [BKP*10]). As noted, edge
midpoints are equivalent to barycenters, so the resulting Laplacian matrix
agrees with the _symmetric negative semi-definite_ linear-FEM / barycentric
dual-volume Laplacian.

For tetrahedral meshes, the choice of tetrahedron center also does not affect
the Laplacian (again, by applying Stokes Theorem against a constant vector
field in each tetrahedron) and of course edge circumcenters are also
barycenters, but obtuse triangle circumcenters snapped according to [AHKS20]
_will_ affect the Laplacian matrix. As a result, the $\mathbf{L}^{\text{A}}$
will be non-symmetric in the presence of obtuse triangles. Indeed, for a
single tetrahderon with an obtuse triangle the resulting
$\mathbf{D}_{t}^{\text{A}}$ does not satisfy
$\mathbf{D}_{t}^{\text{A}}\mathbf{G}_{t}=\mathbf{G}_{t}^{T}(\mathbf{D}_{t}^{\text{A}})^{T}$.
We may solve for its analogous implied “diffusion tensor” _post facto_ as

$\displaystyle\mathbf{A}^{\text{A}}_{t}=-\left(\mathbf{G}^{T}\right)^{+}\mathbf{D}_{t}^{\text{A}}$
(17)

which in this case reveals that $\mathbf{A}^{\text{A}}_{t}$ is, in general,
not symmetric and $\mathbf{A}^{\text{A}}+(\mathbf{A}^{\text{A}})^{T}$ is not
positive definite. This implies that we do not have, in general, a positive
semi-definite Dirichlet energy formed with $\mathbf{L}^{\text{A}}$ (see Figure
4). Across datasets of larger tetrahedral meshes, we experimentally find that
$\mathbf{L}^{\text{A}}$ is consistently significantly asymmetric. Often,
$\mathbf{L}^{\text{A}}$ will have all non-negative eigenvalues, but
$\mathbf{L}^{\text{A}}+(\mathbf{L}^{\text{A}})^{T}$ is indefinite. Meanwhile,
other times $\mathbf{L}^{\text{A}}$ will even have non-trivially imaginary
eigenvalues, but $\mathbf{L}^{\text{A}}+(\mathbf{L}^{\text{A}})^{T}$ will be
negative semi-definite.

###  3.4 [MMdGD11] optimization over orthogonal dual volumes

Orthogonal duals are a subset of simplex center construction where dual faces
are geometrically orthogonal to primal edges. The circumcentric dual-volumes
of Section 3.2 are a special case of orthogonal duals, but the space is much
larger [Gli05, WMKG07, MMdGD11]. Orthogonal dual constructions for triangle
meshes (and less often tetrahedral meshes) have been considered in the past,
primarily discussed through the lens of discrete exterior calculus [DHLM05].

Mullen et al. [MMdGD11] propose to optimize over the space of metrics in each
local neighborhood so that the dual is orthogonal by construction and other
properties such as mass positivity is encouraged by minimization. Mullen et
al. may optimize both the metrics and the primal mesh simultaneously. For our
consideration, we will consider just the metric optimization. Unfortunately,
the optimization problem is non-linear and non-convex, requiring a variant of
gradient descent to find a local minimum. This forgoes the guarantee of
continuity with respect to vertex positions. While they report _reducing_ the
amount of negative volumes compared to circumcentric duals, in their examples
they do not eliminate negative volume entirely. To our knowledge, there is no
available open source implementation of this method to provide explicit
counter-examples here.

###  3.5 Continuous dual-volumes via circumcentric snapping

As a warm-up to our main result, we consider modifying the positive-volume
filtering of Alexa et al. [AHKS20] to ensure continuity as tetrahedra morph
from acute to non-acute shapes (we say a tetrahedron is “acute” if all
dihedral angles are less than $\pi/2$ [ESÜ04]).

Unlike Alexa et al., the original center-filtering of Meyer et al. [MDSB02]
ensures continuity with respect to vertex positions because the triangle
circumcenter is snapped to the _circumcenter_ of its subsimplex (the midpoint
of the edge). We can generalize the choice of Meyer et al. [MDSB02] to 3D
tetrahedral meshes by snapping circumcenters of simplices recursively to
subsimplices. This choice ensures that dual-volumes are continuous with
respect to vertex positions. To see this, consider a non-obtuse simplex
morphing into an obtuse one. At the moment that its circumcenter exits the
simplex it must coincide with a circumcenter of the boundary subsimplex upon
which it lies (because it is also equidistant to that subsimplex’s corners,
see Figure 7).

By design, snapping ensures that the mass matrix is positive, and snapping to
circumcenters ensures that the mass matrix of a regular grid remains unbiased
like circumcentric dual-volumes. Unfortunately, circumcentric snapping — like
[AHKS20] — does not guarantee symmetry or a positive semi-definite Dirichlet
energy.

##  4 Optimized dual-volumes

Drawing inspiration from Meyer et al. [MDSB02] and [AHKS20], we propose to
utilize circumcentric dual-volumes when possible. Unlike those methods, we
will explicitly ensure a positive semi-definite Dirichlet energy _by
constraint_.

Vertex circumcenters are trivially well positioned, and edge circumcenters are
non-troublesome midpoints. Meanwhile, the tetrahedra centers do not affect the
Laplacian matrix, so they can be snapped individually _post facto_. What
remains are the triangle centers.

We propose to optimize the center location of each triangle to be as close as
possible to its circumcenter, but to stay within their respective triangles
_and_ so that the implied diffusion tensor $\mathbf{A}_{t}$ of each
tetrahedron is positive definite. To ensure the continuity of dual-volumes,
two tetrahedra sharing a triangle must agree on the triangle’s center. Thus we
consider the following optimization over the centers of a tetrahedral mesh
with vertex positions in $\mathbf{v}_{i}\in\mathbb{R}^{3}$ for $i=1,\ldots,n$
and triangle indices gathered into $\mathbf{F}\in[1,n]^{k\times 3}$:

$\displaystyle\min_{\mathbf{C}\in\mathbb{R}^{k\times 3}}$
$\displaystyle\quad\sum_{f}\left\|\mathbf{o}_{f}-\mathbf{c}_{f}\right\|^{2}$
(18) s.t. $\displaystyle\quad\mathbf{c}_{f}\in\bullet_{f}$
$\displaystyle\forall\,f$ (19) and
$\displaystyle\quad(\mathbf{G}_{t}^{T})^{+}\mathbf{D}_{t}(\mathbf{C})\preceq
0\quad$ $\displaystyle\forall\,t$ (20)

where $\mathbf{o}_{f}$ is the (known) circumcenter of triangle $f$ and
$\mathbf{c}_{f}$ is the optimized center,
$\mathbf{D}_{t}(\mathbf{C}):\mathbb{R}^{k\times
3}\rightarrow\mathbb{R}^{4\times 3}$ is the deterministic and linear function
which selects the centers of the four triangles incident on tetrahedron $t$
from $\mathbf{C}$ and builds the integrated divergence operator according to
the construction in Eq. (12).

Eqs. 18-20 define a strictly convex semi-definite programming (SDP) problem.
The problem always has a solution: triangle barycenters are feasible.
Therefore, solutions of this feasible strictly convex optimization are a
continuous function of its parameters (mesh vertex positions). The constraint
in Eq. (20) ensures that the Laplacian matrix is symmetric negative semi-
definite, and thus the Dirichlet energy is positive semi-definite. The
constraint in Eq. (19) ensures that the mass matrix is positive. When a
circumcentric dual-volume satisfies both these properties, our solution will
coincide, thus ensuring unbiased dual-volumes on regular grids. Perturbing the
mesh vertices slightly will result in a continuous change, degrading
gracefully while maintaining definiteness and positivity (see Figure 2).

Large sparse semi-definite programs are tractable to solve, but rather slow.
Fortunately, we observe that simply constraining $\mathbf{A}_{t}$ to be
_symmetric_ while ensuring positive dual volumes (via Eq. (19)) appears to
always result in a positive semi-definite $\mathbf{A}_{t}$. While we lack a
formal or symbolic proof, we constructed a numerical experiment on a unit
tetrahedron to hunt for an _negative_ definite $\mathbf{A}_{t}$ while keeping
centers inside their triangles: finding a result would amount to a
counterexample to our claim. However, the SDPT3 solver [TTT03] declared this
counterexample-search problem infeasible: agreeing with our hypothesis that
symmetry and positive dual volumes are sufficient conditions for positive
definiteness. Out of an abundance of caution, for a given mesh we can always
verify whether the resulting symmetric $\mathbf{A}_{t}$ matrices are positive
semi-definite and, if not, resolve the full SDP using a large-sparse-SDP
solver like Mosek [AA00]. We have not encountered the need for this yet.

So, we propose to replace Eq. (20) with the following linear equality
constraints enforcing symmetry:

$\displaystyle\quad\mathbf{D}_{t}(\mathbf{C})\,\mathbf{G}_{t}=\mathbf{G}_{t}^{T}\,\mathbf{D}_{t}(\mathbf{C})^{T}\quad$
$\displaystyle\forall\,t.$ (21)

Eq. (21) appears to be a rank-6 constraint over the entries of
$\mathbf{D}_{t}$. However, after symbolical analysis, we find that this
constraint is in fact only rank 3 over the entries of $\mathbf{C}$. Thus, each
tetrahedron contributes 3 linear equality constraints (Eq. (13)), touching
only its four incident triangles’ centers.

Combined with the quadratic loss in Eq. (18) and the linear inequality
constraints in Eq. (19), we have a convex quadratic program (QP). By re-
parametrizing $\mathbf{C}$ as a linear function of barycenteric coordinates
$\mathbf{B}\in\mathbb{R}^{k\times 3}$, we can express the constraints in Eq.
(19) as non-negative and partition of unity constraints on $\mathbf{B}$. This
large, sparse QP can be efficiently solved with a variety of standard solvers
[AA00, CKKD20, SBG*20, Mat24].

###  4.1 Trivial extension to intrinsic dual-volumes

While the value of intrinsic 2D triangulations has been made exceptionally
clear [FSSB07, BS07, SSC19, GSC21, LGC*23], this value has perhaps not yet
extended to 3D tetrahedral meshes. Nevertheless the extension of our general
centric construction — and thus also our optimized dual-volumes — to intrinsic
dual-volumes is a trivial extension. Our quantities and constraints are
defined per-element and our optimization variables can be expressed in terms
of (intrinsic) barycentric coordinates. To accommodate an intrinsic
tetrahedralization — where mesh edges have lengths but no prescribed vertex
positions — one could simply map each tetrahedron to a canonical tetrahedron
with matching metric, compute relevant quantities and constraints. Each
$\left\|\mathbf{o}_{f}-\mathbf{c}_{f}\right\|^{2}$ in Eq. (18) would split
into contributions from each adjacent tetrahedron (expressed in barycentric
coordinates).

##  5 Numerical Experiments

Figure 8:  Compared to circumcentric, barycentric dual-volumes are known to
produce more variance when used to solve a radially symmetric Laplace problem.
The filtered centers of Alexa et al. [AHKS20] result in even higher variance.
Our optimized centers result in variance just above circumcentric duals. In
all cases, variance decreases with mesh refinement.

In Figure 8, we replicate the “Dirichlet energy minimization” experiment from
[AHKS20], where we generate progressively refined tetrahedral meshes
conforming to concentric spheres of radii 1.0, 0.75 and 0.5. We minimize
$-\mathbf{f}^{T}(\mathbf{L}+\mathbf{L}^{T})\mathbf{f}$ subject to boundary
conditions of 1.0 and 0.0 on the inner and outer shells respectively, then
measure variance of the vertex values on the middle shell. In our experiment,
we generate sphere triangle meshes using centroidal Voronoi tesselation, then
use Tetgen [Si15] to mesh in between them. As shown by Alexa et al.,
barycentric duals result in higher variance than circumcentric. However, the
filtered centers proposed by Alexa et al. result in even higher variance. Our
optimized centers result in variance just above circumcentric duals.

We made almost no effort optimizing the runtime performance of our numerical
implementation. We implemented the optimization of face centers in Matlab with
lots of for loops. Nevertheless, the bottleneck is solving the large-sparse
quadratic program (using Mosek [AA00]). The number of variables in the
optimization (three times the number of triangles $3k$) closely tracks
$30\times$ the number of mesh vertices in the previous experiment. The inset
plot shows that sub-quadratic performance in the number of vertices, far
better than worst-case complexity for convex quadratic programs indicating
that sparsity is paying off. Timings are recorded on a 2020 MacBook Pro laptop
(2.3GHz Quad-core Intel Core i7, 32GB RAM).

Figure 9:  Consider computing $\mathbf{f}^{T}\mathbf{M}\mathbf{f}$ as the
vertex positions of a tetrahedral mesh change with its design (bottom)
Barycentric, circumcentric, and our optimized dual-volumes are continuous
function of mesh vertex positions, and so is this value computed with their
respective mass matrices. The dual volumes of Alexa et al. [AHKS20] are
discontinuous, producing a noisy value as the design changes. The line plot
(top) shows the relative difference of each method’s value compared to that of
barycentric dual volumes as the design changes.

Once constructed numerically, our Laplacian $\mathbf{L}$ and mass matrix
$\mathbf{M}$ have the same sparsity and thus same runtime implications for
downstreams tasks as others methods (but unlike alternatives with larger
stencils, e.g., [BBA21]). In Figure 4, we compare the numerically computed
eigenvectors of the Laplacian built from our dual volumes to that of Alexa et
al. [AHKS20], whose lack of semi-definiteness manifests as numerical debris in
the eigenvectors. Laplacian eigenmodes are a critical downstream task used in
a variety of applications areas in geometry processing [VL08, Rus07, SOG09,
OBS*12, SACO22, BZC*23].

Our optimized dual-volumes are continuous with respect to vertex positions. In
Figure 9, we consider a downstream design analysis task where the tetrahedral
mesh connectivity of a wrench is transported along with procedural changes to
its design _à la_ [SXZ*17]. The squared norm of a sinusoidal function
$\mathbf{f}$ is measured across the changing designs:
$\mathbf{f}^{T}\mathbf{M}\mathbf{f}$. The line plots show relative differences
of each method’s value compared to that of barycentric dual volumes as the
design changes. The dual volumes of Alexa et al. [AHKS20] produce a noisy
discontinuous function, while ours — like barycentric and circumcentric — are
continuous with respect to change in vertex positions.

##  6 Discussion & Conclusion

As casually observed by Alexa et al. [AHKS20], tetrahedral meshes are often
not Delaunay in the wild. _Constrained_ Delaunay meshers (e.g., [Si15,
DPVA23]) do not enforce the local Delaunay criteria at constrained (e.g.,
boundary) faces. Meanwhile, optimization-based meshers (e.g., [HZG*18,
HSW*20]) will trade Delaunay-ness for improved aspect ratios or other
important mesh quality measures. Indeed, considering the 10,000 outputs of
TetWild run on the Thingi10k dataset [ZJ16], we see that most tetrahedra
circumcenters lie outside, and roughly one quarter of face circumcenters lie
outside (inset). Consequently, circumcentric dual-volumes result in negative
mass matrix entries for 74% of the meshes across the dataset. The barycentric
snapping method of Alexa et al. [AHKS20] results in asymmetric Laplacian
matrix $\mathbf{L}$ for 100% of the models, and of these Cholesky
decomposition of $\mathbf{L}+\mathbf{L}^{T}$ fails on 57%.

The general construction considered in this paper explicitly constructs the
dual volume. Besides making constraints and analysis explicitly clear,
ensuring a geometric realization could prove essential for applications which
rely on sampling or interpolation within dual-volumes. Elcott et al. [ETK*07]
use primal-dual meshes for fluids and generalized barycentric coordinates for
interpolation. Batty et al. [BXH10] extend this to non-conforming Delaunay
meshes, where circumcentric dual volumes are positive and convex shaped. Our
optimized dual-volumes are positive but not necessarily convex; on the other
hand, we do not require a Delaunay mesh, so meshes can be conforming. While
convenient for velocity interpolation, convexity is not strictly necessary and
generalized barycentric coordinates have since improved for non-convex
polyhedra [JMD*07, HS08, ZDL*14, CDH23].

In this paper, we considered construction and optimization of dual-volumes
assuming a contract with the user that the primal mesh remains fixed. Prior
works have also considered (simultaneous) optimization of the primal mesh or
its intrinsic connectivity to improve Laplacian and mass-matrix operators for
2D triangulations [dMMD14] and tetrahedral meshes [SMWF22] or both [MMdGD11,
Ale19, ABE*20, Ale20, MKMD23]. Alexa [Ale19] shows how unique the situation is
for triangulations in 2D and that many clear concepts or algorithms do not
extend to 3D and beyond. While we believe our optimization generalizes in
theory to higher dimensional meshes, we leave this exploration to future work.

The space of orthogonal duals is too small to include the barycentric duals
(and thus linear FEM, the _de facto_ standard). Our general construction with
simplex centers and space of optimization explicitly includes these, while
others do not [WMKG07, MMdGD11].

When considering the orthogonal dual construction of the Laplacian for 2D
triangle meshes, the choice of triangle center at the circumcenter seems to
have great importance: Wardetzky et al. [WMKG07] write “the cotan weights …
arise from assigning dual vertices to circumcenters of primal triangles”
[WMKG07]). However, in the more general view of constructing dual-volumes by
connecting simplex centers, the choice of triangle center is demonstrably
irrelevant. For the cotan weights to arise, only the choice of edge centers at
midpoints matters (see also [BKP*10]).

In 3D, the situation is very different. Edge centers remain canonically
midpoints, but triangle centers are neither irrelevant nor canonically
determined. Fortunately, they span an interesting enough space to find dual-
volumes that fulfil a variety of properties. Like the triangle center in 2D,
the tetrahedron center in 3D is irrelevant to the Laplacian matrix, but does
affect the mass matrix.

The taxonomy of Wardetzky et al. [WMKG07] is not quite expressive enough to
capture the locality or lack thereof of our proposed approach. The sparsity
pattern of our Laplacian matrix matches the combinatorial mesh, but the
entries are effectively determined via a global optimization (similar to
[MMdGD11]).

Alexa et al. [AHKS20] muse about linearly blending between barycentric and
circumcentric duals. Using a single parameter for the entire mesh would result
in a space that could ensure positivity, definiteness and continuity, but not
unbiased-ness on (local) grid structures. It is not obvious how to spatially
vary this blend so that properties are ensured locally; our optimization works
on a large space by considering effectively two parameters per triangle
(barycentric coordinates of its center).

Wardetzky et al. [WMKG07] propose a general construction of symmetric 2D
triangle mesh Laplacians by spanning a parameterization over the metrics of
each edge. This is analogous to our observation that once $\nabla f$ is
assumed to b e piecewise constant in an element, what remains is the choice of
divergence operator $\mathbf{D}$ or diffusion tensor $\mathbf{A}$. Perhaps the
construction of Wardetzky et al. could be generalized to tetrahedral meshes,
but it would seem to correspond to directly parameterizing $\mathbf{A}$ rather
than working with simplex centers. By optimizing centers directly we ensure
that dual-volumes are continuous across triangles and embedded. In this paper,
we focus on the effect of these properties on the Laplacian and mass matrices.
It would be interesting to explore our optimized dual-volumes for other
operators or uses (e.g., fluids [BXH10]).

Employing convex-optimization to _build_ the Laplacian and mass matrices will
often be more expensive than a single subsequent sparse linear system or
truncated eigenvalue decomposition. In some applications, ensuring the
features in Table 1 may be worth the wait. In any case, our result also serves
as an existence proof that we hope fuels future research into more efficient
methods. Finally, any invocation of a feature chart like Table 1 should be met
with a critical speculation about which other feature columns should be
considered in the future. We hope the analysis and novel method presented here
will continue to fuel research into dual-volumes. To this end, we have open-
sourced implementation as part of the gptoolbox [Jac*24] library.

## Acknowledgements

Our research is funded in part by NSERC Discovery (RGPIN–2022–04680), the
Ontario Early Research Award program, the Canada Research Chairs Program, a
Sloan Research Fellowship, the DSI Catalyst Grant program and gifts by Adobe
Inc. We are grateful for the feedback of the Banh Mi weekly meeting attendees
at University of Toronto, insightful conversations with Oded Stein & Nick
Sharp, and early draft comments from Towaki Takikawa, Abhishek Madan, & Eitan
Grinspun. The “Fennec Fox” model in Figure 4 is by Physics_Dude by and used
according to the CC BY 4.0 license.

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20XX Vol. X No. XX, 000–000

11institutetext: Institute of Astronomy and National Astronomical Observatory,
Bulgarian Academy of Sciences, BG-1784, Sofia, Bulgaria<EMAIL_ADDRESS>
22institutetext: Department of Physics and Astronomy, Faculty of Natural
Sciences, University of Shumen, 115, Universitetska Str., 9700 Shumen,
Bulgaria
Received 20XX Month Day; accepted 20XX Month Day

# Long-term Photometric Study of the Pre-main Sequence Star V1180 Cas

Asen Mutafov 11 Evgeni Semkov 11 Stoyanka Peneva 11 Sunay Ibryamov 22

###### Abstract

In this paper results from the optical photometric observations of the pre-
main-sequence star V1180 Cas are reported. The star is a young variable
associated with the dark cloud Lynds 1340, located at a distance of 600 pc
from the Sun in the star forming region in Cassiopeia. V1180 Cas shows a large
amplitude variability interpreted as a combination of accretion-induced and
extinction-driven effects. Our data from VRI CCD photometric observations of
the star are collected from September 2011 to February 2022. During our
monitoring, we recorded several brightness dips with large amplitudes of up to
5 mag. (I-band). At the same time, increases in brightness over periods of
several weeks have also been recorded. In this paper, we compare the
photometric data obtained for V1180 Cas with observations of other low-mass
pre-main sequence objects.

###### keywords:

stars: pre-main sequence – stars: variables: T Tauri, Herbig Ae/Be – stars:
individual: V1180 Cas

##  1 Introduction

One of the main fundamental characteristics of the young stellar objects is
their photometric variability in the optical and near infrared range. It
manifests itself as temporary dips in the brightness (eclipses), transient
increases in brightness (outbursts), periodic or non-periodic brightness
changes for a short or long-time scales. Photometric variability with
amplitudes of different magnitudes and periodicity can be observed in both
types of pre-main sequence (PMS) stars - the widespread low-mass ($\it M$
$\leq$ $2M_{\odot}$) T Tauri stars and the more massive Herbig Ae/Be (HAEBE)
stars (Herbst et al. [1994]; Herbst et al. [2007]). It is widely accepted
(Bertout [1989]; Appenzeller & Mundt [1989]) that PMS stars can be divided
into two subclasses based on the presence of accretion disc - classical T
Tauri (CTT) stars surrounded by a massive accretion disk and weak line T Tauri
(WTT) stars without indications of disk accretion. While the variability of
WTT stars is most often explained by rotational modulation of cool spots on
their surface, CTT stars are characterized by variability with large
amplitudes. It can be explained by magnetically channeled accretion from the
circumstellar disk onto the stellar surface. This explanation is summarized in
the review work of Herbst et al. ([2007]).

A significant part of HAEBE stars and some early type CTT stars manifest
strong photometric variability with sudden quasi-Algol drops in brightness and
amplitudes up to $2\fm 5$ ($V$) (Natta et al. [1997]; van den Ancker et al.
[1998]; Herbst & Shevchenko [1996]). This group of PMS objects with masses
similar to or greater than the solar mass are named UXors after their
prototype UX Orionis. They show an increase in polarization and specific color
variability (“blueing effect”) at the deep minimums in brightness. One of the
most widespread explanations of its variability is a variable extinction from
dust clumps or filaments passing through the line of sight to the star
(Dullemond et al. [2003]; Grinin et al. [1991]). Normally, when the star is
covered by clouds of dust located along the line of sight, it becomes redder.
But when the obscuration rises sufficiently the scattered part of the light in
the total observed light become considerable and the star color gets bluer.

Kun et al. ([1994]) first recognized V1180 Cas as a young variable object with
a strong Hα emission, associated with the dark cloud Lynds 1340 in the star
forming region in Cassiopeia (Lynds [1962]). Lynds 1340 is a small dark cloud
with area 0.001 square degrees and opacity class 5, located at a distance of
600 pc from the Sun. The photometric observations performed by Kun et al.
([2011]) during the period from October 1999 to February 2011 show variability
with an amplitude of about 6 mag. (I C band). Variability with such a large
amplitude is consistent with that of known eruptive young stellar objects. But
the observed characteristic timescales of the faint and bright phases differ
from these of the eruptive PMS stars from FU Ori (FUor) and EX Lupi (EXor)
type.

Kun et al. ([2011]) noted that the color magnitude diagram (IC versus RC \-
IC) shows reddening while a weakening of the brightness of the star occurs.
They determined the spectral type of V1180 Cas as K7-M0; the luminosity as
$L/L_{\sun}$ $\approx$ 0.07 with an effective temperature Teff = 4060 K of the
K7 spectral type; the equivalent widths of Hα range from 300 to 900 $\AA$ and
the mass accretion rate as $>$1.6x10-7M${}_{\sun}$/yr-1 (Ca II $\lambda$
8542). The authors suggest that it is necessary to refine the current picture
of the mechanisms of episodic accretion bursts at the time. They presume that
V1180 Cas may be a new member of an unclassified eruptive young stars.

Near-IR monitoring of the object for the period September-August 2013 was
reported by Antoniucci et al. ([2013]). The authors recognized two major
outburst events and plot the [J-H] vs. [H-K] two-color diagram. According to
them the description of the color variation cannot be pure extinction alone,
which is in agreement with the conclusions of Kun et al. ([2011]). Using their
optical and near-IR emission observations Antoniucci et al. ( [2014])
calculated a mass accretion rate of 3 x10-8M${}_{\sun}$/yr-1. The first X-ray
detections of V1180 Cas are reported in the paper of Antoniucci et al.
([2015]). The authors have performed simultaneous observations – X-ray, JHK
photometry and J-band spectroscopy. From the observed Chandra signal the
authors estimate the X-ray luminosity LX(0.5-7 keV) in the range 0.8 $\div$
2.2 x1030 $\rm erg~{}s^{-1}$. Based on the spectral energy distribution the
authors inferred a stellar luminosity of 0.8-0.9 L${}_{\sun}$. Furthermore,
the measured X-ray flux emission levels are in the order of 5 x1030 $\div$ 1
x1031 $\rm erg~{}s^{-1}$. The authors concluded that the photometric behavior
of V1180 Cas might be explained by a combination of accretion-induced and
extinction-driven effects.

In our first paper (Mutafov et al. [2018]) we reported results from VRI
photometric monitoring of V1180 Cas in the period from September 2011 to April
2018. During this period we observed drops in stellar brightness with
amplitude up to 3 mag. (Ic). In this paper, we present data from the
continuation of this multicolor optical monitoring until February 2022 and an
analysis of the observed photometric variability of the star.

##  2 Observations

The CCD observations of V1180 Cas were performed in two observatories with
four telescopes: the 2-m Ritchey-Chrétien-Coudé, the 50/70-cm Schmidt and the
60-cm Cassegrain telescopes of the Rozhen National Astronomical Observatory
(Bulgaria) and the 1.3-m Ritchey-Chrétien telescope of the Skinakas
Observatory of the University of Crete (Greece)111Skinakas Observatory is a
collaborative project of the University of Crete, the Foundation for Research
and Technology - Hellas, and the Max-Planck-Institut für Extraterrestrische
Physik.. The observations were performed with five different types of CCD
cameras: ANDOR iKon-L and Vers Array 1300B at the 2-m RCC telescope, ANDOR
DZ436-BV at the 1.3-m RC telescope, FLI PL16803 at the 50/70-cm Schmidt
telescope, and FLI PL9000 at the 60-cm Cassegrain telescope. The technical
parameters and specifications of the used CCDs are summarized in Table 1.
Observational procedure and data reduction process are described in Ibryamov
et al. ( [2015]). All frames were taken through a standard Johnson-Cousins set
of filters. The observations in all filters (VRI) are not simultaneous and
have total duration of 20 - 30 minutes. During this time there is no
significant change in the brightness of the star which could lead to
significant changes of the color index.

Table 1: CCD cameras and optical specifications Telescope | CCD Camera Type | Size | Field | Pixel size | Scale | RON | Gain
---|---|---|---|---|---|---|---
|  | (px) | (arcmin) |  ($\mu$m) |  (/px) |  ($e^{-}$rms) |  ($e^{-}$/ADU)
2m RCC | Vers Array 1300B |  1340$\times$1300 |  5.8$\times$5.6 | 20.0 | 0.26 | 2.00 | 1.0
2m RCC | ANDOR iKon-L |  2048$\times$2048 |  6.0$\times$6.0 | 13.5 | 0.17 | 6.90 | 1.1
Schmidt | FLI PL16803 |  4096$\times$4096 |  73.8$\times$73.8 | 9.0 | 1.08 | 9.00 | 1.0
60cm Cass | FLI PL9000 |  3056$\times$3056 |  16.8$\times$16.8 | 12.0 | 0.33 | 8.50 | 1.0
1.3m RC | ANDOR DZ436-BV |  2048$\times$2048 |  9.6$\times$9.6 | 13.5 | 0.28 | 8.14 | 2.7

Table 2: Coordinates and the photometric data for the VRI comparison sequence

Standard | $\alpha(2000)$ | $\delta(2000)$ | V | $\sigma_{V}$ | R | $\sigma_{R}$ | I | $\sigma_{I}$
---|---|---|---|---|---|---|---|---
A | 02 33 25.2 | 72 43 54.9 | 16.84 | 0.04 | 15.78 | 0.03 | 14.39 | 0.03
B | 02 32 58.0 | 72 42 54.5 | 17.64 | 0.03 | 16.29 | 0.04 | 14.68 | 0.03
C | 02 33 06.0 | 72 41 58.8 | 18.68 | 0.05 | 17.58 | 0.03 | 16.46 | 0.04
D | 02 33 11.3 | 72 43 37.2 | - | - | 18.79 | 0.07 | 16.54 | 0.07
E | 02 32 48.7 | 72 42 45.0 | 18.69 | 0.05 | 17.60 | 0.03 | 16.56 | 0.03
F | 02 33 06.7 | 72 45 20.9 | 19.63 | 0.03 | 18.59 | 0.07 | 17.51 | 0.07
G | 02 33 03.2 | 72 42 11.9 | - | - | 19.15 | 0.07 | 17.67 | 0.06
H | 02 32 52.0 | 72 44 32.9 | 19.73 | 0.05 | 18.72 | 0.06 | 17.68 | 0.05

In order to facilitate transformation from instrumental measurement to the
standard Johnson-Cousins system eight stars in the field of V1180 Cas were
calibrated in VRI bands. Calibration was made during six clear nights in 2011,
2012 and 2015 with the 1.3-m RC telescope of the Skinakas Observatory.
Standard stars from Landolt ( [1992]) were used as reference. The finding
chart of the comparison sequences is presented in Fig. 1. The comparison stars
are labeled from A to H in order of their V magnitudes. The field is
$15\arcmin\times 15\arcmin$, centered around V1180 Cas. North is at the top
and east to the left. The chart is retrieved from the STScI Digitized Sky
Survey Second Generation Red. Table 2 contains the coordinates and the
photometric data for the VRI comparison sequence. Our data was analyzed using
fixed apertures for V1180 Cas which was chosen to be 4radius. The background
annulus was taken from 13to 17.

Figure 1: The finding chart of the comparison sequences.

##  3 Results and Discussion

The results of our multicolor photometric CCD observations of V1180 Cas in the
period from September 2011 to February 2022 are summarized in Table 3. The
columns provide the Julian date (JD) of observation, VRI magnitudes and the
telescope used. In the column Telescope, the abbreviation 2-m denotes the 2-m
Ritchey-Chrétien-Coudé telescope, Sch - the 50/70-cm Schmidt telescop, 60-cm -
the 60-cm Cassegrain telescope and 1.3-m - the 1.3-m Ritchey-Chrétien
telescope. The values of instrumental errors are in the range $0\fm 01$-$0\fm
02$ (for I and R) and $0\fm 01$-$0\fm 03$ (for V). The VRI-light curves of
V1180 Cas during the period of our observations are plotted in Fig. 2. In most
cases the size of error bars are smaller than the size of the symbols used.

Table 3: VRI photometric observations of V1180 Cas J.D. (24…) | V | R | I | Tel | J.D. (24…) | V | R | I | Tel
---|---|---|---|---|---|---|---|---|---
55824.45 | 18.68 | 17.26 | 15.86 | 1.3-m | 57603.49 | 17.84 | 16.50 | 15.21 | 2-m
55842.39 | 18.66 | 17.33 | 16.02 | 1.3-m | 57605.49 | 18.07 | 16.57 | 15.25 | Sch
55848.40 | 18.60 | 17.16 | 15.76 | 1.3-m | 57607.47 | 17.85 | 16.45 | 15.14 | Sch
55865.40 |  | 17.42 | 16.06 | 2-m | 57664.46 | 17.84 | 16.36 | 15.14 | Sch
55866.50 | 18.58 | 17.19 | 15.85 | 2-m | 57698.36 | 18.01 | 16.66 | 15.33 | Sch
55892.34 |  | 17.15 | 15.76 | 2-m | 57714.43 | 17.81 | 16.59 | 15.31 | 2-m
55896.35 |  | 16.88 | 15.58 | Sch | 57715.42 | 18.01 | 16.76 | 15.45 | 2-m
55925.33 |  | 16.87 | 15.55 | Sch | 57716.43 | 17.93 | 16.73 | 15.48 | 2-m
56122.52 | 18.46 | 16.62 | 15.30 | Sch | 57756.33 | 17.87 | 16.45 | 15.20 | Sch
56123.51 |  | 16.98 | 15.60 | Sch | 57781.26 | 17.96 | 16.55 | 15.20 | Sch
56141.54 |  | 16.77 | 15.44 | 1.3-m | 57782.33 | 17.85 | 16.60 | 15.23 | 2-m
56159.50 |  | 17.20 | 15.86 | Sch | 57785.28 | 19.05 | 17.61 | 16.25 | 2-m
56160.50 |  | 17.25 | 15.88 | Sch | 57786.31 | 18.80 | 17.32 | 15.87 | 2-m
56161.52 |  | 17.15 | 15.79 | Sch | 57801.26 |  | 16.60 | 15.29 | Sch
56173.47 | 18.72 | 17.06 | 15.74 | 1.3-m | 57817.23 |  | 16.43 | 15.16 | Sch
56174.48 | 18.90 | 17.15 | 15.77 | 1.3-m | 57893.54 | 17.56 | 16.36 | 15.10 | 2-m
56182.41 | 18.40 | 16.87 | 15.54 | 1.3-m | 57904.54 | 17.66 | 16.30 | 15.10 | Sch
56183.46 | 17.86 | 16.51 | 15.23 | 1.3-m | 57968.49 | 18.80 | 17.24 | 15.84 | Sch
56193.45 | 18.01 | 16.59 | 15.28 | 1.3-m | 57969.47 | 18.56 | 16.98 | 15.57 | Sch
56193.47 | 18.04 | 16.60 | 15.27 | Sch | 58011.43 | 18.70 | 17.23 | 15.82 | Sch
56194.46 |  | 16.75 | 15.43 | Sch | 58012.43 | 19.40 | 17.91 | 16.47 | Sch
56214.33 | 17.86 | 16.59 | 15.26 | 2-m | 58013.46 | - | 18.52 | 17.31 | Sch
56249.37 |  | 16.98 | 15.65 | Sch | 58043.41 |  | 17.85 | 16.43 | Sch
56275.37 | 18.22 | 16.87 | 15.57 | 2-m | 58080.39 | 19.65 | 18.23 | 16.95 | Sch
56328.37 |  | 17.76 | 16.44 | Sch | 58081.40 | 20.24 | 18.58 | 17.21 | Sch
56329.29 |  | 17.65 | 16.30 | Sch | 58109.42 |  | 18.44 | 17.80 | Sch
56356.29 |  | 17.62 | 16.14 | 60-cm | 58113.34 | 20.50 | 18.72 | 17.81 | Sch
56357.29 |  | 17.53 | 16.10 | 60-cm | 58114.35 |  | 18.77 | 18.00 | Sch
56371.29 | 18.49 | 17.13 | 15.85 | 2-m | 58220.58 | 18.14 | 16.75 | 15.34 | Sch
56428.52 |  | 17.36 | 16.11 | 60-cm | 58220.59 | 18.37 | 16.80 | 15.39 | 2-m
56430.50 |  |  | 15.98 | 60-cm | 58278.52 | 18.00 | 16.73 | 15.47 | Sch
56432.49 |  | 17.32 | 15.83 | 60-cm | 58343.47 | 18.00 | 16.60 | 15.24 | Sch
56444.55 |  | 18.02 | 16.41 | Sch | 58344.47 | 17.97 | 16.62 | 15.26 | Sch
56478.53 | 18.85 | 17.36 | 16.00 | 2-m | 58346.49 | 18.05 | 16.75 | 15.34 | 2-m
56507.52 | 19.76 | 18.32 | 17.30 | 2-m | 58363.50 | 17.88 | 16.54 | 15.16 | Sch
56508.51 | 19.53 | 18.01 | 16.95 | 2-m | 58364.44 |  | 16.74 | 15.38 | Sch
56509.47 |  | 17.71 | 16.59 | Sch | 58365.45 | 18.30 | 16.99 | 15.63 | 2-m
56510.49 |  | 17.74 | 16.71 | Sch | 58428.32 |  | 17.90 | 17.08 | Sch
56510.55 |  | 17.68 | 16.64 | 60-cm | 58492.26 |  | 18.41 | 17.75 | Sch
56511.51 |  |  | 16.99 | Sch | 58496.28 |  |  | 18.01 | Sch
56512.49 |  | 17.94 | 17.18 | Sch | 58543.38 |  |  | 19.65 | Sch
56540.46 |  | 19.17 | 18.66 | Sch | 58544.24 |  |  | 19.58 | 2-m
56541.49 |  | 19.23 | 18.73 | Sch | 58604.56 |  |  | 17.20 | Sch
56543.48 | 20.82 | 19.62 |  | 2-m | 58606.56 | 19.32 | 18.11 | 16.86 | 2-m
56544.45 | 20.81 | 19.33 | 18.20 | 2-m | 58666.52 |  | 17.34 | 15.92 | Sch
56553.45 | 19.63 | 17.87 | 16.44 | 1.3-m | 58667.55 |  | 17.48 | 15.92 | Sch
56577.54 |  | 16.89 | 15.67 | 60-cm | 58690.52 | 18.08 | 16.73 | 15.32 | 2-m
56578.55 |  | 17.02 | 15.68 | 60-cm | 58691.56 | 17.96 | 16.68 | 15.25 | 2-m
56604.48 |  | 16.98 | 15.57 | 60-cm | 58704.50 | 18.90 | 17.39 | 15.84 | Sch
56655.32 |  | 17.98 | 16.93 | Sch | 58705.49 | 18.60 | 17.27 | 15.72 | Sch
56656.47 |  | 17.95 | 16.88 | Sch | 58706.50 | 18.80 | 17.34 | 15.70 | Sch
56657.27 |  | 18.05 | 17.00 | Sch | 58707.52 | 18.80 | 17.29 | 15.71 | Sch
56681.32 |  | 16.83 | 15.55 | Sch | 58726.48 | 18.80 | 17.40 | 15.95 | 2-m
56694.26 | 17.93 | 16.63 | 15.40 | 2-m | 58727.51 |  | 17.54 | 16.03 | 2-m
56715.40 | 17.83 | 16.42 | 15.18 | Sch | 58728.59 |  | 17.82 | 16.30 | 2-m
56738.36 |  | 16.57 | 15.28 | Sch | 58729.48 | 19.34 | 17.76 | 16.23 | 2-m
56899.46 | 18.24 | 16.69 | 15.40 | 1.3-m | 58730.51 | 19.10 | 17.58 | 16.09 | 2-m
56988.29 |  | 17.07 | 15.66 | Sch | 58730.57 |  | 17.65 | 16.13 | Sch
57005.34 |  | 17.00 | 15.58 | Sch | 58758.26 |  | 18.52 | 17.29 | Sch
57006.43 |  | 16.71 | 15.39 | Sch | 58759.41 |  | 18.27 | 17.06 | Sch
57016.39 | 19.95 | 18.21 | 16.78 | 2-m | 58864.32 |  |  | 18.95 | Sch
57074.23 |  |  | 15.83 | Sch | 58865.33 |  |  | 18.35 | Sch
57164.55 |  | 17.88 | 16.21 | Sch | 58867.29 | 20.75 | 19.62 | 18.90 | 2-m
57187.55 | 18.65 | 16.97 | 15.55 | 2-m | 58870.32 |  | 19.30 | 19.00 | Sch
57190.54 | 18.44 | 17.01 | 15.55 | 2-m | 58909.25 |  | 19.37 | 18.20 | Sch
57220.52 |  | 17.05 | 15.67 | Sch | 58993.55 |  |  | 18.22 | Sch
57221.54 |  | 17.10 | 15.67 | Sch | 58993.53 |  | 19.32 | 18.27 | 2-m
57223.54 | 18.36 | 16.92 | 15.56 | 2-m | 59041.51 | 19.10 | 17.46 | 16.11 | Sch
57224.53 | 18.27 | 16.84 | 15.45 | 2-m | 59042.50 | 19.37 | 17.91 | 16.79 | Sch
57246.53 | 18.88 | 17.41 | 15.84 | 1.3-m | 59059.44 |  | 18.39 | 17.24 | Sch
57247.48 | 18.64 | 17.19 | 15.65 | 1.3-m | 59060.51 |  | 18.05 | 16.92 | Sch
57259.45 |  | 16.84 | 15.33 | Sch | 59075.34 | 18.83 | 17.46 | 16.14 | 2-m
57260.47 |  | 16.92 | 15.48 | Sch | 59101.50 |  | 15.55 | 14.29 | Sch
57271.49 | 18.64 | 17.24 | 15.79 | 2-m | 59102.40 | 16.43 | 15.52 | 14.30 | 2-m
57272.47 | 18.25 | 17.04 | 15.59 | 2-m | 59103.43 | 17.50 | 16.38 | 15.08 | 2-m
57330.34 | 17.93 | 16.55 | 15.24 | Sch | 59104.45 |  | 16.80 | 15.51 | Sch
57331.36 | 17.89 | 16.60 | 15.23 | Sch | 59105.47 | 17.22 | 16.13 | 14.83 | Sch
57332.35 | 18.03 | 16.65 | 15.28 | Sch | 59109.44 | 17.75 | 16.55 | 15.37 | Sch
57333.33 | 17.93 | 16.52 | 15.22 | Sch | 59176.34 | 18.65 | 17.22 | 15.60 | Sch
57334.32 | 17.87 | 16.53 | 15.20 | Sch | 59177.36 | 18.58 | 17.06 | 15.48 | Sch
57369.31 | 18.42 | 17.13 | 15.49 | 2-m | 59220.31 | 20.10 | 18.68 | 17.21 | 2-m
57370.37 | 19.08 | 17.60 | 16.24 | 2-m | 59250.32 | 18.88 | 17.34 | 15.86 | 2-m
57371.34 | 18.65 | 17.22 | 15.79 | 2-m | 59251.27 | 19.02 | 17.48 | 16.02 | 2-m
57372.33 | 19.20 | 17.57 | 16.20 | Sch | 59314.58 | 18.16 | 16.81 | 15.45 | 2-m
57374.34 | 18.68 | 17.17 | 15.77 | Sch | 59402.53 | 17.97 | 16.76 | 15.41 | 2-m
57390.23 | 18.24 | 16.76 | 15.43 | Sch | 59403.51 | 17.33 | 16.15 | 14.90 | Sch
57425.39 | 17.92 | 16.45 | 15.21 | Sch | 59426.50 | 16.91 | 15.95 | 14.69 | 2-m
57426.40 | 17.94 | 16.46 | 15.21 | Sch | 59428.52 | 17.32 | 16.30 | 14.99 | 2-m
57427.37 | 17.86 | 16.35 | 15.16 | Sch | 59441.50 | 16.97 | 15.83 | 14.57 | Sch
57483.60 | 17.75 | 16.36 | 15.03 | 2-m | 59468.45 |  | 16.23 | 14.93 | Sch
57484.60 | 17.81 | 16.41 | 15.10 | 2-m | 59523.33 |  | 16.90 | 15.58 | Sch
57522.54 | 17.93 | 16.37 | 15.22 | Sch | 59549.37 | 17.37 | 16.15 | 14.91 | Sch
57523.54 | 18.22 | 16.50 | 15.32 | Sch | 59550.34 | 17.36 | 16.16 | 14.86 | Sch
57565.53 |  | 16.12 | 14.95 | Sch | 59582.32 | 17.42 | 16.26 | 14.99 | Sch
57581.52 | 18.03 | 16.42 | 15.17 | Sch | 59605.30 | 18.24 | 17.10 | 15.80 | 2-m
57582.55 | 17.89 | 16.46 | 15.16 | Sch | 59636.25 |  | 17.55 | 16.60 | Sch
57583.53 | 17.72 | 16.35 | 15.09 | Sch |  |  |  |  |
Table 3: VRI photometric observations of V1180 Cas. Continued. Figure 2: VRI
light curves of V1180 Cas in the period September 2011 - February 2022.

During the period from 2011 to 2020 our data shows very strong photometric
variability with a large amplitude variations ($\Delta$ I $\sim$ 5 mag). The
same brightness variability was also registered in the previous studies of Kun
et al. ([2011]), Antoniucci et al. ([2013], [2014]) and Lorenzetti et al.
([2015]). Most of the time, the I-band brightness of the star is kept in the
range of 15-16 mag., which is considered to be the maximum light in the
previous studies. However, during these periods there are changes in
brightness with small amplitudes, which is characteristic of T Tau stars.

Our photometric data over the same time period also show another type of
variability, deep declines in brightness for which no periodicity is observed.
We have registered four deep minimums in brightness in the light curve of
V1180 Cas: the first deep minimum is registered on September 2013, the second
on December 2017, the third on February-March 2019 and fourth on January 2020.
Therefore, the long-term light curve of V1180 Cas is similar to that of other
low-mass young stars as GM Cep (Semkov and Peneva [2012]; Semkov et al.
[2015]; Mutafov et al. [2019]), 2MASS J22534654+6234582 (Ibryamov et al.
[2020b]) and FHO 27 (Findeisen et al. [2013], Ibryamov et al. [2020a]). All of
these objects show typical UXor variability, but are T Tauri stars from late
spectral types.

The color-magnitude diagrams of V1180 Cas V versus (V-R), V versus (V-I) and R
versus (R-I) are shown in Fig. 3. The collected multicolor photometric data
shows the typical for UXors color reversal during the minimums in brightness.
This is a manifestation of the so-called “blueing” effect, the color of the
star gets bluer in the minima of its brightness, in accordance with the model
of dust-clump obscuration. The figure shows that, for each of the color
diagrams a point of color reversal is observed at different star brightnesses:
in the V to (V-R) diagram, the point of reversal is observed at V about 19.0
mag, in the V to (V-I) diagram at V about 19.2 mag and in the R to (R-I)
diagram at R about 17.5 mag. In the color-magnitude diagrams of V1180 Cas like
in the case of GM Cep and V1184 Tau the ”blueing” effect is observed, besides
in V-R color, in the R-I color, too (Semkov et al. [2013], Semkov et al.
[2015], Mutafov et al. [2019], Mutafov et al. [2022]).

Since the fall of 2020, there has been a significant change in the photometric
behavior of V1180 Cas. We registered two increases in brightness (local
maximums of brightness): the first one on September 2020 and the second on
July/August 2021. In this case, the increase in brightness seems to be caused
by increased accretion. Evidence of this is the decrease in the color indices
(V-R and V-I) of the star (Fig. 3), during the increase in brightness.

PMS stars are characterized by different types of variability. In many cases,
two or more different types of variability can be observed for the same star.
We know objects that show both an increase in brightness due to increased
accretion and a decrease in brightness caused the variable extinction such as
V582 Aur (Semkov et al. [2013], Ábrahám et al. [2018]). The assumption that
the observed variations in the brightness of V1180 Cas are a combination of
variable accretion and variable extinction in the line of sight was made by
Kun et al. ([2011]) and Antoniucci et al. ([2015]). But using data from
multicolor photometry we can distinguish the two phenomena over different
periods of time.

The UXor phenomenon has been discovered and defined in young stars with high
and medium masses (HAEBE stars and early type CTT stars). But recently, a
large number of small-mass objects have been discovered that also show this
type of variability (Semkov et al. [2013], Ibryamov et al. [2015], Semkov et
al. [2015], Mutafov et al. [2022]). This phenomenon is especially common among
stars from young stellar clusters and stellar associations (Findeisen et al.
[2013], Barsunova et al [2015]). This shows that the processes of formation of
stars with different masses proceed similarly. A significant part of the
protostellar gas-dust cloud remains in the vicinity of the newly formed stars
and causes variable extinction.

Figure 3: The color-magnitude diagrams of V1180 Cas in the period of
observations September 2011 - February 2022.

##  4 Conclusions

The last collected multicolored photometric data confirms that outside the
deep minimums V1180 Cas shows significant variations of brightness lasting
days and months. The same data, again, confirms that the variability of the
star is dominated by the variable extinction. The VRI light curves of the
V1180 Cas are similar to other objects - GM Cep (Semkov and Peneva [2012];
Semkov et al. [2015]; Mutafov et al. [2022]), V1184 Tau (Mutafov et al.
[2019]), 2MASS J22534654+6234582 (Ibryamov et al. [2020b]) and FHO 27
(Findeisen et al. [2013]; Ibryamov et al. [2020a]). The ”blueing effect” is
observed during the minimums of brightness which is typical for the UXor
variables. Color-magnitude diagrams of V1180 Cas, like the ones of GM Cep and
V1184 Tau, clearly show the ”blueing” effect both in V-R and R-I colors
(Mutafov et al. [2019];Mutafov et al. [2022]). In our previous paper (Mutafov
et al. [2019]) we concluded that the photometric properties of V1180 Cas can
be explained by a superposition of highly variable accretion from the
circumstellar disk onto the stellar surface and occultation from circumstellar
clumps of dust, planetesimals or from other features of the circumstellar
disk. The new observational data confirms this conclusion but, additionaly, we
observed significant change of the behaviour of the variability of the star,
too - in the increase of its brightness, probably caused by increased
accretion. An attempt of finding a periodicity of light curves was made but
without a satisfactory result. The detailed explanation of this effect
requires further observations.

###### Acknowledgements.

The authors thank the Director of Skinakas Observatory Prof. I. Papamastorakis
and Prof. I. Papadakis for the award of telescope time. We thank Dr. Valentin
D. Ivanov (ESO) for disscusions on the ideas for planetesimals. This work was
partly supported by the National RI Roadmap Project (contracts
D01-383/18.12.2020 and D01-176/29.07.2022) of the Ministry of Education and
Science of the Republic of Bulgaria. This research has made use of the NASA’s
Astrophysics Data System Abstract Service, the SIMBAD database and the VizieR
catalogue access tool, operated at CDS, Strasbourg, France.

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