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An efficient iterative method for the formulation of flow networks
Wei Hua, Dong Wangb, Xiao-Ping Wanga,b,∗
aDepartment of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, HongKong.
bSchool of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China.
Abstract
We propose an efficient iterative convolution thresholding method for the formulation of flow networks where
the fluid is modeled by the Darcy–Stokes flow with the presence of volume sources. The method is based
on the minimization of the dissipation power in the fluid region with a Darcy term. The flow network is
represented by its characteristic function and the energy is approximated under this representation. The
minimization problem can then be approximately solved by alternating: 1) solving a Brinkman equation
to model the Darcy–Stokes flow and 2) updating the characteristic function by a simple convolution and
thresholding step. The proposed method is simple and easy to implement. We prove mathematically
that the iterative method has the total energy decaying property. Numerical experiments demonstrate the
performance and robustness of the proposed method and interesting structures are observed.
Keywords: topology optimization, convolution, thresholding, Darcy–Stokes flow
1. Introduction
The flow networks are the cornerstone infrastructure of transporting substances and information. They
are classified into two categories, social transport networks [1] and biological transport networks [2], based
on whether they are created and designed by humans. Examples of social transport networks include but not
limited to electrical power-grids, railway and highway networks, information networks such as the internet,
and supply and demand networks. Biological transport networks are widely present in the organism, e.g.leaf
venation, blood vessel systems, neural networks etc. The primary design purpose of a flow network is to
balance the cost of constructing and maintaining the structure, the efficiency of transportation, and resilience
to local failure and disturbance.
Explosive growing demand for transferring matter and messages put urgent needs in studies on transport
networks. Material properties are closely related to their structure. Studies on networks’ structures not
∗Corresponding authorEmail addresses: [email protected] (Wei Hu), [email protected] (Dong Wang), [email protected]
(Xiao-Ping Wang )
1
only help us build efficient and robust networks but also shed insights into the understanding of biological
transport networks where natural selection plays the role of optimization processes. For example, the cubic
laws for blood vessel which was discovered in experiments can be explained by balancing the delivering
energy cost of blood flow and the metabolic cost of blood cells [3, 4].
The difficulties and challenges of designing flow networks are the non-convex optimization property and
ill-posedness (no solution, e.g.Steiner tree model [5] or multiple solutions [6–8]) of the formulated mathe-
matical models. It is necessary to adopt filtering techniques and/or design suitable objective functionals
to resolve the well-posedness. To tackle the non-convexity, stability, convergence and ability to avoid local
optima are critical for a practical numerical algorithm.
There are many algorithms developed for constructing transport networks. In [2], discrete adaption
dynamics, which reacts to local stimulus for network formation, is proposed. The continuous analogy has
also been studied and analyzed [9–12]. Tree-like structures have been observed using their models. They
also discover that loops emerge when strong fluctuations of flow sources present. In [13], a flow network
system consists of Darcy and Stokes region is analyzed, and optimal layout is obtained through sensitivity
analysis and the first-order method MMA [14]. Many other methods have also been developed in the context
of shape/topology/structure optimization: density interpolation [15]; level-set method approach [16, 17];
topological derivatives [18]; phase field method [19]; evolutionary structural optimization (ESO) approaches
[20], and several others. To ensure well-posedness and mesh-independent solutions, regularization is usually
needed in topology optimization. Many robust regularization approaches were introduced. For example,
regularizing the optimization problem by introducing fictitious interface energy [21] or penalizing the original
objective function by perimeter control functions based on phase field methods [22].
In this paper, we formulate the formation of flow networks as a constrained optimization problem. To
be specific, we assume the fluid in the network is either a Stokes flow or governed by Darcy’s law. The
interface is implicitly represented by characteristic functions of corresponding domains and the network is
formed by minimizing the total potential energy of the system or the dissipation of the fluids under volume
constraints. It is motivated by the MBO method for approximating mean curvature flow using characteristic
functions [23] and Esedoglu and Otto’s [24] novel interpretation using minimizing movement for problems
of multiphase flow with arbitrary surface tensions. The MBO method has attracted much attention due
to its simplicity and unconditional stability. It has subsequently been extended to deal with many other
applications, including image processing [25–27], problems of anisotropic interface motions [28, 29], the
wetting problem on solid surfaces [30, 31], and so on.
The main observation is that the approximate energy functional with perimeter regularization (using
Esedoglu and Otto’s formula in [24]) is concave with respect to the characteristic function of the domain
and convex with respect to the fluid velocity field. This allows us to design an iterative method to minimize
the energy functional effectively and robustly.
2
In our method, we minimize the total potential power with a perimeter regularization, subject to some
constraints (i.e.; 1. Stokes region volume is fixed, 2. Brinkman equation is used to model the velocity field in
the whole domain, and 3. The presence of constant or delta volume sources, which implies the divergence of
the velocity field is a delta function or constant). Borrowing the idea from threshold dynamics [24, 26], the
perimeter term is based on the diffusion of the characteristic functions of the fluid regions. We introduce the
Brinkman equation to “interpolate” between the Stokes equation in the flow region and Darcy flow through
a porous medium (a weakened nonfluid region). In each iteration of the proposed algorithm, we solve the
Brinkman equation for the whole domain by the finite-element method and update the fluid-solid regions
by a convolution step followed by a thresholding step (See Algorithm 1). The proposed scheme is simple,
easy to implement, and stable.
From both theoretical well-posedness and numerical stability concerns, we apply diffusion as a filter to
the fluid-solid region. It couples well with the diffusive approximation of perimeter. The diffusion step size
is an adjustable parameter in our method, based on which we manipulate this parameter and improve our
algorithm to a self-warm-starting version (See Algorithm 2). Numerical results show that the method is
efficient, robust, and insensitive to the initial guess and parameters. In particular, we further prove the
unconditional energy-decaying property of the proposed algorithm.
The paper is organized as follows. In Section 2, we introduce the model and derive the approximate energy
for the total energy. Based on this approximation, we derive an efficient iterative convolution thresholding
method and prove its unconditional stability (i.e.; the energy decaying property). We discuss the numerical
implementation and demonstrate the efficiency and the performance of the proposed method in Section 3.
Finally, we draw some conclusions and discussions in Section 4.
2. Derivation of the method
In this section, we discuss the derivation of the proposed method. We first introduce the mathematical
model for modeling the fluid flow with fluid generation and consumption. Then, we discuss the relaxed and
the approximate model. In Section 2.3 we derive the iterative convolution thresholding method to solve the
approximate problem iteratively. The unconditional stability is proved in Section 2.4.
2.1. The mathematical model
To introduce the optimization problem, we first introduce the following forward problem. In a fixed
domain Ω consisting of fluid region Ωf and solid region Ωs (i.e.; Ω = Ωf ∪ Ωs), we consider a generalized
Stokes equation, which is similar to the Brinkman system, to “interpolate” between the Stokes equation in
3
Ωf and the Darcy’s law in Ωs:
αu− 2∇ · (µε(u)) +∇p = g,
∇ · u = s,
u|ΓD= uD,
−pn + 2µε(u)n|Γt = t
(1)
where α denotes inverse permeability, µ is dynamic viscosity, p is pressure, ε(u) = ∇u+(∇u)T
2 is strain, g is
body force, and s is volume source in the system. The boundary is splitted into two parts ∂Ω = ΓD ∪ Γn,
where Dirichlet and Neumann boundary conditions are specified respectively. In the fluid region, α = 0, the
partial differential equation reduces to Stokes equation. In solid region, µ = 0, the equation is the Darcy’s
law. In the extreme case where α = ∞ with no body force, the differential equation leads to u = 0. It is
easy to check that, when Ωf and Ωs are fixed, the solution of this system of partial differential equations is
the minimizer of the following total potential energy functional,
E(u,Ωf ,Ωs) =1
2
∫Ω
αu · u + 2µε(u) : ε(u) dx−∫
Ω
g · u dx−∫
Γt
t · u dS (2)
Note here E depends on Ωf ,Ωs through α, µ as well as u which varies with α, µ.
Now, we consider the topology optimization problem to find the optimal interface between the fluid
domain Ωf and the solid domain Ωs in the sense of minimizing E(u,Ωf ,Ωs) subject to the constraints in (1)
and an additional volume constraint. Certain regularization techniques are required from both theoretical
and numerical points of view.
Theoretically, it is well-known that such topology optimization problems lack the existence of solutions
property without suitable restrictions on admissible configurations or appropriate regularization. Numer-
ically, discontinuity brings difficulty in convergence and stability of the numerical scheme. We adopt two
techniques to address this issue. First, we will apply a Gaussian filter on the solutions (interpolation, see
Section 2.2.1). Also, we may add perimeter penalty to energy functional (2) to smooth the problem. To be
specific, the goal of this paper is to develop an efficient method to solve approximately,min
u,Ωf ,Ωs
E(u,Ωf ,Ωs) + γ|∂Ωf |,
s.t.,u solves (1), |Ωf | = ω|Ω|(3)
where |∂Ωf | denotes the perimeter or surface area of the fluid-solid interface, |Ωf | denotes the volume of the
fluid domain, and ω is the fraction for the volume constraint.
2.2. The relaxed and approximate model
Motivated by the idea from the threshold dynamics methods developed in [24, 26], we use the character-
istic functions for the fluid region and the solid region to implicitly represent the interface.
4
Define an admissible set B as follows:
B :=(v1, v2) ∈ BV (Ω) | vi(x) = 0, 1, v1(x) + v2(x) = 1 a.e. in Ω, and
∫Ω
v1 dx = ω|Ω|, (4)
where BV (Ω) is the space of functions with bounded variation in Ω, and ω is the fixed volume fraction of
the fluid region. We introduce χ1(x) to denote the characteristic function of the fluid region Ωf , i.e.;
χ1(x) = χΩf:=
1, if x ∈ Ωf ,
0, otherwise,
and χ2(x) as the characteristic function of Ωs, i.e.χ2(x) = 1 − χ1(x). The interface Γ is then implicitly
represented by χ1 and χ2. Let χ = (χ1, χ2) and we have χ ∈ B. The problem (3) can be reformulated as,minu,χ∈B
E(u, χ) + γTV (χ),
s.t.,u solves (1),
(5)
where the total variation TV (χ) is defined as,
TV (χ) = supφ∈C1
c (Ω),|φ|≤1
∫Ω
χ1∇ · φdx.
2.2.1. Interpolation
As we have mentioned, smooth interpolation between solid and fluid is needed. Suggested by the thresh-
old dynamics algorithm, we shall consider the following interpolation:
Xτ = Gτ ∗ χ1 : χ ∈ B, (6)
where Gτ is the heat kernel,
Gτ (x) =1
(4πτ)d/2exp(−|x|
2
4τ),
here d = 2, 3 is the dimension of space.
We next interpolate inverse permeability and viscosity as,
α(χ) = αs + (αf − αs)Gτ ∗ χ1, (7)
µ(χ) = µs + (µf − µs)Gτ ∗ χ1, (8)
where subscripts s, f correspond to parameters in solid and fluid respectively.
5
2.2.2. Weak and variational formulation
The weak form of (1) is: find u ∈ [H1(Ω)]d,u|ΓD= uD such that for all v ∈ [H1(Ω)]d,v|ΓD
= 0 on ΓD,
a(u,v)− b(p,v) = L(v),
b(p,u) = (s, p)L2(Ω),
(9)
where a, b are the bilinear form,
a(u,v) =
∫Ω
αu · v + 2µε(u) : ε(v) dx, (10)
b(p,v) =
∫Ω
p∇ · v dx, (11)
L is a linear functional,
L(v) =
∫Ω
g · v dx+
∫Γt
t · v dS. (12)
We note here that the energy functional (2) can be written as,
E(u,Ωf ,Ωs) =1
2a(u,u)− L(u).
The variational formulation of (1) is: find u such that it solves the minimization problem,
minv∈[H1(Ω)]d,v|ΓD
=uD
E(u,Ωf ,Ωs). (13)
2.2.3. The approximate energy
Motivated by [24, 26, 32], we approximate the TV (χ) in (6) by
|TV (χ)| ≈√π
τ
∫Ω
χ1Gτ ∗ χ2 dx, (14)
which measures the heat content “escaped” from the fluid domain χ1 to the solid domain χ2 in a short time
τ .
Now, we arrive at the following approximate optimization problem to (5):minu,χ∈B
Eτ (u, χ) : = 12a(u,u)− L(u) + γ
√πτ
∫Ωχ1Gτ ∗ χ2 dx,
s.t.,u solves (9),
(15)
Remark 2.1. For simplicity, we use the same τ in interpolation of α, µ and approximation of |TV (χ)|.
Indeed, one can use different values of τ in different terms and the algorithm is similar.
6
2.3. The iterative convolution thresholding method
In this section, we solve the approximate problem (15) iteratively. We decouple the dependence on u
and χ to design an efficient and unconditional stable scheme to minimize Eτ (u, χ). We use a coordinate
descent algorithm to minimize the approximate energy Eτ (u, χ) with constraints (9) on the admissible set
B. That is, given an initial guess χ0 = (χ01, χ
02), we compute a series of minimizers
u0, χ1,u1, χ2, · · · ,uk, χk+1, · · ·
such that
uk = arg minu∈SEτ (u, χk), (16)
χk+1 = arg minχ∈BEτ (uk, χ), (17)
for k = 0, 1, 2, · · · . Here, B is defined in (4) and the admissible set S is defined as
S :=
u ∈ [H1(Ω)]d | u|ΓD
= uD,
∫Ω
p(∇ · u− s) dx = 0
where s is a fixed function.
Given the k-th iteration χk, we first solve (16) to get the uk. As we discussed in Section 2.2.2, the
minimum is attained at uk which solves (9) when χk is fixed.
That is, uk can be obtained by finding u ∈ [H1(Ω)]d,u|ΓD= uD such that for all v ∈ [H1(Ω)]d,v|ΓD
= 0
on ΓD, ak(u,v)− b(p,v) = L(v),
b(p,u) = (s, p)L2(Ω),
(18)
where ak(u,v) is the corresponding form of a(u,v) at χk,
ak(u,v) =
∫Ω
α(χk)u · v + 2µ(χk)ε(u) : ε(v) dx. (19)
Given uk, we now rewrite the objective functional Eτ (χ,u) into Eτ,k(χ) as follows:
Eτ,k(χ) : =1
2
∫Ω
α(χ)uk · uk + 2µε(uk) : ε(uk)− 2g · uk dx−∫
Γt
t · uk dS + γ
√π
τ
∫Ω
χ1Gτ ∗ χ2 dx
=1
2
∫Ω
(αGτ ∗ χ1)uk · uk + (2µGτ ∗ χ1)ε(uk) : ε(uk) dx+ γ
√π
τ
∫Ω
χ1Gτ ∗ χ2 dx+ C
where C =∫
Ωαsu
k · uk + 2µsε(uk) : ε(uk)− g · uk dx−
∫Γt
t · uk dS, α = αf − αs, and µ = µf − µs.
The next step is to find χk+1 such that
χk+1 = arg minχ∈BEτ,k(χ). (20)
7
We note that, because χ2 = 1 − χ1, Eτ,k(χ) is strictly concave in χ1. Thus, (20) is a minimization of
a concave functional on a nonconvex admissible set B. However, we can relax it to a problem defined on a
convex admissible set by finding rk+1 such that
rk+1 = arg minr∈HEτ,k(r), (21)
where H is the convex hull of B defined as follows:
H :=(v1, v2) ∈ BV (Ω) | vi(x) ∈ [0, 1], and v1(x) + v2(x) = 1 a.e. in Ω,
∫Ω
v1 dx = ω|Ω|. (22)
The following lemma shows that the relaxed problem (21) is equivalent to the original problem (20).
Therefore, we can solve the relaxed problem (21) instead.
Lemma 2.2. Let u ∈ H1uD
(Ω,Rd) be a given function and r = (r1, r2). Then
arg minr∈HEτ,k(r) = arg min
r∈BEτ,k(r). (23)
Proof. We refer a similar proof to [26].
Now, we show that (21) can be iteratively solved with a simple thresholding step. Because Eτ,k(r) is
quadratic in r and concave, we first linearize the energy Eτ,k(r) at rk by
Eτ,k(r) ≈ Eτ,k(rk) + Lτ,krk
(r − rk), (24)
where
Lτ,krk
(r) =
∫Ω
(γ
√π
τr1Gτ ∗ (1− 2rk1 ) +
1
2αr1Gτ ∗ (uk · uk) + µr1Gτ ∗ (ε(uk) : ε(uk))
)dx (25)
=
∫Ω
r1φ1 dx.
Here
φ1 = γ
√π
τGτ ∗ (1− 2rk1 ) +
α
2Gτ ∗ (uk · uk) + µGτ ∗ (ε(uk) : ε(uk)). (26)
Then (21) can be approximately reformulated into
χk+1 = arg minr∈HLτ,krk
(r) = arg minr∈H
∫Ω
r1φ1 dx. (27)
It is easy to see that (27) can be solved in a point-wise manner byχk+11 (x) = 1, if φ1(x) < δ,
χk+11 (x) = 0, otherwise,
(28)
where δ is chosen as a constant such that∫
Ωχk+1
1 dx = V0, and χk+12 = 1− χk+1
1 .
The method is summarized in Algorithm 1.
8
Algorithm 1: An iterative convolution thresholding method for (15)
Input: Let Ω be a fixed domain, τ > 0, ω > 0 and initial χ0.
Output: A function χ ∈ B that approximately minimizes (15) .
Set k = 0
while not converged do
1. Find u ∈ [H1(Ω)]d,u|ΓD= uD such that for all v ∈ [H1(Ω)]d,v|ΓD
= 0 on ΓD,ak(u,v)− b(p,v) = L(v),
b(p,u) = (s, p)L2(Ω).
2. Evaluate
φ1 = γ
√π
τGτ ∗ (1− 2χk1) +
α
2Gτ ∗ (uk · uk) + µGτ ∗ (ε(uk) : ε(uk)).
3. Choose a proper δ and set χk+11 (x) = 1, if φ1(x) < δ,
χk+11 (x) = 0, otherwise,
such that∫
Ωχk+1
1 dx = V0.
Set k = k + 1
To determine the value of δ, one can treat∫
Ωχk+1
1 dx−V0 as a function of δ and use an iteration method
(e.g., bisection method or Newton’s method) to find the root of∫
Ωχk+1
1 dx− V0 = 0.
For the uniform discretization of Ω, a more efficient method is the quick-sort technique proposed in [30].
Assume we have a uniform discretization of Ω with grid size h, we can approximate∫
Ωχk+1
1 dx by Mh2,
sort the values of φ1 in an ascending order and simply set χk+11 = 1 on the first M points and χk+1
2 = 1 on
the other points.
In many implementations, one may discretize the domain using nonstructural meshes. To preserve the
volume for the discretization on nonuniform grids, although the volume cannot be simply approximated by
the number of grid points times the size of each cell, a similar technique can be applied. One can still sort
the values of φ1 in ascending order, save the index into S, calculate the integrating weight at each grid point
into V, and set V = 0 and i = 0. Then, δ can be simply found by:
while V < V0; i← i+ 1; V = V + V(S(i)); end; δ = φ1(S(i+ 1)).
Besides, we note that in the original threshold dynamics method, on one hand, the algorithm can be
easily stuck when τ is very small because, in the discretized space, τ is so small that no point can switch
9
Algorithm 2: An iterative convolution thresholding method with warm start
Input: Given Ω as a fixed domain, an initial τinit > 0, a target τ > 0, ω > 0 and an initial shape χ0.
Output: A function χ ∈ B that approximately minimizes (5) .
Set a rough convergence criterion
Set k = 0, τcurrent = τinit
while τ < τcurrent do
1. Run Algorithm 1 with τcurrent, initial shape χk and prescribed convergence criterion to get
new shape χk+1.
2. Set k = k + 1, τcurrent = 0.5× τcurrent
from one phase to another (i.e., χ1 changes from 0 to 1 or 1 to 0) at one iteration step. On the other hand,
with a large τ , the interface can easily move but creates a large error. Hence, we apply the adaptive in time
technique [30] in numerical experiments by modifying Algorithm 1 into an adaptive algorithm by adjusting
τ during the iterations, which is summarized in Algorithm 2.
2.4. Stability analysis
In this section, we prove the unconditional stability property of the proposed algorithm. Specifically, we
have the following theorem.
Theorem 2.3. For the series of minimizers
u0, χ1,u1, χ2, · · · ,uk, χk+1, · · · ,
calculated with Algorithm 1, we have
Eτ (χk+1,uk+1) ≤ Eτ (χk,uk) (29)
for all τ > 0. Furthermore, the equality holds only when χk+1 = χk.
Proof. As we discussed in Section 2.2.2, we have that uk+1 attains the minimum value of Eτ (χk+1,u) when
χk+1 is fixed. In other words, for all u ∈ S, we have
Eτ (χk+1,uk+1) ≤ Eτ (χk+1,u). (30)
In particular, we have
Eτ (χk+1,uk+1) ≤ Eτ (χk+1,uk). (31)
10
For a fixed uk, from the linearization of Eτ,k(χk) in (24), we have
Eτ (χk,uk) =Lτ,kχk (χk) + γ
√π
τ
∫Ω
χk1Gτ ∗ χk1 dx,
Eτ (χk+1,uk) =Lτ,kχk (χk+1) + γ
√π
τ
∫Ω
(χk+1
1 Gτ ∗ (1− χk+11 )− χk+1
1 Gτ ∗ (1− 2χk1))dx.
Because Lτ,kχk (χk+1)− Lτ,k
χk (χk) ≤ 0, we have
Eτ (χk+1,uk)− Eτ (χk,uk) =Lτ,kχk (χk+1)− Lτ,k
χk (χk)
− γ√π
τ
∫Ω
(χk+1
1 Gτ ∗ χk+11 − 2χk+1
1 Gτ ∗ χk1 + χk1Gτ ∗ χk1)dx
≤− γ√π
τ
∫Ω
(χk+11 − χk1)Gτ ∗ (χk+1
1 − χk1) dx (32)
=− γ√π
τ
∫Ω
[Gτ/2 ∗ (χk+11 − χk1)]2 dx
≤0
where the penultimate equality is a consequence of the semi-group property of heat kernel.
Combining (31) and (32), we have
Eτ (χk+1,uk+1) ≤ Eτ (χk+1,uk) ≤ Eτ (χk,uk).
Furthermore, because
−γ√π
τ
∫Ω
[Gτ/2 ∗ (χk+11 − χk1)]2 dx
is 0 only when χk1 = χk+11 , we have that the equality in (29) holds only when χk1 = χk+1
1 .
3. Numerical experiments
In this section, we will first introduce the implementation of Algorithm 2 using the finite element method.
Then, we will display a variety of numerical examples to show the performance of the method. In all
numerical examples, we set Ω = [0, 1] × [0, 1]. All numerical results were run on a single machine with two
E5-2683v4 (16 cores each) and 128GB memory.
3.1. Finite element discretization
Based on weak formulation (9) in section 2.2.2, we will solve numerically (1) by mixed finite element
method. We will use a uniform mesh with triangular elements. Let h be parameter associated to discretiza-
tion size of the domain, Th be uniform triangulation of Ω. See Figure 1 for a typical discretization consists
of 2× 10× 10 triangles.
11
Figure 1: A sturctural mesh with 2 × 10 × 10 triangles. See Section 3.1.
In light of the LBB stability condition [33–36], we use standard P1−element for approximation of pressure
and P2−element for velocity, i.e. for pressure,
Ph = vh ∈ C(Ω) : vh|T ∈ P1(T ), T ∈ Th; vh|ΓD= 0,
for velocity,
Uh = vh = ((vh)1, (vh)2), (vh)1,2 ∈ C(Ω) : (vh)1,2|T ∈ P2(T ), T ∈ Th;vh|ΓD= uD,
Pk denotes polynomial of order up to k. Test space of pressure is the same as trial space Ph while test space
for velocity is,
Wh = vh = ((vh)1, (vh)2), (vh)1,2 ∈ C(Ω) : (vh)1,2|T ∈ P2(T ), T ∈ Th;vh|ΓD= 0.
To approximate the solution to the system (1) or its weak form (9) numerically, we solve the linear system:
find (uh, ph) ∈ Uh × Ph such that for all (vh, qh) ∈Wh × Ph,a(uh,vh)− b(ph,vh) = L(vh),
b(qh,uh) = (s, qh)L2(Ω).
(33)
3.2. Choices of parameters
In the following experiments, we set the external force g to be 0. α and µ are set as displayed in the
Table 1 which are consistent with those in [13]. We will first use a relatively large τ as an initial choice and
decrease it gradually with iterations. The initial τ of all experiments are set to be 0.01. All examples are
computed by Algorithm 2 using the initial shape as shown in Figure 2. The width of the initial shape is
12
α µ
Fluid Region 0 1.04× 10−3
Solid Region 1.04× 107 1.04× 10−11
Table 1: The parameters α and µ used in numerical experiments. See Section 3.2.
Figure 2: Initial shape for all examples. See Section 3.2.
determined by volume fraction ω. Actually, we observe that with a warm start, initial shape is not important,
even randomized initial shapes can lead same final results.
3.3. Application 1: A constant source
In this application, we consider the case where s = 1 which is a constant source. Part of the boundary
is set to be traction free so that fluids can flows out. Specifically, as shown in Figure 3, we let,
Γt = |x1 − 0.5| ≤ 0.05, x2 = 0,ΓD = ∂Ω− Γt,
uD = 0, t = 0.
This example can be understood as: the domian Ω is continuously creating fluid at each point uniformly,
this fluid then unleashes through part of the boundary, Γt. The domain is consists of fluid region and solid
(porous media) region. The fraction of fluid region is fixed. The flow network is formed to minimize total
energy needed in carrying out produced fluid. This problem has also been studied in [13].
In the implementation, we set the volume fraction ω = 0.3 and use 1024 × 1024 meshes to discretize
the domain. As the problem is symmetric about the line x = 0.5, we run our algorithm in the domain
Ω ∩ x ≥ 0.5 and extend it to the whole Ω symmetrically.x
13
Figure 3: A diagram for the boundary consisting Γu and Γt. See Section 3.3.
In this situation, we observe that τ plays an important role in the resulting scales of the results. Figure 4
displays the optimal networks obtained from Algorithm 2 with the choices of the smallest τ = 3.9×10−5, 9.8×
10−6, and 2.4 × 10−6, respectively. We observe relatively regular structures for all choices of τ and three
results are consistent with each other. Furthermore, as τ decreases, we observe that it generates more
fractals and the scale gets smaller.
To get some insight into it, we recall that by Algorithm 1, a new shape is obtained by thresholding after
diffusion (convolution with heat kernel) of L2 function at certain value. Namely, the boundaries of these
shapes are the level set of the diffused functions. To be more specific, we write
Wτ = Gτ ∗ f : f ∈ L2(Ω). (34)
Then the boundary of resulting shape is a level set of some function φ ∈ Wτ . The following Proposition
demonstrates the fertility of Wτ increase as τ decrease.
Proposition 3.1. If ν > τ , then Wν $Wτ .
Proof. We can write,
Gν ∗ f = Gτ ∗ (Gν−τ ∗ f),
and obviously Gν−τ ∗ f ∈ L2(Ω), thus Wν ⊂Wτ . On the other hand, if
Gτ ∗ f1 = Gν ∗ f2,
then
Gτ ∗ f1 = Gτ ∗ (Gν−τ ∗ f2),
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which implies
f1 = Gν−τ ∗ f2 a.e.
However, the right hand side of last equality is a function of C∞ while the left hand side is just L2. Therefore,
Wν 6= Wτ .
Figure 4: The optimal configuration obtained from Algorithm 2 with a constant source and different choices of the smallest τ .
Left: τ = 3.9 × 10−5. Middle: τ = 9.8 × 10−6. Right: τ = 2.4 × 10−6. See Section 3.3.
3.4. Effect of perimeter and convergence
As we have mentioned, adding the perimeter penalty is another way of regularizing solutions. Figure 5
lists the approximate optimal structures for γ = 1 × 102, 1 × 103, 1 × 104. It shows that adding the weight
of the perimeter term limits the growth of braches.
Figure 5: The optimal configuration obtained from Algorithm 2 with a constant source and different choices of the γ. τ =
2.4 × 10−6 for all three cases. Left: γ = 1 × 104. Middle: γ = 1 × 103. Right: γ = 1 × 102. See Section 3.4.
In Section 3.3, we observed that the Gaussian filter (interpolation) puts an effective constraint on the
admissible solutions. However, without perimeter term in the objective functional (γ = 0), different τ ’s lead
15
to distinct optimal shapes due to the ill-posedness of the problem. The following experiments demonstrate
that the optimization problem with perimeter regularizer will converge as τ tends to zero. We set γ = 1×102
and discretize the domain with 2× 1024× 1024. With different values of τ , all experiments give almost the
same shapes when τ is small (about < 4.9 × 10−6 in this case). Figure 6 displays the structures obtained
via using different τ . We note that the (local) optimal solution to the optimization problem with prescribed
γ can be well expressed under the filter with relatively small τ .
Figure 6: The optimal configuration obtained from Algorithm 2 with a constant source and different choices of the final τ .
γ = 1 × 102 for all three cases. Left: τ = 4.9 × 10−6. Middle: τ = 2.4 × 10−6. Right: τ = 1.2 × 10−6. See Section 3.4.
Besides, we observe that different meshes might lead to slightly different results due to the non-convexity
of the problem. Nevertheless, those optimal shapes obtained from one mesh are the stationary points of
that from another mesh. To see this, we take the final shape obtained from 2 × 512 × 512 mesh as the
initial shape to simulate over 2 × 1024× 1024 mesh. See Figure 7. We can also achieve a similar result on
2× 1024× 1024 meshes with the optimal shape from 2× 512× 512 being the initial data.
Figure 7: The optimal configuration obtained from Algorithm 2 with a constant source and varieties of meshes. γ = 1×102, τ =
4.9×10−6 for all three cases. Left: 2×1024×1024. Middle: 2×512×512. Right: 2×1024×1024, use solution from 2×512×512
as initial. See Section 3.4.
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3.5. Application 2: Discrete delta sources
In this example, we consider the case where s is chosen to be a linear combination of discrete delta
sources [9]. We consider the solely zero Dirichlet condition, i.e.; ΓD = ∂Ω,uD = 0. We first write the source
as a weighted summation of several delta functions,
s =
k∑i=1
ckδxk, (35)
where δxkis an unit point measure at xk and ck is chosen such that
k∑i=1
ck = 0, (36)
which preserves the total volume of the fluid and makes Ω a closed system. In this case, the∫
Ωα|u|2dx
term in total potential energy 2 is considered as penalty for having fluid in solid region. The problem can
be viewed as: there are multiple fluid production and consumption locations in the region, one need to
design flow network with fixed volume that minimizing dissipative energy (total potential energy reduces to
dissipation in this case) in transportation.
In the implementation, we discretize the domain Ω by 2 × 512 × 512 triangles. For the numerical
consideration, we admit the Gaussian with a proper choice of variance as an approximation to the delta
function,
δxk(x) ≈ 1
4πle−|x−xk|
2
4l = Gl(x− xk). (37)
For the 2 × 512 × 512 mesh, we take l = 1 × 10−4. With this approximation, (more than 99.95% of) unit
mass is spreading to surrounding 50× 50 grid points. If not stated otherwise, in this application, we let,
ω = 0.1, τ = 4.9× 10−6.
For example in this section, we end iteration if there is no change between two consecutive iterations. As
we will later demonstrate, a much loose convergence criterion can be applied to get a satisfactory result.
To get results that are visually indistinguishable from what we present here, the number of iterations can
be much less than reported in these experiments. Besides, we note that when running Algorithm 2, by one
iteration we mean one iteration in Algorithm 1, i.e. solve (16) and (17) once.
3.5.1. 1 source with 4 sinks
In this section, we consider the case where a source is located at the center of the domain surrounded
by 4 sinks along with different directions (see the left figures in Figures 8 and 9). Specifically, the source is
located at (0.5, 0.5), and the 4 sinks lie equidistantly on the circle centered at the source with a radius of
0.25. The coefficients ck for sinks are set to −1 and thus the corresponding ck for the source is 4. In Figure 9,
17
the distribution of the sinks is a rotation of that in Figure 8, 30 around the source counterclockwise. Right
figures in Figures 8 and 9 are the optimal flow networks we obtained using the Algorithm 2 after 45 and
54 iterations, respectively. We observe that the configuration is invariant to the rotation, implying that the
results are reliable and self-consistent.
Also, we plot the change of energy with iterations of the first source distribution in Figure 10. It
demonstrates the energy decaying property which was proved in Theorem 2.3. Besides, we mark those steps
at which τ decrease as red. We see significant energy drops of several steps after each τ decrease, followed
by a steady change of energy. Significant shape change is observed after decreasing τ, see Figure 11.
Figure 8: Left: The distribution of the source and sinks. Right: The optimal configuration obtained from Algorithm 2 after 45
iterations. See Section 3.5.1.
Figure 9: Left: The distribution of the source and sinks. Right: The optimal configuration obtained from Algorithm 2 after 54
iterations. See Section 3.5.1.
3.5.2. 1 source with 8 sinks
In this example, as shown in Figure 12, we first change the radius in the example in Section 3.5.1 to
0.15 and then add another 4 sinks (with coefficient ck = 0.2) on the circle with radius 0.3 without rotation
18
Figure 10: Log total energy versus step. At red points, the τ used in Algorithm 2 is halved. See Section 3.5.1
Figure 11: Left: Shape at τ = 2.5e−3. Right: Shape at τ = 1.3e−3. See Section 3.5.1.
(i.e.; totally 2 sinks at each direction of the N, S, W, and E directions.). Then the coefficient ck at the
source is modified to 4.8 accordingly to satisfy the mass balance condition (36). Figure 12 displays the
optimal flow network we obtained through Algorithm 2 with different choices of volume fractions. We run
the experiments for volume fraction ω = 0.1 and ω = 0.05 with 52 and 70 iterations. As expected, the pipe
gets thinner after fluids passing the sinks on the first level (i.e.; sinks located at the radius 0.15). We also
observe that, as the volume of fluid decreases, the structure gets even thinner which is also consistent with
the expectation.
However, we notice that the shape around the source/sink in the fluid region is unexpected fat. We further
refine the mesh to 2× 1024× 1024 and decrease l in approximation (37) to 2e−5 to study the dependence
of the configuration on the Gaussian approximation. Figure 13 list the results using different Gaussian
approximations. In the middle one, we choose the volume fraction ω = 0.1, and the number of iteration is
19
Figure 12: Left: The distribution of the source and sinks. Middle: The optimal configuration obtained from Algorithm 2 after
52 iterations with ω = 0.1. Right: The optimal configuration obtained from Algorithm 2 after 70 iterations with ω = 0.05..
See Section 3.5.2.
122. In the right one, we set ω = 0.05 and the algorithm takes 217 iterations to reach the equilibrium shape.
In Figure 13, we observe that there are no unexpected fat regions when using a Gaussian with a smaller
variance, which provides a better approximation to the delta functions. One can also approximate the delta
functions by a piecewise linear hat function, a cosine approximation, a piecewise cubic function, or so on.
We refer to [37] and the references therein for more discussions on the approximation to delta functions.
Figure 13: Left: The distribution of the source and sinks. Middle: The optimal configuration obtained from Algorithm 2 after
122 iterations with ω = 0.1. Right: The optimal configuration obtained from Algorithm 2 after 217 iterations with ω = 0.05.
See Section 3.5.2.
3.5.3. 1 source with 2 sinks
In this example, we consider the source and sinks distributed as shown in Figures 14 and 15. To be
specific, two sinks with a source on their bisection. In Figure 14, the distance of two sinks is 0.2, the
distance from the source to the line of two sinks is 0.4. In Figure 15, the distance of two sinks is 0.4, the
distance from the source to the line of two sinks is 0.2. Strengths at sinks are set to 1. As we can see, when
20
the angle formed by these three points at the source is small, the network that the flow first merges then
bifurcates is preferred. As the angle gets larger, e.g.; 90 in Figure 14, two channels from source to two
sinks are formed.
Figure 14: Left: The distribution of the source and sinks. Middle: The optimal flow network obtained from Algorithm 2 after
158 iterations with ω = 0.1. Right: The optimal flow network obtained from Algorithm 2 after 77 iterations with ω = 0.05.
See Section 3.5.3.
Figure 15: Left: The distribution of the source and sinks. Middle: The optimal flow network obtained from Algorithm 2 after
105 iterations with ω = 0.1. Right: The optimal flow network obtained from Algorithm 2 after 114 iterations with ω = 0.05.
See Section 3.5.3.
3.5.4. Tree sturcture
In this example, we consider the source and sinks distributed as shown in Figure 16. Specifically, we put
three sinks with strength 1 (i.e.; ck = 1) surrounding a source at (0.5, 0.5) with distance 0.15, symmetrically.
For each sink, we put another two sources around it at a distance of 0.1 with a strength of 0.3. The strength
at the central source is 1.2 from a direct calculation using the balance condition (36). Figure 16 displays
the optimal shape obtained by Algorithm 2 after 78 iterations. It is consistent with the distribution of the
source and sinks. Because at such choice of the locations and setup of the strength at the source and sinks,
21
the flux along each edge between source and sink is the same. In Figure 16, we observe that all branches
are almost at the same scale, which is consistent with the setup and the expectation.
Figure 16: Left: The distribution of the source and sinks. Right: The optimal configuration obtained from Algorithm 2 after
78 iterations. See Section 3.5.4.
4. Conclusions and discussions
In this paper, we proposed an efficient iterative convolution thresholding method for the topology op-
timization for fluids with sources. We theoretically proved that this method enjoys the energy decaying
property in each iteration. In numerical experiments, we considered two general cases; i) the fluid is uni-
formly created all over the space and flows out through part of the boundary; ii) the sources and sinks are
described by discrete delta functions with zero boundary condition. A variety of numerical examples were
shown to verify the performance, efficiency, and energy-decaying property of the method. We observed in-
teresting fractal structures and the dependence of the configurations on parameters. It would be interesting
to have more theoretical studies on flow networks with more complicated structures.
Acknowledgement
D. Wang acknowledges support from National Natural Science Foundation of China grant 12101524 and
the University Development Fund from The Chinese University of Hong Kong, Shenzhen (UDF01001803).
X.-P. Wang was supported in part by the Hong Kong Research Grants Council (GRF grants 16302715,
16324416, 16303318 and NSFC-RGC joint research grant N-HKUST620/15). The authors would like to
thank Dan Hu for introducing the model where the source and sinks are discrete delta functions.
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