topology optimization, theory, numerical methods and...

Topology optimization, theory, numerical methodsand applications

Xiao-Ping Wang

Department of MathematicsThe Hong Kong University of Science and Technology

IMS workshop, NUSDec. 26-30, 2019

Xiao-Ping Wang (HKUST) Topology optimization, theory, numerical methods and applications 1 / 44

Collaborators

Huangxin Chen (Xiamen University)Haitao Leng (HKUST)Dong Wang (Univ of Utah)


Introduction

Topology optimization (TO) is a mathematical (computational) methodthat optimizes material layout within a given design space and constraintswith the goal of maximizing the performance of the system.

New industrial technologies like metal 3D printing opens up a new era forthe applications of topology optimization.


Structure OptimizationStructure optimization by distribution of isotropic material2 1 Topology optimization by distribution of isotropic material

·1 IIII X 1 =t>! 1 iIi 11 '1 1 =t>¥SZS7'i Fig. 1.1. Three categories of structural optimization. a) Sizing optimization of a truss structure, b) shape optimization and d) topology optimization. The initial problems are shown at the left hand side and the optimal solutions are shown at the right

The topology, shape, and size of the structure are not represented by stan-dard parametric functions but by a set of distributed functions defined on a fixed design domain. These functions in turn represent a parametrization of the stiffness tensor of the continuum and it is the suitable choice of this parametrization which leads to the proper design formulation for topology optimization.

1.1.1 Minimum compliance design

In the following, the general set-up for optimal shape design formulated as a material distribution problem is described. The set-up is analogous to well known formulations for sizing problems for discrete and continuum structures [1], and to truss topology design formulations that are described later in this monograph. It is important to note that the problem type we will consider is from a computational point of view inherently large scale, both in state and in the design variables. For this reason the first problems treated in this area employed the simplest type of design problem formulation in terms of objec-tive and constraint, namely designing for minimum compliance (maximum global stiffness) under simple resource constraints. This is also conceptually a natural starting point for this exposition as its solution reflects many of the fundamental issues in the field.

Consider a mechanical element as a body occupying a domain nmat which is part of a larger reference domain n in R 2 or R 3 . The reference domain n is chosen so as to allow for a definition of the applied loads and boundary con-ditions and the reference domain is sometimes called the ground structure, in parallel with terminology used in truss topology design. Referring to the reference domain n we can define the optimal design problem as the problem


Darcy-Stokes Topology Optimization

1392 N. WIKER, A. KLARBRING AND T. BORRVALL

4. Solve the state problem for the new design and calculate the value of the objective functionand constraint. Pass these values on to the termination step.

Termination: Check for convergence by comparing the relative difference of two successiveobjective values. Also check the difference between the relative change in design. If the stoppingcriteria are fulfilled, then save data and exit. If not, then return to the main iteration step foranother run.

7. NUMERICAL EXAMPLES

7.1. Preliminaries

As a demonstration of the proposed design method, we have studied a type of area-to-point flowproblem discussed in Reference [12]. A similar problem was also treated in Reference [11], but inthat work the goal was to find optimal regions of Darcy flow with different permeability. Here weconsider the square shaped domain with side length L , shown in Figure 2. Most of the boundaryhas a non-slip condition, i.e. u! ≡ 0 on !u , and on the remaining part a traction condition hasbeen prescribed. For simplicity the tractions have also been set to zero, i.e. t! ≡ 0 on !t . At everypoint inside the domain, fluid is added (like rain falling down on a bed of sand), and the goal isto transport this fluid out from the domain and through the only part of the boundary possible,through !t , with as little effort as possible.

With one exception (right picture in Figure 3, below) the symmetry was utilized when solvingthese problems, in order to reduce the computational time. That is, the actual computational domain

Figure 2. The design domain for the area-to-point flow problem. The dashed line in the middle indicatesthe line of symmetry used in computations.

Copyright 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 69:1374–1404DOI: 10.1002/nme

TOPOLOGY OPTIMIZATION OF REGIONS OF DARCY AND STOKES FLOW 1393

Figure 3. Optimal regions with different permeability of Darcy flow (left) and regions of Stokes (white)and Darcy (black) types of flow (right). The prescribed volume fraction was ! = 0.3 in both cases. Note

the non-symmetric solution due to not using symmetry conditions during the computations.

was only the part to the right of the dashed line of symmetry in Figure 2. The discretization wasmade with a finite element grid of 54× 108 velocity elements and 27× 54 pressure elements. Forthe implementation, the in-house C++ class library TO++ written by Borrvall [22] has been used.This library contains, among other options, a direct solution method using LU factorization withpivoting for solving the state problem (36), and an implementation of MMA for the optimizationproblem (42).

The examples presented in the forthcoming sections have been focused on studying the influenceof various parameters, including the volume fraction !, the filter radius R, the size of the outflowboundary !t , as well as the relative difference of the permeability and the viscosity, "/#, and thelower limit #. The only parameter kept unchanged throughout all examples is the volume sourceterm s, which for simplicity has been set to s = 1, representing an inflow of fluid to the domain.Regarding other specific parameter values, these will be presented in conjunction with each specificexample. A note to be made here, is that the value of the filter radius has always been set relativeto the side-length of the velocity elements. We indicate this relation by writing, e.g. R = 2× [e.s.l.](element-side-length) instead of writing out the actual numerical value of R. This since the filterradius will have no effect if it is too small, i.e. smaller than the side length of an element.

In order to avoid local optimal solutions as much as possible, it is suggested in References [5, 23]that the computations should be carried out by using an iterative procedure, where the penaltyparameter q is altered successively. This involves to solve the problem in several steps, where oneuses a low value of q in the first step in order to make the problem more convex. Since the solverrequires an initial guess of the design variable, x is in this step set to a constant value !, where !is the prescribed volume fraction of Stokes flow. In the next step, the obtained design solution isused as an initial guess when re-solving the problem with a new, increased value on q. All otherparameters are kept unchanged during the whole procedure. Here, we have adopted a strategyinvolving three steps using the following values on q:

q∈ 0.00001, 0.01, 10.0


TOPOLOGY OPTIMIZATION OF REGIONS OF DARCY AND STOKES FLOW 1393

Figure 3. Optimal regions with different permeability of Darcy flow (left) and regions of Stokes (white)and Darcy (black) types of flow (right). The prescribed volume fraction was ! = 0.3 in both cases. Note

the non-symmetric solution due to not using symmetry conditions during the computations.

was only the part to the right of the dashed line of symmetry in Figure 2. The discretization wasmade with a finite element grid of 54× 108 velocity elements and 27× 54 pressure elements. Forthe implementation, the in-house C++ class library TO++ written by Borrvall [22] has been used.This library contains, among other options, a direct solution method using LU factorization withpivoting for solving the state problem (36), and an implementation of MMA for the optimizationproblem (42).

The examples presented in the forthcoming sections have been focused on studying the influenceof various parameters, including the volume fraction !, the filter radius R, the size of the outflowboundary !t , as well as the relative difference of the permeability and the viscosity, "/#, and thelower limit #. The only parameter kept unchanged throughout all examples is the volume sourceterm s, which for simplicity has been set to s = 1, representing an inflow of fluid to the domain.Regarding other specific parameter values, these will be presented in conjunction with each specificexample. A note to be made here, is that the value of the filter radius has always been set relativeto the side-length of the velocity elements. We indicate this relation by writing, e.g. R = 2× [e.s.l.](element-side-length) instead of writing out the actual numerical value of R. This since the filterradius will have no effect if it is too small, i.e. smaller than the side length of an element.

In order to avoid local optimal solutions as much as possible, it is suggested in References [5, 23]that the computations should be carried out by using an iterative procedure, where the penaltyparameter q is altered successively. This involves to solve the problem in several steps, where oneuses a low value of q in the first step in order to make the problem more convex. Since the solverrequires an initial guess of the design variable, x is in this step set to a constant value !, where !is the prescribed volume fraction of Stokes flow. In the next step, the obtained design solution isused as an initial guess when re-solving the problem with a new, increased value on q. All otherparameters are kept unchanged during the whole procedure. Here, we have adopted a strategyinvolving three steps using the following values on q:

q∈ 0.00001, 0.01, 10.0


No-slip on Γu and zero traction on Γt .At every point inside the domain, fluid is added (like rain falling down on abed of sand), and the goal is to transport this fluid out from the domainand through the only part of the boundary possible Γt , with as little effortas possible.Optimal regions with different permeability of Darcy flow (left) and regionsof Stokes (white)and Darcy (black) types of flow (right). The prescribedvolume fraction is ν = 0.3 in both cases.Xiao-Ping Wang (HKUST) Topology optimization, theory, numerical methods and applications 5 / 44

Topology Optimization of Fluid and Solid

Solution for volume fraction 1/3

? outflows

inflows ?

Figure 3: The system of channels giving minimum pressure loss for prescribed in- and out-flow isfound. It is concluded that the length of the domain influences the topology.

REFERENCES

[1] T. Borrvall. Computational Topology Optimization of Elastic Continua by Design Restriction.Linkoping Studies in Science and Technology. Thesis No. 848, (2000).

[2] T. Borrvall and J. Petersson, Topology optimization of fluids in Stokes flow, InternationalJournal for Numerical Methods in Fluids, 41, 77-107, (2003).

[3] R. Karch, F. Neumann, F. Neumann and W. Schreiner. A three-dimensional model for arterialtree representation, generated by constrained constructive optimization. Computers in Biologyand Medicine, 29, 19-38, (1999).

[4] A. Klarbring, J. Petersson, B. Torstenfelt and M. Karlsson, Topology optimization of flow net-works, Computer Methods in Applied Mechanics and Engineering, 192, 3909-3932, (2003).

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An optimal arterial tree problem

inflow and outflow is prescribed; the tree represents that of minimum pressureloss (Borrvall, Klarbring, Petersson and Torstenfelt)


Topology Optimization of Heat SinkNEXPERiA

Tijs Van Oevelen & Martine Baelmans, 2014Xiao-Ping Wang (HKUST) Topology optimization, theory, numerical methods and applications 8 / 44

Reconstruction of the shape of an obstacle

An obstacle of unknown shape ω is immersed in a fixed domain D filledby the considered fluid.Given a mesure umeas of the velocity uΩ of the fluid inside a smallobservation area O, one aims at reconstructing the shape of ω.The optimized domain is Ω := D/ω and only the part ∂ω of ∂Ω isoptimized. One then minimizes the least-square criterion:

J(Ω) =

∫

O|uΩ − umeas|2dx .

Introduction Examples Shape derivatives Numerics Other methods

Shape optimization in fluid mechanics (III)

Model problem II: Reconstruction of the shape of an obstacle.• An obstacle of unknown shape ! is immersed in a fixed domain D

filled by the considered fluid.• Given a mesure umeas of the velocity u of the fluid inside a small

observation area O, one aims at reconstructing the shape of !.• The optimized domain is := D \ !, and only the part @! of @ is

optimized. One then minimizes the least-square criterion:

J() =

Z

O|u umeas |2 dx .

!in out

D

O

18 / 91


Image segmentation


Mathematical framework

A topology optimization problem writes as the minimization of a cost (orobjective) function J of the domain Ω:

minΩ∈Uad

J(Ω)

where Uad is a set of admissible shapes (e.g. that satisfy constraints).

In most applications, the relevant objective functions J(Ω) depend on Ω via astate uΩ, which arises as the solution to a PDE posed on Ω (e.g. the linearelasticity system, or Stokes equations).


Mathematical model for structure mechanicsIntroduction Examples Shape derivatives Numerics Other methods

Shape optimization in structure mechanics (I)

We consider a structure Rd , which is• fixed on a part D @ of its boundary,• submitted to surface loads g , applied onN @, D \ N = ;.

The displacement vector field u : ! Rd isgoverned by the linear elasticity system:8>><>>:

div(Ae(u)) = 0 in u = 0 on D

Ae(u)n = g on N

Ae(u)n = 0 on := @ \ (D [ N)

,

where e(u) = 12(ruT +ru) is the strain tensor

field, and A is the Hooke’s law of the material.

D

N

A ‘Cantilever’ beam

The deformed cantilever

13 / 91Minimizing the compliance C(Ω) subject to the volume constraint Vol(Ω) = V0

Introduction Examples Shape derivatives Numerics Other methods

Shape optimization in structure mechanics (II)

Examples of objective functions:

• The work of the external loads g or compliance C () of domain :

C () =

Z

Ae(u) : e(u)dx =

Z

N

g .u ds

• A least-square discrepancy between the displacement u and atarget displacement u0 (useful when designing micro-mechanisms):

D() =

Z

k(x)|u u0|↵dx

1↵

,

where ↵ is a fixed parameter, and k(x) is a weight factor.

14 / 91


Mathematical model for fluid and solidΩ ∈ Rd (d = 2,3) : computational domain

Ω0 ⊂ Ω: fluid domainΩ \ Ω0 ∈ Ω: solid domain

Our goal is to find an optimal fluid regionΩ0 which minimizes

min(Ω0,u)

J0(Ω0,u) =

∫

Ω

(µ2|Du|2 − u · f

)dx + γ|Γ|

subject to

∇ · u = 0, in Ω,

∇p −∇ · (µ∇u) = f, in Ω0,

u = 0, in Ω \ Ω0,

u|∂Ω = uD, on|Ω0| = β|Ω| with a fixed parameter β ∈ (0,1).

|Γ| is the perimeter of the boundary Γ = ∂Ω0, and γ > 0 is a weightingparameter.


Numerical Methods

Parametric representation (e.g. domains are represented by a set ofcontrol points)Region based: domains are described by density functions, level setfunctions or phase field functions (Borrval and Petersson, 2003, Challisand Guest, 2009; Garcke, et al 2015)

Gradient algorithem: using shape derivativeMethod of Moving Asymptotes (MMA) (Svanberg, 1998)

In our approach, domains are represented by the characteristic functions( or indicator functions). We introduce an efficient and simple strategy toupdate the topology of fluid-solid regions, based on the thresholddynamics method


Threshold dynamics method

Motion by mean curvatureInterface between different materialsDriven by the minimization of the interfacial energy (e.g. L2 Gradient flowof the energy)


Phase field formulation

Total surface energy minimization

Iε(φε) =

∫

Ω

ε

2|∇φε|2 +

1ε

f (φε)dxdy (1)

where f (φ) = (φ2−φ)2

4 .min Iε(φε)

In the sharp interface limit Iε(φε)→ I(φ) (L. Modica, ARMA. 1987,1989)L2-gradient flow: Allen-Cahn equation

φt − ε∆φ+ f ′(φ)/ε = 0in Ω,

Sharp interface limit shows that interface motion converges to motion bymean curvature


MBO Threshold Dynamics MethodMerriman, Bence, Osher (1992) introduced a threshold dynamics method for interfacemotion by mean curvature. Consider the Allen-Cahn equation

φt − ε∆φ+ 1εf ′(φ) = 0, in Ω,

∂nφ = 0, on ∂Ω,f (φ) =

(φ2 − φ)2

4

Operator splitting:

Step 1: Solve a heat equation for δt :φt − ε∆φ = 0, in Ω,φ(0) = χDk .∂nφ = 0, on ∂Ω,

⇒ φ(x , t) = Gδt ? χDk

Step 2: Solve the equation:

φt = −f ′(φ)/ε, in Ω.

When ε→ 0, Step 2 turns into thresholding

φ(x) ≈

1, if φ(x) > 1/2,0, if φ(x) < 1/2. ⇒ Dk+1 = x ∈ Rn;φ(x) >

12

Dk

Dk+1

MBO converges to moton by mean curvature as δt → 0 (Barles-Georgelin, 95).Xiao-Ping Wang (HKUST) Topology optimization, theory, numerical methods and applications 17 / 44

The threshold dynamics - a new formulation

D1 D1

D2 D2δt

Esedoglu and Otto; CPAM, 2015It was shown that the following functional defined on sets, with the Gaussiankenel G, to be a non-local approximation to (isotropic) perimeter, is dissipatedby the MBO scheme at every step, regardless of time step size:

J(D) =1√δt

∫χDc Gδt ∗ χDdx (2)

Thus, it is a Lyapunov functional for MBO scheme above, establishing itsunconditional gradient stability. MBO has the following minimizing movementsinterpretation involving

Dk+1 = arg minD

J(D) +1√δt

∫(χD − χDk )Gδt ∗ (χD − χDk )dx (3)


An iterative schemeL(uk ,u) is the linearization of J(D) at uk = χDk ,

Lδt (uk ,u) : =

∫

Ω

uφ dx , where φ =

√π

τGτ ∗ (1− 2uk ).

Since u(x) ∈ [0,1], the minimization can be carried out in a pointwisemanner by checking whether φ(x) > 0 or not. That is, the minimum canbe attained at

uk+1(x) =

1 if φ(x) ≤ 0,0 otherwise.

(4)

Linearization⇒ Accelerating the convergence


Energy decay property

TheoremLet uk , k = 0,1,2, ... obtained by the above procedure. we have

J(uk+1) ≤ J(uk ), (5)

for all δt > 0.


Benefits of the scheme:Implicit representation of the interface (can handle topological changingcases)easy to generalize to muti-phases (extremely large number of phases)Unconditional stable, eqsy to implement

Extensions:Wetting on solid surface [Xu, Wang and Wang, J. Comput. Phys., 2017,Wang, Wang and Xu, J. Comput. Phys., 2019]Image segmentation [ Wang et al., J. Comput. Phys., 2017, Wang andWang, 2019]


Mathematical model

Ω ∈ Rd (d = 2,3) : computational domainΩ0 ⊂ Ω: fluid domainΩ \ Ω0 ∈ Ω: solid domainOur goal is to find an optimal shape of Ω0 which minimizes

min(Ω0,u)

J0(Ω0,u) =

∫

Ω

(µ2|Du|2 − u · f

)dx + γ|Γ| (6)

subject to

∇ · u = 0, in Ω,

∇p −∇ · (µ∇u) = f, in Ω0,

u = 0, in Ω \ Ω0,

u|∂Ω = uD, on ∂Ω,

|Ω0| = β|Ω| with a fixed parameter β ∈ (0,1).

|Γ| is the perimeter of the boundary of Γ = ∂Ω0, and γ > 0 is a weightingparameter.


The porous medium approach

The idea is to interpolate between the Stokes equation in the fluid domain andu = 0 in the solid domain by introducing an additional penalization term,

∇ · u = 0, in Ω, (8a)∇p −∇ · (µ∇u) + α(x)u = f, in Ω, (8b)

u|∂Ω = uD, on ∂Ω. (8c)

Here, α(x) is a smooth function varying from 0 to ατ (large) through a thininterface layer and α−1

τ is the permeability.

α = ατφ = ατGτ ∗ χ2. (9)

χ2 is the indicator function of the solid region.

This is also called SIMP method (Solid Isotropic Material Penalization)


Approximate energy

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

v1dx = V0,

(10)

We introduce χ1(x) to denote the indicator function of the fluid region Ω0, i .e.,

χ1(x) :=

1, if x ∈ Ω0,

0, otherwise,

and χ2(x) as the indicator function of Ω \ Ω0, i .e., χ2(x) = 1− χ1(x). Theperimeter of the interface Γ can be approximated by,

|Γ| ≈√π

τ

∫

Ω

χ1Gτ ∗ χ2dx, (11)

where Gτ (x) =1

(4πτ)d2

exp(−|x|

2

4τ

)is the Gaussian kernel.


Modified objective functional

We add a penalty term ατ

2 Gτ ∗ χ2|u|2 to the objective functional and considerthe following approximate objective functional

Jτ (χ,u) =

∫

Ω

(µ

2|Du|2 +

α

2|u|2Gτ ∗ χ2 − u · f + γ

√π

τχ1Gτ ∗ χ2

)dx. (12)

The minimization problem:

min(χ,u)

Jτ (χ,u), (13)

subject to χ = (χ1, χ2) ∈ B and u satisfies the Stokes equation (8).


The iterative scheme.We use a coordinate descent algorithm to minimize the approximate energy(12) with constraints (8). Given an initial guess χ0 = (χ0

1, χ02), we compute a

series of minimizers

u0, χ1,u1, χ2, · · · ,uk , χk+1, · · ·

Step 1: given χk , update u

uk = arg minu∈S

Jτ (χk ,u), (14)

S :=

u ∈ H1uD

(Ω,Rd ) | ∇ · u = 0

(15)

This is equivalent to solving the Brinkman equations.Step 2: Fix uk , update χ

χk+1 = arg minχ∈B

Jτ (χ,uk ), (16)

B is defined in (10).This is done by volume preserving threshold dynamics


Energy decay

TheoremFor the sequence of minimizers

u0, χ1,u1, χ2, · · · ,uk , χk+1, · · · ,

we have

Jτ (χk+1,uk+1) ≤ Jτ (χk ,uk ) (17)

for all τ > 0.


Numerical example: The optimal design of a diffuser

we assume that the Dirichlet boundary condition with a parabolic profile andthe magnitude of the velocity is set as |uD| = g(1− (2t/l)2) with t ∈ [−l/2, l/2]where l is the length of the part of boundary where the inflow/outflow velocityis imposed.Let g = 1 and 3 for the inflow and outflow velocities respectively. We set thefluid region fraction is β = 0.5 and test the problem on a 128× 128 grid.


The optimal design of a diffuser

10 HUANGXIN CHEN, DONG WANG, AND XIAO-PING WANG

6.1. Numerical results. For all the examples in this section, we assume that the Dirichlet boundarycondition with a parabolic profile and the magnitude of the velocity is set as |uD| = g(1 (2t/l)2) witht 2 [l/2, l/2] where l is the length of the part of boundary where the inflow/outflow velocity is imposed.The direction of the inflow/outflow velocity will be illustrated in the following examples.

Example 7.1. The first example shown in Figure 6.1 is the optimal design of a di↵user which was testedfor topology optimization for fluids using the MMA in [7]. Here we apply Algorithm 1 to obtain the optimaldesign of the di↵user. Let g = 1 and 3 for the inflow and outflow velocities respectively. We set the fluidregion fraction is = 0.5 and test the problem on a 128 128 grid.

Figure 6.1. (Example 7.1) Design domain for the di↵user example.

Figure 6.2. (Example 7.1) Left (Case 1): Initial distribution of 1. Right (Case 2): Initial distribution of 1.

We first perform the simulations with ↵ = 2.5 104, = 0.01, = 0.1 and with two types of initialdistribution of 1 as shown in Figure 6.2, i.e., the initial fluid region is restricted in the middle of the domainin the left graph of Figure 6.2 (Case 1), and the initial fluid region satisfies a random distribution in theright graph of Figure 6.2 (Case 2). In both cases, we always get the same optimal design result shown inthe left graph of Figure 6.3, which also shows the quiver plot of the approximate velocity in the fluid region.The optimal design result seems to be similar to the result obtained by the MMA in [7]. Energy decayingproperty can be observed from the right graph of Figure 6.3 which shows the energy curves for the abovetwo cases of initial distribution of 1. The iteration converges in about 25 steps in both cases.

Next, we test the case ( initial fluid region of Case 1) for di↵erent parameters. We first fix ↵ = 2.5 104, = 0.01 and vary = 0.01, 0.005, 0.001. We then test the cases for fixed = 0.001, and di↵erent choices of = 0.05, 0.01, 0.001. The optimal design of the di↵user is similar to the result in the left graph of Figure

THRESHOLD DYNAMICS METHOD FOR TOPOLOGY OPTIMIZATION FOR FLUIDS 11

Figure 6.3. (Example 7.1) Left: Optimal di↵user for the case ↵ = 2.5 104 and the approximate

velocity in the fluid region. Right: Plot of energy curves for two cases of distribution of 1. In this case the

parameters are set as ↵ = 2.5 104, = 0.01, = 0.1.

Figure 6.4. (Example 7.1) Plot of energy curves for case 1 of distribution of 1 with ↵ = 2.5 104.

Left: For fixed = 0.01, energy curves for the cases of = 0.01, 0.005, 0.001. Right: For fixed = 0.01,

energy curves for the cases of = 0.05, 0.01, 0.001.

6.3. From Figure 6.4, the energy decaying property can be observed for all the above cases. In all cases, theiteration converges in less than 30 steps.

In the next example, we increase ↵ = 2.5 105. Again, we use the initial fluid region of Case 1 with = 0.001, = 0.01. The optimal design of the di↵user and the approximate velocity in the fluid region areshown in the left graph of Figure 6.5. It seems that the fluid region at the left boundary reaches top andbottom bounaries in this case. The energy decaying property is also observed in Figure 6.5. The iterationconverges even faster at about 15 steps.

We also test the problem with the same inflow Dirichlet boundary condition as above, but the outflowDirichlet boundary condition is replaced with homogeneous Neumann boundary. Then the similar optimaldesign of di↵user is obtained as above for the cases of ↵ = 2.5 104 and ↵ = 2.5 105.

Example 7.2. In this example we test the double pipes problem shown in Figure 6.6. The inflow andoutflow Dirichlet boundaries are located with centers [0, 1/4], [0, 3/4], [1, 1/4], [1, 3/4] as shown in Figure 6.6.Let g = 1 for the inflow and outflow velocities respectively and the fluid region fraction be = 1/3. We testthe problem with ↵ = 2.5 104 on a 128 256 grid for d = 0.5, and on a 192 128 grid for d = 1.5.

For the case d = 0.5, we choose a random initial distribution 1 as shown in the left graph of Figure 6.7.We remark that can also be chosen as zero in Algorithm 1. For fixed = 0.001, we test = 0.01, 0.001, 0.The optimal design result is almost the same for the three choices of which is shown in the middle graph


Efficiency Comparison

Methods Grid Number of iterations

MMA 100 × 100 33Level set 96 × 96 197

Our algorithm 128 × 128 21

Table: Comparison of the number of iterations of different methods to obtain the optimal designresult for Example 6.1 with α = 2.5 × 104. The parameters used in our algorithm are set asτ = 0.001, γ = 0.01.


The double pipes


Figure 6.5. (Example 7.1) Left: The associated optimal di↵user and the approximate velocity in the

fluid region. Right: Plot of energy curve for Case 1 of distribution of 1. In this case the parameters are

set as ↵ = 2.5 105, = 0.001, = 0.01.

Figure 6.6. (Example 7.2) Design domain for the double pipes example.

Figure 6.7. (Example 7.2) For the case d = 0.5. Left: Initial distribution of 1. Middle: Optimal double

pipes and the approximate velocity in the fluid region. Right: For fixed = 0.001, energy curves for the

cases of = 0.01, 0.001, 0.


Figure 6.5. (Example 7.1) Left: The associated optimal di↵user and the approximate velocity in thefluid region. Right: Plot of energy curve for Case 1 of distribution of 1. In this case the parameters are

set as ↵ = 2.5 105, = 0.001, = 0.01.



pipes and the approximate velocity in the fluid region. Right: For fixed = 0.001, energy curves for thecases of = 0.01, 0.001, 0.


The double pipes


Figure 6.5. (Example 7.1) Left: The associated optimal di↵user and the approximate velocity in thefluid region. Right: Plot of energy curve for Case 1 of distribution of 1. In this case the parameters are

set as ↵ = 2.5 105, = 0.001, = 0.01.



pipes and the approximate velocity in the fluid region. Right: For fixed = 0.001, energy curves for thecases of = 0.01, 0.001, 0.

THRESHOLD DYNAMICS METHOD FOR TOPOLOGY OPTIMIZATION FOR FLUIDS 13

of Figure 6.7, and the energy decaying property is observed from the energy curves in the right graph ofFigure 6.7.

Figure 6.8. (Example 7.2) For the case d = 1.5, and the parameters are set as = 0.01 and = 0.0001.

Left: Optimal double pipes and the approximate velocity in the fluid region. Right: Energy curve.

For the case d = 1.5, we choose an initial distribution 1 with fluid region located in the middle of thedomain as Case 1 of Example 7.1. We set = 0.01 and = 0.0001. The optimal design result and theapproximate velocity are shown in the left graph of Figure 6.8, and the energy decaying property is alsoobserved from the energy curve in the right graph of Figure 6.8. Compared with the computational costused by the MMA in [7], we find that our algorithm converges faster to the optimal result (cf. Table 2).

Example 7.3. We consider another example studied in [7] that includes a body fluid force term imposedin the local circular region with center [1/2, 1/3] and radius r = 1/12. We show the design domain in Figure6.9. The inflow and outflow Dirichlet boundaries locate with centers [0, 2/3] and [1, 2/3] respectively. Letg = 1 for the inflow and outflow velocities and the fluid region fraction be = 1/4. We test the problemwith di↵erent choices of body fluid force on a 128 128 grid, and we always choose ↵ = 2.5 104, = 0.01, = 0.0001 in this example.

Figure 6.9. (Example 7.3) Design domain for for the example with a force term.

We test the cases for three di↵erent force terms f = [1125, 0], [562.5, 0], [1687.5, 0]. We choose the initial

distribution 1 with fluid region located in a circular region with center [1/2, 1/2] and radius 1/p

3. Optimalresults and energy curves are plotted in Figures 6.10-6.12 for di↵erent force f , and the new algorithm alsoconverges faster to the optimal results than the MMA shown in [7]. For the cases f = [1125, 0] andf = [1687.5, 0], one can observe the fluid flows in the clockwise direction and counter clockwise direction inthe center roundabout respectively.




MMA 150 × 100 236Level set 216 × 144 681

Our algorithm 192 × 128 35

Table: Comparison of the number of iterations of different methods to obtain the optimal designresult for Example 6.2 with d = 1.5. The parameters used in our algorithm are set as τ = 0.01,γ = 0.0001.


3D results


3D results

Figure: Left: Optimal design result on a 64 × 64 × 64 grid. Right: Energy curve. In this case theparameters are set as α = 2.5 × 104, τ = 0.05, γ = 0.01.




Level set 36 × 36 × 36 31660 × 60 × 60 647

Our algorithm 32 × 32 × 32 3364 × 64 × 64 48

Table: Comparison of different methods for Example 6.5. The parameters used in our algorithmare set as α = 2.5 × 104, τ = 0.05, γ = 0.01.


Darcy Stokes (W.Hu)

No-slip on Γu and zero traction on Γt .At every point inside the domain, fluid is added (like rain falling down on afield), and the goal is to transport this fluid out from the domain andthrough the only part of the boundary possible Γt , with as little effort aspossible.Optimal regions regions of Stokes (yellow)and Darcy (black) types of flow.The prescribed volume fraction is ν = 0.3 in both cases.


Topology optimization of artery flow networks

rand(0,1) is a random number between 0 and 1.

The nodes belong to β are randomly distributed with the allowed domain one by one

according to the following rules: if the distance between a new node and any already existing

node is less than a value lmin, then the new node is not accepted; if the distance of two different

nodes is less than a value lmax, then two nodes are connected by a pipe.

2.2.1. The network on a circular area

In Part 2.2.1, V = 0.003, x = 1; β = 1, α = i : 2 ≤ i ≤ 1000, i ∈ ℕ; nα = 999,nβ = 1.

The nodes are on a circle with a radius of 0.5. The node belong to α is on the top of the circle.

lmin = 0.019, lmax = 0.044, which results in m = 3237.

Fig. 6 is the ground structure for arterial tree-type networks constructed on a circular

area.When η = 0.2, the corresponding optimum generated after 78 iterations is shown in Fig. 7.

When η = 0.1, 0.2, 0.3,0.4, 0.5, 0.6, the number of required iterations to obtain the

corresponding optimum was 353, 78, 104, 188, 236, 512, and the solution is shown in the following

six figures(Fig. 8 ~ Fig. 13), respectively.

6

rand(0,1) is a random number between 0 and 1.

The nodes belong to β are randomly distributed with the allowed domain one by one

according to the following rules: if the distance between a new node and any already existing

node is less than a value lmin, then the new node is not accepted; if the distance of two different

nodes is less than a value lmax, then two nodes are connected by a pipe.

2.2.1. The network on a circular area

In Part 2.2.1, V = 0.003, x = 1; β = 1, α = i : 2 ≤ i ≤ 1000, i ∈ ℕ; nα = 999,nβ = 1.

The nodes are on a circle with a radius of 0.5. The node belong to α is on the top of the circle.

lmin = 0.019, lmax = 0.044, which results in m = 3237.

Fig. 6 is the ground structure for arterial tree-type networks constructed on a circular

area.When η = 0.2, the corresponding optimum generated after 78 iterations is shown in Fig. 7.

When η = 0.1, 0.2, 0.3,0.4, 0.5, 0.6, the number of required iterations to obtain the

corresponding optimum was 353, 78, 104, 188, 236, 512, and the solution is shown in the following

six figures(Fig. 8 ~ Fig. 13), respectively.

6

Ground structure for arterial tree-type networks constructed on a circular area.


Topology optimization of artery flow networks

observed from Fig. 8x ~ Fig. 13x and their corresponding histograms Fig. 8x hist ~ Fig. 13x hist(the

following six figures).

11


Topology Optimization Heat Sink

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TopologyOptimizationofHeatsink


Topology Optimization of Heat Sink

Figure: (Example 1.1) Left: The temperature distribution for g · n = 10000. Right: Thetemperature distribution for g · n = 50000.


Image Segmentation for the intensity inhomogeneityimagesLocally Statistical Active Contour (LSAC) Model for image segmentation withintensity inhomogeneity

# of iteration steps of ICTM 8 7 7 7 7# of iteration steps of level set method [1] 7 13 35 186 239

Figure:First row: initial contour on the same image with different intensity inhomogeneity.Second row: the segmented region.Table: Comparison of the number of iteration steps for each case from the left to theright between ICTM and the level set method used in [1]. In this simulation, we setρ = 15, γ = 0.1, and τ = 0.001.


Noisy intensity inhomogeneity images

# of iterations of the ICTM 5 30 28 35 18# of iterations of the level-set method [1] 57 219 670 290 230

Figure: Initial contour and segmented region by using ICTM in LSAC model. Theparameters are: 1. ρ = 15, γ = 0.1, and τ = 0.02, 2. ρ = 5, γ = 0.15, and τ = 0.03, 3.ρ = 10, γ = 0.02, and τ = 0.01, 4. ρ = 10, γ = 0.7, and τ = 0.03, 5. ρ = 10,γ = 0.035, and τ = 0.002, respectively.


Conclusions

We proposed a novel iterative convolution-thresholding method that isapplicable to a range of problems in toplogy optimization.The method enjoys the energy-decaying property under certainconditionsThe method is simple, efficient and insensitive to parameters

Thanks! Questions?


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