STRUCTURAL TOPOLOGY OPTIMIZATION USING FINITE
ELEMENT BASED LEVEL SET METHOD
STRUCTURAL TOPOLOGY OPTIMIZATION
FINITE ELEMENT BASED LEVEL SET METHOD
USING
Xianghua Xing; Michael Yu Wang
Department of Mechanical and Automation Engineering
The Chinese University of Hong Kong
THEME
Optimization
KEYWORDS
Structural topology optimization, level set methods, finite element, streamline
diffusion.
SUMMARY
A finite element based level set method is proposed for structural topology
optimization. Because both the level set equation and the reinitialization
equation are advection dominated partial differential equations, the standard
Galerkin finite element method may produce oscillating results. In this paper,
both equations are solved using a streamline diffusion finite element method
(SDFEM). The Dirichlet boundary condition is enforced using the penalty
method to preserve the mass during reinitialization.
The advantage of the proposed method is that structural optimization on
irregular design domains can be carried out easily. Furthermore, this method
integrates the stress analysis and the boundary evolution within the framework
of finite element methods, which facilitates the programming and combination
with other finite element software.
1. INTRODUCTION
The level set method has become an emerging technique for structural shape
and topology optimization because it can handle topology changes easily and
describe smooth boundaries [1, 2, 20, 25]. In most of the applications, the level
set method is implemented with the finite difference technique, such as upwind
scheme, ENO, and WENO [14, 19]. These methods work well on a structured
grid, but difficulties happen if the problem involves complex geometries and
boundaries, where spatial discretization with the structured grid is impossible.
However, the finite element method (FEM) handles these problems flexibly.
This is one of our motivations for implementing the level set method with the
FEM.
STRUCTURAL TOPOLOGY OPTIMIZATION USING FINITE
ELEMENT BASED LEVEL SET METHOD
There are generally two stages in a level set based structural optimization
process: the stress analysis stage and the boundary evolution stage involving
level set methods. The first one is typically carried out with FEMs in industrial
applications. Therefore, our second motivation is to unify the techniques of
both stages within a uniform framework.
The level set equation is a first order hyperbolic equation. It is well known that
the standard Galerkin FEM may produce spatial instability when pure
advection equations or advection dominated equations are considered. Hence,
stabilized finite element techniques must be used. Barth and Sethian are the
first to discretize the level set equation on unstructured triangular meshes using
finite element techniques [3], and their method is subsequently employed in
[9]. In addition, some other finite element techniques have been implemented
for level set methods, such as the Streamline upwind/Petrov-Galerkin (SUPG)
[10, 23, 26], the Galerkin least squares (GLS) [8, 18, 24], the Taylor Galerkin
[5, 22], the characteristic Galerkin [7, 11, 17], the discontinuous Galerkin (DG)
[12, 16], and the least squares finite element method (LSFEM) [15]. Another
approach [13, 27] is to eliminate the advection term using the so called
assumed gradient method, and then the equation can be solved using the
standard Galerkin FEM.
In this paper, a level set method based on the streamline diffusion finite
element method (SDFEM) [23] is employed into structural topology
optimization problems. This paper is organized as follows. In Section 2, we
introduce structural optimization problems taking the mean compliance
problem for example. The implicit representation and propagation of the
boundary with level set methods is described in Section 3. Section 4 presents
the SDFEM for level set methods, which is followed by numerical examples in
Section 5. Conclusions are given in the last section.
2. STRUCTURAL OPTIMIZATION PROBLEMS
There are several kinds of structural optimization problems, such as the
minimum mean compliance problem, the maximum natural frequency
problem, and the minimum stress problem. In this paper, the first one is taken
for example.
In general, the minimum mean compliance problem can be specified as:
Minimize J (u, Ω) = ∫ F (u)dΩ
Ω
(1)
subject to : ∫ ε(u)T Dε( v )dΩ = ∫ p ⋅ vdΩ + ∫ τ ⋅ vdΓ
(2)
u = u 0 on ΓD
(3)
∫ dΩ ≤ vol
(4)
Ω
Ω
Ω
ΓN
STRUCTURAL TOPOLOGY OPTIMIZATION USING FINITE
ELEMENT BASED LEVEL SET METHOD
where J is the objective function, Ω is the domain occupied by the structure,
u the displacement vector, ε the strains, D the elasticity matrix, p the body
forces, τ the tractions applied on the boundary ΓN , u 0 the prescribed
displacement on the boundary ΓD , and v the virtual displacements. The
boundary of Ω is denoted by ∂Ω . The strain energy density F (u) is
F (u) = ε(u) T Dε(u) 2 .
(5)
Inequality (4) describes an upper limit on the amount of material in terms of
the maximum admissible volume vol of the structure. One can combine it with
the objective function using the augmented Lagrange multiplier method to
construct an augmented objective function:
(
)
(
J (u, Ω) = ∫ F (u)dΩ + λ ∫ dΩ − vol + 0.5η ∫ dΩ − vol
Ω
Ω
Ω
)
2
(6)
where λ is the Lagrange multiplier and η the penalty parameter. Following
[1, 25], we write the shape derivative of J as follows:
J ′ = ∫ (λ − β )Vn dΩ
(7)
∂Ω
with
β = F (u) − p ⋅ u
λ = λ + η ∫ dΩ − vol
(
Ω
)
(8)
(9)
and Vn means the normal velocity on the boundary. It should be noted that, we
have assumed that the part of the boundary where τ is applied can not move
for simplicity. Now we can define a descent direction by moving the boundary
with the normal velocity:
Vn = β − λ .
(10)
3. LEVEL SET METHODS
In the level set method, a boundary is represented implicitly through a level set
function φ (x) , which is a scalar function defined on a fixed design domain D
that always contains the structure domain. In this study, we define the
boundary as the zero level set, the interior of the structure as the domain where
φ > 0 , and the exterior as the domain where φ < 0 (Figure 1).
The level set method adds dynamics to implicit boundary and the evolution is
governed by the Hamilton-Jacobi equation
∂φ ∂t − Vn ∇φ = 0
(11)
where Vn is the normal velocity on the boundary and is also the important link
between the level set method and the structural optimization problem. Once the
velocity is computed using Eq. (10), one can update the boundary of the
structure by solving Eq. (11).
STRUCTURAL TOPOLOGY OPTIMIZATION USING FINITE
ELEMENT BASED LEVEL SET METHOD
Figure 1: Level set representation: structure, boundary and the design domain
(left) and the level set function (right).
During the evolution, the level set function frequently becomes too flat or too
steep, which may introduce numerical errors into spatial discretization. In order
to regularize the level set function, we reinitialize it periodically by solving the
reinitialization equation:
∂ψ ∂t + w ⋅ ∇ψ = S , w = S (∇ψ ∇ψ )
(12)
where S is the sign function:
S =ψ
ψ 2 + α 2 ∇ψ
2
(13)
The parameter α can be specified as the element size.
4. FINITE ELEMENT BASED LEVEL SET METHOD
In this section, the SDFEM is used to discretize Eqs. (11) and (12) as described
in [23]. The weak form of Eq. (11) is
(φ n+1 − φ n , v ) − ∆t (Vn ∇φ , v + δV ⋅ ∇v ) = 0
(14)
where v is the test function and
V = Vn n = Vn (− ∇φ ∇φ )
(15)
is the advection velocity vector. The parameter δ is simply chosen as
2
δ = 0.5⎛⎜ ∆t − 2 + J −1 V ⎞⎟
−1
(16)
⎝
⎠
and J −1 is the inverse Jacobian matrix for the transformation from the natural
coordinates to the global coordinates.
It is worthwhile to note that only the test function for the advection term is
modified while the transient term is the same as that in the standard Galerkin
FEM. This is unlike the SUPG [6], which uses modified test function for both
terms. An advantage of the presented method is that the coefficient matrix is
symmetric and constant, so the matrix is easily to be decomposed and it only
STRUCTURAL TOPOLOGY OPTIMIZATION USING FINITE
ELEMENT BASED LEVEL SET METHOD
needs one time of decomposition. Furthermore, this matrix is similar to the
mass matrix in computational dynamics, so it can be lumped to a diagonal
matrix. In this way, the system of equations (14) is decoupled and solving it
becomes very economical.
In structural optimization problems, velocity obtained from the shape
derivative (7) is only defined on the boundary. However, velocity Vn must be
defined on the whole computation domain or at least on a narrow band around
the boundary to solve Eq. (14). This needs a velocity extension and various
techniques have been reported in [14, 19]. The extension can be performed in a
natural way in structural optimization problems—because β is defined on the
whole computation domain and λ is constant everywhere, extended velocity
can be calculated using Eq. (10) directly.
The temporal discretization in (14) uses the forward Euler scheme, which is
conditionally stable. Therefore, the size of time step ∆t should satisfy the CFL
condition:
∆t = c(h V max )
(17)
with h the size of the smallest element, V
max
the largest absolute velocity,
and c is a coefficient between 0 and 1. We use c = 0.5 in this study.
The weak form of Eq. (12) is
(ψ m+1 −ψ m , v ) + ∆t (ε∇ψ m+1 , ∇v ) =
(
∆t S − w ⋅ ∇ψ m , v + δw ⋅ ∇v
)
(18)
with ψ 0 = φ n +1 . An extra numerical diffusion has been added ( ε > 0 ) because
the streamline diffusion gives an insufficient diffusion effect close to the
boundary where S is small. The reinitialization process causes loss of volume
due to numerical diffusions and a Dirichlet boundary condition should be
enforced to fix the boundary. Here, following the method introduced in [8], we
derive the constraint for a 4-node quadrilateral element that is intersected by
the boundary as shown in Figure 2. The constraint for other element types may
be derived similarly.
For each edge that is intersected by ∂Ω , we need to satisfy the constraint:
4
(19)
∑i =1 N i (ξ ∗ ,η ∗ )φi = 0
where ξ ∗ and η ∗ are natural coordinates of the intersection as shown in Figure
2. Actually, shape functions N 3 and N 4 vanish on edge 1-2 and Eq. (19)
reduces to the follows:
N 1 (ξ ∗ ,η ∗ )φ1 + N 2 (ξ ∗ ,η ∗ )φ 2 = 0.
(20)
Then the submatrix G I for the considered intersection is:
STRUCTURAL TOPOLOGY OPTIMIZATION USING FINITE
ELEMENT BASED LEVEL SET METHOD
[
]
G I = N 1 (ξ ∗ , η ∗ ) N 2 (ξ ∗ , η ∗ ) .
(21)
The global matrix G can be assembled in the same way as the global stiffness
matrix:
G ( I , dofs ) = G I
(22)
with dofs indicating the positions of node 1 and 2 in the global system.
Adding the constraint into the original system of equations gives the final form
of the discretized reinitialization equation as
(C + ρG T G )ψ = f
(23)
with C is the original coefficient matrix, f the right hand side, and ψ the
unknown vector.
Figure 2:
An element intersected by the boundary.
5. NUMERICAL EXAMPLES
Two numerical examples are tested to illustrate the reliability of the presented
method. The design domain is discretized with a finite element mesh and this
mesh remains unchanged during the optimization process. Both the stress
analysis stage and the level set evolution stage are implemented on this mesh.
It should be noted that, in the stress analysis, the “ersatz material” approach is
employed, which replaces holes and margins, where actually no material
exists, by a weak material with small Young’s modulus of elasticity E. In this
study, E of the solid and weak material are 1 and 1e-3 respectively. Young’s
modulus of an intersected element is calculated according to the amount of the
solid material in this element. Poisson ratio is 0.3.
Figure 3:
Cantilever beam
STRUCTURAL TOPOLOGY OPTIMIZATION USING FINITE
ELEMENT BASED LEVEL SET METHOD
The first example is a cantilever beam (Figure 3) with the following geometric
sizes: H = 1 and L = 2 . A concentrated force P = 1 acts on the middle of the
right edge. The maximum volume is 0.5 of the volume of the design domain.
First, a uniform mesh with 100-by-50 4 node elements is used, and the size of
each element is 0.02. The optimization process terminates if rJ ≤ 1e − 6 and rJ
is defined as
rJ = J n +1 − J n J n .
(24)
In practice, this criterion may be too strict, so the maximum number of steps is
always specified. In this example, the maximum number is 200. Figure 4
shows the initial design and the final design after 200 steps. Figure 5 shows the
convergence of the objective function and the volume ratio. Both functions
become stable after 130 steps, which mean that the optimization has converged
at that time.
Figure 4: Cantilever beam: Initial design (left) and final design with the uniform
mesh (right)
Figure 5: Convergence of the objective function (left) and the volume ratio (right)
of the cantilever beam with the uniform mesh.
Next, a free mesh shown in Figure 6 is used, which consists of 4864 elements
and 5015 nodes. In this case, the optimization process terminates after 154
steps and Figure 7 shows the final design, whose initial design is the same as
the initial design in Figure 4. Figure 8 shows the convergence of the objective
function and the volume ratio.
STRUCTURAL TOPOLOGY OPTIMIZATION USING FINITE
ELEMENT BASED LEVEL SET METHOD
Figure 6:
Free mesh of the cantilever beam
Figure 7:
Cantilever beam: The final design with the free mesh
Figure 8: Convergence of the objective function (left) and the volume ratio (right)
of the cantilever beam with the free mesh.
The second numerical example is a cantilever beam with a fixed hole (Figure
9). The geometric parameters are as follows: L = 9 , H = 6 , W = 3 , D = 3 and
R = 2 . A concentrated force P = 1 is applied at the lower right corner. The
design domain in this case is the rectangle excluding the hole. The maximum
volume is 0.5 of the design domain. The maximum number of steps is 100.
Figure 10 shows the mesh, which contains 5400 elements and 5670 nodes.
Figure 11 shows the initial and final designs. The optimal result is very similar
STRUCTURAL TOPOLOGY OPTIMIZATION USING FINITE
ELEMENT BASED LEVEL SET METHOD
to what are reported in [4, 21]. Figure 12 shows the convergence of the
objective function and the volume ratio.
Figure 9:
Cantilever beam with a fixed hole
Figure 10: Mapped mesh for the cantilever beam with a fixed hole
Figure 11: Cantilever beam with a fixed hole: Initial design (left) and the final
design (right)
STRUCTURAL TOPOLOGY OPTIMIZATION USING FINITE
ELEMENT BASED LEVEL SET METHOD
Figure 12: Convergence of the objective function (left) and the volume ratio (right)
of the cantilever beam with a fixed hole.
6. CONCLUSIONS
A finite element based level set method is employed for structural topology
optimization problems. Both the level set equation and the reinitialization
equation are solved with the streamline diffusion FEM for stabilization.
Dirichlet boundary condition is enforced during reinitialization to preserve the
mass. Numerical Results show that this proposed method is reliable and
effective. This method facilitates optimization problems on irregular design
domains. Moreover, both stress analysis stage and boundary evolution stage
are unified within the same framework of finite element techniques.
ACKNOWLEDGEMENTS
This research work was supported in part by the Research Grants Council of
Hong Kong SAR (Project No. CUHK416507).
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