4.6. Topology Optimization

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4.5.4.3. Domain velocity computation

Perturbation of shape design variables results in the movement of the design boundary, as discussed in the previous section. Interior material points, or finite element nodes, in the structural domain must also move following certain rules. Among numerous approaches for domain velocity field computations, we briefly discuss two representative methods: the boundary displacement method and the isoparametric mapping method (Choi and Chang 1994).

4.5.4.3.1. Boundary displacement method

In the boundary displacement method, movements of nodal points at the design boundary are treated as the prescribed displacements. The domain velocity field that corresponds to the design perturbation can be obtained by solving an auxiliary elasticity problem with prescribed displacements specified at design boundary nodes as well as prescribed rollers at the nondesign boundary, as depicted in Figure 4.23. Note that rollers are added to constrain the nodal point movements at the nondesign boundary, as shown in Figure 4.23b, in accordance with the design intent.

FIGURE 4.23 Illustration of the boundary displacement method for domain velocity field computation. (a) The original structure with boundary conditions. (b) The boundary velocity field added to the design boundary and additional rollers added to the nondesign boundary.

To form the auxiliary problem for the domain velocity computation, both the velocity field of the design boundary and the displacement constraints (e.g., rollers) that define nodal point movements on the nondesign boundary must be imposed to the finite element model. The discretized equilibrium equation of the auxiliary finite element model can be written as

$KV = f$

(4.262)

where K is the reduced stiffness matrix of the auxiliary structure (e.g., the structure shown in Figure 4.23b, which is different from that of the original structure (e.g., the one shown in Figure 4.23a. V is the design velocity vector and f is the unknown vector of boundary forces, which produce the prescribed boundary velocity at both the design and nondesign boundary. In a partitioned form, Eq. 4.262 can be written as

$[\begin{matrix} K_{bb} & K_{db} \\ K_{db} & K_{dd} \end{matrix}] [\begin{matrix} V_{b} \\ V_{d} \end{matrix}] = [\begin{matrix} f_{b} \\ 0 \end{matrix}]$

(4.263)

where V_b is the prescribed velocity of nodes on the boundary, V_d is the nodal velocity vector in the interior domain (domain velocity), and f_b is the unknown boundary force that acts on the structural boundary. Equation 4.263 can be rearranged as

$K_{dd} V_{d} = - K_{db} V_{b}$

(4.264)

This defines a linear relationship between the boundary and domain velocity fields.

The boundary displacement method is independent of the way in which the boundary velocity field is computed. The same finite element code used for analysis can be used to compute the displacement field of the auxiliary model; yielding a domain velocity field that naturally satisfies the linearity requirement (because the finite element matrix equations are a system of linear equations). An important characteristic of the elasticity problem is that the solution trajectory tends to maintain orthogonality. As a result, the updated mesh obtained using the velocity field, as stated in Eq. 4.242, tends to be more regular (see Figure 4.26 as an example).

One drawback of the boundary displacement method is that finite element matrix equations must be formulated and solved to generate the velocity field for each shape design variable. For this, the reduced stiffness matrix of the auxiliary finite element model must be formed and decomposed for each design variable. Once the design velocity field is calculated, it can be used throughout the design iterations.

4.5.4.3.2. Isoparametric mapping method

The isoparametric mapping method is far more efficient than the boundary displacement method because the former needs only a few matrix multiplications. However, the essence of the isoparametric method is the availability of a mapping function N that maps nodal points from parametric coordinates (u,w) of the geometric model to Cartesian coordinates (x,y,z), as illustrated in Figure 4.24.

In Figure 4.24, the structural domain is meshed into 5 × 3 finite elements. If the nodes are evenly distributed in the (u,w) space, the parametric coordinates of individual nodes are readily available. For example, the parametric coordinates of node 8 are (u₈,w₈) = (0.2, 0.667). To simplify the mathematical expressions in our discussion, we assume a 2-D planar structure for discussing the parametric mapping method.

FIGURE 4.24 Concept of isoparametric mapping.

Finding an appropriate mapping function N for accurate velocity field computation is difficult when the geometric shape of the structure is complicated. However, for a simple geometric model or a smaller subdomain of the structure, for which an accurate mapping function N can be found, the isoparametric mapping method is attractive. For such a model or subdomain that can be modeled by a single geometric entity, velocity fields can be computed using the isoparametric mapping method.

Geometric modeling software, such as MSC/PATRAN and HyperMesh, employs a standard patch to represent geometric surfaces. For example, in PATRAN, a surface patch is modeled in Coons patch (Mortenson 2006). A Coons patch, as shown in Figure 4.25a, is a bicubic parametric surface in terms of parametric coordinates u and w, where u and w ∊ [0,1]. Obviously, one single patch is not able to represent a complicate geometric entity. Therefore, a 2-D structural domain is often decomposed into several smaller patches in the modeling process.

FIGURE 4.25 A bicubic parametric patch. (a) Corner points and tangent vectors. (b) Adjusting tangent vector P_11,u. (c) Data formulated in a 4 × 4 × 2 B-matrix that defines the patch.

FIGURE 4.26 The two-dimensional fillet example modeled in two patches. (a) Original mesh. (b) Remesh using the boundary displacement method. (c) Remesh the using mapping method (Choi and Chang 1994).

The edges of a Coons patch are cubic curves in u and w, respectively. The shape of the cubic curve can be controlled by adjusting tangent vectors in both direction and magnitude. For example, changing the direction of the tangent vector P_11,u will alter the geometry of edge 3 and, therefore, the geometry of the patch, as illustrated in Figure 4.25b. A mesh of quad-elements can be generated by specifying the number of elements along the u- and w-parametric directions, such as the mesh shown in Figure 4.24, in which the number of elements along the u- and w-parametric directions are 5 and 3, respectively.

The data representation of a Coons patch is in a so-called B-matrix, as shown in Figure 4.25c, defined as

$B = {[\begin{matrix} P_{00} & P_{01} & P_{00, w} & P_{01, w} \\ P_{10} & P_{11} & P_{10, w} & P_{11, w} \\ P_{00, u} & P_{01, u} & P_{00, u w} & P_{01, u w} \\ P_{10, u} & P_{11, u} & P_{10, u w} & P_{11, u w} \end{matrix}]}_{4 \times 4 \times 2}$

(4.265)

where P₀₀ = S(0,0), P₀₁ = S(0,1), P₁₀ = S(1,0), and P₁₁ = S(1,1) are the four corner points; P_00,u = ∂S/∂u|_u=w=0, P_01,u, P_10,u, and P_11,u are the tangent vectors in the u-direction at the four corner points; P_00,w = ∂S/∂w|_u=w=0, P_01,w, P_10,w, and P_11,w are the tangent vectors in the w-direction at the four corner points; and P_00,uw = ∂²S/∂u∂w|_u=w=0, P_01,uw, P_10,uw, and P_11,uw are the twister vectors at the four corner points. Note that a twister vector represents changes of tangent vector in the u (or w)-direction at a corner point along a boundary curve in the w (or u)-direction. For example, P_10,uw = ∂/∂w(∂S/∂u)|_{u=1, w=0} = ∂P_10,u/∂w is the derivative of the tangent vector along the u-direction at P₁₀ with respect to w. The same twister vector can also be interpreted as P_10,uw = ∂/∂u(∂S/∂w)|_{u=1, w=0} = ∂P_10,w/∂u, which is the derivative of the tangent vector along the w-direction at P₁₀ with respect to u. Note that for a planar patch, all twister vectors vanish. Also, the first two columns of the B matrix are boundary edges 1 and 3, respectively; and rows 1 and 2 are boundary edges 2 and 4, respectively.

Recall that the mathematical expression (algebraic format) for a bicubic parametric surface was given in Eq. 4.226. Similar to a Bézier surface, a Coons patch can also be written in a form like that of Eq. 4.227:

$S (u, w) = U N^{C} B N^{C^{T}} W^{T} = [\begin{matrix} u^{3} & u^{2} & u & 1 \end{matrix}] [\begin{matrix} 2 & - 2 & 1 & 1 \\ - 3 & 3 & - 2 & - 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}] {[\begin{matrix} P_{00} & P_{01} & P_{00, w} & P_{01, w} \\ P_{10} & P_{11} & P_{10, w} & P_{11, w} \\ P_{00, u} & P_{01, u} & P_{00, u w} & P_{01, u w} \\ P_{10, u} & P_{11, u} & P_{10, u w} & P_{11, u w} \end{matrix}]}_{4 \times 4 \times 3} [\begin{matrix} 2 & - 3 & 0 & 1 \\ - 2 & 3 & 0 & 0 \\ 1 & - 2 & 1 & 0 \\ 1 & - 1 & 0 & 0 \end{matrix}] [\begin{matrix} w^{3} \\ w^{2} \\ w \\ 1 \end{matrix}]$

(4.266)

Here, (u,w) ∊ [0,1] × [0,1], and

$N^{C} = [\begin{matrix} 2 & - 2 & 1 & 1 \\ - 3 & 3 & - 2 & - 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}]$

(4.267)

The boundary edges of a Coons patch are Hermit cubic curves, defined as

$P (u) = U N^{C} H = [\begin{matrix} u^{3} & u^{2} & u & 1 \end{matrix}] [\begin{matrix} 2 & - 2 & 1 & 1 \\ - 3 & 3 & - 2 & - 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}] [\begin{matrix} P_{0} \\ P_{1} \\ P_{0, u} \\ P_{1, u} \end{matrix}]$

(4.268)

Here, H = [P₀, P₁, P_0,u, P_1,u]^T, in which P₀ and P₁ are the end points and P_0,u and P_1,u are tangent vectors at the end points. Entries in the matrix H can be used for shape design variables, depending on which boundary edge the curve represents. Any change in H alters the shape of the Coons patch:

$δ S (u, w) = U N^{C} δ B N^{C^{T}} W^{T}$

(4.269)

where

$δ B = {[\begin{matrix} δ P_{00} & δ P_{01} & δ P_{00, w} & δ P_{01, w} \\ δ P_{10} & δ P_{11} & δ P_{10, w} & δ P_{11, w} \\ δ P_{00, u} & δ P_{01, u} & 0 & 0 \\ δ P_{10, u} & δ P_{11, u} & 0 & 0 \end{matrix}]}_{4 \times 4 \times 2}$

(4.270)

For example, if the y-component of the tangent vector along the u-direction at (u,w) = (1,1)—that is, P_11y,u—is varied by δP_11y,u = 1, then δB is

$δ B_{y} = {[\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}]}_{4 \times 4}$

(4.271)

In this case, δB_x(u,w) = 0 because no change occurs in the x-direction. The domain velocity can be calculated by plugging the (u,w) of the node into Eq. 4.269. Note that in Eq. 4.270, we set the variation of the twister vectors to zero for planar structures.

EXAMPLE 4.22

Assume that δP_11y,u = 1. Calculate the velocity for node 8 (n₈) shown in Figure 4.24, where (u₈,w₈) = (0.2, 0.667).

Solutions

Using Eq. 4.269 with δB_y shown in Eq. 4.271, we have

$V_{y}^{8} = δ S_{y} (u_{8}, w_{8}) = δ S_{y} (0.2, 0.667) = [\begin{matrix} {0.2}^{3} & {0.2}^{2} & 0.2 & 1 \end{matrix}] [\begin{matrix} 2 & - 2 & 1 & 1 \\ - 3 & 3 & - 2 & - 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}] {[\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}]}_{4 \times 4} [\begin{matrix} 2 & - 3 & 0 & 1 \\ - 2 & 3 & 0 & 0 \\ 1 & - 2 & 1 & 0 \\ 1 & - 1 & 0 & 0 \end{matrix}] [\begin{matrix} 0 {.667}^{3} \\ 0 {.667}^{2} \\ 0 .667 \\ 1 \end{matrix}] = - 0.0237$

Hence, the velocity is V⁸ = [0,−0.0237].

If the boundary curve is not parameterized using a Hermit cubic curve, the change in the curve must be transformed to a Hermit cubic form and then the perturbed B matrix δB is created accordingly. For example, a cubic curve defined in Eq. 4.216 can be written as Bézier curve or a Hermit cubic curve, as follows:

$P (u) = U (u) A = U N^{B} G^{B} = U N^{C} H$

(4.272)

Here, G^B contains the four control points of the cubic Bézier curve, and H was defined in Eq. 4.268, consisting of the end points and tangent vectors of a Hermit cubic curve. Therefore, control point vector of a Bézier curve can be transformed into an H vector of a Hermit cubic curve as

$H = {(N^{C})}^{- 1} N^{B} G^{B}$

(4.273)

Then, the changes in a Bézier curve can be transformed into a change in the Hermit cubic curve:

$δ H = {(N^{C})}^{- 1} N^{B} δ G^{B}$

(4.274)

EXAMPLE 4.23

Assume that the design boundary of the patch that contains nodes n₁ to n₆ shown in Figure 4.24 is parameterized by a Bézier curve identical to that of Example 4.18. As stated in Example 4.18, the design change is δb = δP_2y = 1; that is, the design variable is the y-direction movement of control point P₂. Note that the design boundary is edge 3 of a Coons patch shown in Figure 4.25b. Calculate the velocity for node 8 inside the patch.

Solutions

From Example 4.18, the

$\partial G_{y}^{B}$

of the Bézier curve is

$δ G_{y}^{B} = [\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}] δ P_{2 y} = [\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}]$

(4.275)

From Eq. 4.274, we have

$δ H_{y} = {(N^{C})}^{- 1} N^{B} δ G_{y}^{B} = {[\begin{matrix} 2 & - 2 & 1 & 1 \\ - 3 & 3 & - 2 & - 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}]}^{- 1} [\begin{matrix} - 1 & 3 & - 3 & 1 \\ 3 & - 6 & 3 & 0 \\ - 3 & 3 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}] [\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ - 3 \\ 0 \end{matrix}]$

(4.276)

Using Eq. 4.269 with the second column of δB_y in Eq. 4.271 (with edge 3 as the design boundary in this case) replaced by the vector δH_y in Eq. 4.276, we have

$V_{y}^{8} = δ S_{y} (u_{8}, w_{8}) = δ S_{y} (0.2, 0.667) = [\begin{matrix} {0.2}^{3} & {0.2}^{2} & 0.2 & 1 \end{matrix}] [\begin{matrix} 2 & - 2 & 1 & 1 \\ - 3 & 3 & - 2 & - 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}] {[\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & - 3 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]}_{4 \times 4} [\begin{matrix} 2 & - 3 & 0 & 1 \\ - 2 & 3 & 0 & 0 \\ 1 & - 2 & 1 & 0 \\ 1 & - 1 & 0 & 0 \end{matrix}] [\begin{matrix} 0 {.667}^{3} \\ 0 {.667}^{2} \\ 0 .667 \\ 1 \end{matrix}] = - 0.2846$

Hence, the domain velocity at node 8 is obtained as V⁸ = [0,−0.2846].

As mentioned before, the isoparametric mapping method is much more efficient than the boundary displacement method; however, the velocity field obtained using the boundary displacement method tends to keep the finite element mesh more regular because tracing of the solution to an elastic problem, governed by partial differential equations, is more orthogonal. Figure 4.26a shows a 2-D fillet example, in which the finite element mesh updated using the velocity obtained from the boundary displacement method (Figure 4.26b) is of better quality than those updated using isoparametric mapping method. Figure 4.26c shows that, using isoparametric mapping, the finite element mesh could be severely distorted after a large design change (e.g., element A).

4.5.5. Shape Sensitivity Analysis Using Finite Difference or Semianalytical Method

As with the sensitivity analysis for sizing or material design variables, shape DSA can be carried out using the overall finite difference or semianalytical method, in which a design variable is perturbed and a finite element model is regenerated with the design change. These methods are general and widely employed in support of gradient-based optimization, despite the drawbacks pointed out previously.

For shape sensitivity analysis using the finite difference method or semianalytical method, it is desirable to update the finite element model (using Eq. 4.242) at the perturbed design using a velocity field obtained from the methods discussed above. In the overall finite difference method, an FEA is carried out for the FE model at the perturbed design, then the sensitivity coefficient is approximated using Eq. 4.35. In the semianalytical method, the stiffness matrix and load vector of the perturbed design must be generated, and Eq. 4.41 is followed to calculate the sensitivity coefficients.

In many situations, the design velocity field may not be readily available. Overall, the finite difference method is widely implemented for shape sensitivity analysis using CAD software equipped with a mesh generator and FEA solver, such as SolidWorks Simulation. A new dimension value with a design perturbation can be entered in CAD and the solid model can be rebuilt and remeshed. Thereafter, an FEA model for the perturbed design can be created (usually meshed by an automatic mesh generator), and an FEA can be carried out. The results obtained from FEA of the current and perturbed designs can then be used for sensitivity calculations using Eq. 4.35.

Although the approach is easy to implement, it has two potential pitfalls in addition to the problems of step size determination and computation efficiency, as discussed in Section 4.3. For shape design, the added pitfalls are altering topology of finite element mesh or geometric features at the perturbed design, as illustrated in Figure 4.27.

The pitfalls affect local performance measures, such as the stress measure defined at node A shown in Figure 4.27a. We assume a design change pulls the holes further apart. As a result, mesh is regenerated due to a design perturbation, as shown in Figure 4.27b. Due to the change in mesh topology, there is no clear trace in locating node A in the perturbed design: Is it at node A1, A2, or A3 that the stress performance is to be measured for sensitivity calculation using finite difference method? Another such example is the disappearing of nodes where performance measures are defined. As shown in Figure 4.27c, a performance measure, such as displacement, is defined at node B. Due to a design perturbation, the hole is moved to the right, such that it intersects the right edge of the rectangular structural boundary. As a result, part of the right edge disappears, as well as node B. Such a topology change in geometric features cause problems in calculating shape sensitivity coefficients using finite difference method with CAD software.

4.5.6. Material Derivatives

Besides finite difference and semianalytical methods, the continuum approach offers options for shape sensitivity analysis with a rigorous mathematical basis and better computational efficiency. Moreover, no step size is required for design perturbation, such as in the finite difference and semianalytical methods.

The key theory in continuum approach for shape sensitivity analysis is the material derivative. In general, the material derivative describes the time rate of change of a physical quantity, such as heat or momentum, for a material element subjected to a space- and time-dependent velocity field. For structural shape design, time is replaced by design change, and the time rate of change of a physical quantity becomes the rate of a performance measure change with respect to design (i.e., gradient or sensitivity coefficient).

Recall the energy bilinear form and load linear form discussed in Section 4.4.1, which governs the behavior of the structure under static load:

$a_{Ω} (z, \bar{z}) = ℓ_{Ω} (\bar{z}), \forall \bar{z} \in S$

(4.277)

FIGURE 4.27 Pitfalls in shape sensitivity analysis using finite difference or semianalytical methods with the mesh generator in CAD software: (a) mesh of current design, (b) mesh topology altered after design change, (c) Node B at the right edge of the current design, and (d) topology of geometric features altered eliminating Node B.

Note that the subscript Ω is added to the energy and load forms in Eq. 4.277 to emphasize their dependence on the structural domain Ω, which varies. In Eq. 4.277, z is the solution, or displacement, of the governing equation of the structure at the current design, in which the structural domain is Ω, as illustrated in Figure 4.28.

Let z_τ(x_τ) be the solution of the same governing equation but at a perturbed domain Ω_τ due to a design change:

$a_{Ω_{τ}} (z_{τ}, {\bar{z}}_{τ}) = ℓ_{Ω_{τ}} (\bar{z}), \forall {\bar{z}}_{τ} \in S_{τ}$

(4.278)

Here, the energy bilinear form and load linear form are formulated at the new structural domain Ω_τ, and S_τ is the space of kinematically admissible virtual displacement at the new design. Note that, in general, S = S_τ because the essential boundary conditions do not change with design and the smoothness of the solution function is not affected by the shape change. Again, τ is a scalar parameter that defines a shape change. In a material derivative for a physical problem, τ plays the role of time.

FIGURE 4.28 Illustration of material derivative concept.

It is important to note that the displacement z_τ(x_τ) at the new design is a new function z_τ and measured at the new location x_τ, which is determined by the design velocity field V, as discussed in Section 4.5.4. As discussed before, for a linear design velocity field, x_τ is determined by

$x_{τ} = x + τ V$

(4.279)

Expanding z_τ(x_τ) using Taylor's series, we have

$z_{τ} (x_{τ}) = z_{τ} (x + τ V) \approx z_{τ} (x) + \nabla [z_{τ}^{T} (x)] (τ V) + \dots$

(4.280)

Here, the gradient operator is employed—that is, ∇ = [∂/∂x¹, ∂/∂x², ∂/∂x³]^T, which is a 3 × 1 vector for three-dimensional structures. The material derivative of a function, such as the displacement, is defined as

$\dot{z} (x) \equiv {\frac{ⅆ z_{τ} (x_{τ})}{ⅆ τ} |}_{τ = 0} = {\frac{ⅆ z_{τ} (x + τ V)}{d τ} |}_{τ = 0}$

(4.281)

in which the dot on top of the function z denotes the material derivative on z. Equation 4.281 can be rewritten in a limiting form as

$\begin{matrix} \dot{z} (x) = {\frac{ⅆ z_{τ} (x + τ V)}{ⅆ τ} |}_{τ = 0} = lim_{τ \to 0} \frac{z_{τ} (x + τ V) - z (x)}{τ} \\ = lim_{τ \to 0} \frac{z_{τ} (x + τ V) - z_{τ} (x)}{τ} + lim_{τ \to 0} \frac{z_{τ} (x) - z (x)}{τ} \end{matrix}$

(4.282)

The first term on the right-hand side of Eq. 4.282 defines the change of the same function z_τ evaluated at different locations (x_τ and x respectively). This can be further derived using the expansion shown in Eq. 4.280 as

$lim_{τ \to 0} \frac{z_{τ} (x + τ V) - z_{τ} (x)}{τ} = lim_{τ \to 0} \frac{\nabla [z_{τ}^{T} (x)] (τ V) + \dots}{τ} = \nabla (z^{T}) V = \nabla z^{T} V$

(4.283)

The second term on the right-hand side of Eq. 4.282 defines the change due to different functions (z_τ and z) evaluated at the same location x, which is nothing but the variation of z from its definition discussed in Section 4.4.2:

$lim_{τ \to 0} \frac{z_{τ} (x) - z (x)}{τ} = z^{'}$

(4.284)

Hence, Eq. 4.282 becomes

$\dot{z} (x) = z^{'} + \nabla z^{T} V$

(4.285)

The physical meaning of Eq. 4.285 can be illustrated using a cantilever beam example shown in Figure 4.29, in which z = z₃ = z (subscript 3 is omitted for simplicity), x = x₁ = x (subscript 1 is omitted), and V = V₁ = V (subscript 1 is omitted). Hence, Eq. 4.285 is reduced to

$\dot{z} (x) = z^{'} + z_{, 1} V$

(4.286)

in which V is the design velocity that determines the material point movement due the beam length change. As shown in Figure 4.29, the difference between z(x) and z_τ(x_τ) consists of two parts, z′(x) ≈ z_τ(x) − z(x) and θ·(x_τ − x) = z_,1 τV.

FIGURE 4.29 Deformed cantilever beam at current and perturbed designs due to a length change.

Recall that the FEA solution for the cantilever beam with an evenly distributed load shown in Figure 4.13 is obtained in Eq. 4.200 as

$z_{3} (x) = \frac{q ℓ}{24 E I} (5 x^{2} ℓ - 2 x^{3})$

(4.287)

In the next example, we use this solution to further illustrate the concept of material derivatives for the cantilever beam example shown in Figure 4.13.

EXAMPLE 4.24

Calculate

$\dot{z}$

, z′, and z_,1V at

$x = ℓ$

for the cantilever beam shown in Figure 4.13, using the finite element solution of Eq. 4.200.

Solutions

We first evaluate z′ from the definition stated in Eq. 4.284. The displacement at the perturbed design due to the change

$δℓ$

in the length of beam can be found as

$z_{τ} (x) = \frac{q (ℓ + τ δ ℓ)}{24 E I} (5 x^{2} (ℓ + τ δ ℓ) - 2 x^{3}) = \frac{q}{24 E I} (5 x^{2} {(ℓ + τ δ ℓ)}^{2} - 2 x^{3} (ℓ + τ δ ℓ))$

(4.288)

Hence from Eq. 4.284, z′ can be calculated as

$\begin{matrix} z^{'} = \lim_{τ \to 0} \frac{z_{τ} (x) - z (x)}{τ} = \frac{\frac{q}{24 E I} (5 x^{2} {(ℓ + τ δ ℓ)}^{2} - 2 x^{3} (ℓ + τ δ ℓ)) - \frac{q}{24 E I} (5 x^{2} ℓ^{2} - 2 x^{3} ℓ)}{τ} \\ = \lim_{τ \to 0} \frac{q}{24 E I} \frac{2 x^{2} (5 ℓ - x) τ δ ℓ + 5 x^{2} {(τ δ ℓ)}^{2}}{τ} = \frac{q x^{2}}{12 E I} (5 ℓ - x) δ ℓ \end{matrix}$

(4.289)

For simplicity in shape sensitivity analysis, we set

$δℓ = 1$

. Note that Eq. 4.289 is nothing but

$\frac{\partial z_{3} (x)}{\partial ℓ}$

shown in Eq. 4.208. Evaluating Eq. 4.289 at

$x = ℓ$

, we have

$z^{'} = \frac{q ℓ^{3}}{3 E I}$

(4.290)

which is identical to Eq. 4.209. Now, we take the derivative of Eq. 4.200 with respect to x:

$z_{, 1} (x) = \frac{q ℓ}{12 E I} (5 x ℓ - 3 x^{2})$

(4.291)

$x = ℓ$

$V = δℓ$

. Hence,

$z_{, 1} V = \frac{q ℓ^{3}}{6 E I} δ ℓ = \frac{q ℓ^{3}}{6 E I}$

(4.292)

Adding Eqs 4.290 and 4.292, we have

$\dot{z} (x) = z^{'} + z_{, 1} V_{1} = \frac{q ℓ^{3}}{3 E I} + \frac{q ℓ^{3}}{6 E I} = \frac{q ℓ^{3}}{2 E I}$

(4.293)

which is

$\frac{\partial z_{3}^{ℓ}}{\partial ℓ} = \frac{q ℓ^{3}}{2 E I}$

, as obtained in Eq. 4.202 (and Eq. 4.207).

As demonstrated in Example 4.24, for the cantilever beam example, the sensitivity coefficient obtained using the discrete approach discussed at the beginning of this section is the same as the result of material derivative, in which the measurement point is assumed to be moving with the design. On the other hand, the result obtained in Eq. 4.209 is nothing but z′, in which the measurement point is assumed to be stationary. As discussed, the movement of the material point or the measurement point is determined by the design velocity field. At the boundary, the velocity field is

$δℓ$

. How do the interior points move due to the design change

$δℓ$

? How does the domain velocity field affect the shape sensitivity coefficient?

Now, we discuss further on the design velocity field. At the tip, the boundary velocity is defined as

$δℓ$

. The interior or domain velocity can be defined in different ways, depending on the design intent. For example, let us consider two design velocity fields V^A and V^B, shown in Figures 4.30a and 4.30b, respectively. V^A is a linear function, and V^B is a quadratic function. Both assume a length change at the right end by

$δℓ$

. Mathematically, they are defined as

$V^{A} = \frac{x}{ℓ} δ ℓ$

(4.294)

$V^{B} = {(\frac{x}{ℓ})}^{2} δ ℓ$

(4.295)

No matter how the velocity field is defined, the z′ term is unchanged because the functions z_τ and z are evaluated at the same point. The same is true for z_,1. The only term that is affected is the velocity field V in the second term of Eq. 4.287. We assume the displacement is measured at midpoint

$x = ℓ / 2$

, the design velocities at the midpoint are

$δℓ / 2$

and

$δℓ / 4$

, respectively, using V^A and V^B. Hence, the material derivatives of the displacement at the midpoint become

FIGURE 4.30 Midpoint movement with design change $δℓ$ . (a) $δℓ / 2$ : linear velocity. (b) $α δℓ \neq δℓ / 2$ : other than linear velocity. (c) $δℓ$ : beam length change is zero.

${\dot{z}}_{ℓ / 2}^{A} = z^{'} + z_{, 1} V_{1} = \frac{3 q ℓ^{3}}{32 E I} δ ℓ + \frac{7 q ℓ^{3}}{48 E I} (\frac{δ ℓ}{2}) = \frac{q ℓ^{3}}{6 E I} δ ℓ$

(4.296)

and

${\dot{z}}_{ℓ / 2}^{B} = z^{'} + z_{, 1} V_{1} = \frac{3 q ℓ^{3}}{32 E I} δ ℓ + \frac{7 q ℓ^{3}}{48 E I} (\frac{δ ℓ}{4}) = \frac{25 q ℓ^{3}}{192 E I} δ ℓ$

(4.297)

in which

${\dot{z}}_{ℓ / 2}^{A}$

predicts the displacement at the midpoint of the beam if the movement of the displacement measurement point is determined by velocity field V^A when a beam length is changed by

$δℓ$

. Similarly,

${\dot{z}}_{ℓ / 2}^{B}$

predicts the displacement at the midpoint of the beam if the movement of the displacement measurement point is determined by velocity field V^B. Therefore, in general, domain velocity is not unique, as long as it satisfies the requirements discussed in Section 4.5.4 and adequately captures the design intent.

We now take a look at a design velocity field V^C shown in Figure 4.30c, which is defined as a quadratic function with zero at both ends and maximum at the midpoint. Mathematically, V^C is defined as

$V^{C} = \frac{4 x}{ℓ^{2}} (ℓ - x) δ ℓ$

(4.298)

In this case, the shape sensitivity coefficient at the tip is zero because V = 0 at x = l. However, at the midpoint, the sensitivity is not zero; instead

${\dot{z}}_{ℓ / 2}^{C} = z^{'} + z_{, 1} V = 0 + \frac{7 q ℓ^{3}}{48 E I} (δ ℓ) = \frac{7 q ℓ^{3}}{48 E I} δ ℓ$

(4.299)

Here, z′ = 0 because the displacement function is unchanged (length stays constant). The second term of Eq. 4.299 is nonzero, which characterizes the change of displacement due to the change of the measurement point.

With a basic understanding of the concept of shape design, we move on to introduce the shape sensitivity analysis method based on the continuum approach. To simplify the mathematical derivations, we focus only on the beam example.

4.5.7. Shape Sensitivity Analysis Using the Continuum Approach

We start by stating one of the major material derivative equations that are essential for the continuum approach. Readers are referred to Choi and Kim (2006a) for details and mathematical proof of the equation.

For an integral function defined as

$Φ = \int_{Ω_{τ}} f_{τ} (x_{τ}) Ω_{τ}$

(4.300)

in which f is a scalar function, the material derivative of Φ is given as

$\dot{Φ} = (\int_{Ω_{τ}} f {(x_{τ})}^{ċ} Ω_{τ}) = \int_{Ω} (f^{'} (x) + div (f V)) Ω$

(4.301)

in which the divergence is defined as div(fV) = (fV)_,1 + (fV)_,2 + (fV)_,3 for 3-D structures.

This equation is applied to the energy bilinear form and load linear form of Eq. 4.277. We rewrite the energy bilinear form as

$a_{Ω_{τ}} (z_{τ}, {\bar{z}}_{τ}) = \int_{Ω_{τ}} c (z_{τ}, {\bar{z}}_{τ}) Ω_{τ}$

(4.302)

For a beam structure,

$c (z, \bar{z}) = E I z_{3,11} {\bar{z}}_{3,11}$

. The material derivative of

$a_{Ω_{τ}} (z_{τ}, {\bar{z}}_{τ})$

can be written using Eq. 4.301:

$[a_{Ω_{τ}} \dot{(z_{τ}}, {\bar{z}}_{τ})] = \int_{Ω} {[c (z, \bar{z})]}^{'} Ω + \int_{Ω} div [c (z, \bar{z}) V] Ω = \int_{Ω} [c (z^{'}, \bar{z}) + c (z, {\bar{z}}^{'})] Ω + \int_{Ω} div [c (z, \bar{z}) V] Ω = \int_{Ω} [c (\dot{z} - \nabla z^{T} V, \bar{z}) + c (z, \dot{\bar{z}} - \nabla {\bar{z}}^{T} V)] Ω + \int_{Ω} div [c (z, \bar{z}) V] Ω = \int_{Ω} c (\dot{z}, \bar{z}) Ω - \int_{Ω} [c (\nabla z^{T} V, \bar{z}) + c (z, \nabla {\bar{z}}^{T} V)] Ω + \int_{Ω} div [c (z, \bar{z}) V] Ω = a_{Ω} (\dot{z}, \bar{z}) + a_{V}^{'} (z, \bar{z})$

(4.303)

in which we employed the relation

$z^{'} = \dot{z} (x) - \nabla z^{T} V$

and set

$\dot{\bar{z}} (x) = 0$

because

$\bar{z}$

is virtual displacement. We define

$a_{Ω} (\dot{z}, \bar{z}) \equiv \int_{Ω} c (\dot{z}, \bar{z}) Ω$

(4.304)

and

$a_{V}^{'} (z, \bar{z}) \equiv - \int_{Ω} [c (\nabla z^{T} V, \bar{z}) + c (z, \nabla {\bar{z}}^{T} V)] Ω + \int_{Ω} div [c (z, \bar{z}) V] Ω$

(4.305)

Repeating the same for the load linear form, we have

$\overset{\cdot}{[ℓ_{Ω_{τ}} (\bar{z})]} = \int_{Ω} \overset{\cdot}{(f \bar{z})} Ω = \int_{Ω} {(f \bar{z})}^{'} Ω + \int_{Ω} div (f \bar{z} V) Ω = - \int_{Ω} f \nabla {\bar{z}}^{T} V Ω + \int_{Ω} div (f \bar{z} V) Ω \equiv ℓ_{V}^{'} (\bar{z})$

(4.306)

in which we assume f′ = 0. Hence, equating

$\overset{\cdot}{[a_{Ω_{τ}} (z_{τ}, {\bar{z}}_{τ})]}$

and

$\overset{\cdot}{[ℓ_{Ω_{τ}} ({\bar{z}}_{τ})]}$

, we have

$a_{Ω} (\dot{z}, \bar{z}) = - a_{V}^{'} (z, \bar{z}) + ℓ_{V}^{'} (\bar{z}), \forall \bar{z} \in S$

(4.307)

Similar to sizing sensitivity analysis, in implementation, the right-hand side of Eq. 4.307 is calculated as a fictitious load, which is employed to solve for

$\dot{z}$

. Equation 4.307 represents the direct differentiation method of the continuum approach. For the adjoint variable method of the continuum approach, readers are referred to Choi and Kim (2006a).

4.5.7.1. Shape sensitivity analysis for a cantilever beam

For one-dimensional beam bending problems, we have

$c (z, \bar{z}) = E I z_{3,11} {\bar{z}}_{3,11}$

, z = z₃, V^Tn = V, ∇z = z_3,1, and div f = f_,1. Hence, from Eqs 4.304, 4.305, and 4.306, we have

$a_{Ω} (\dot{z}, \bar{z}) = \int_{0}^{ℓ} E I {\dot{z}}_{3,11} {\bar{z}}_{3,11} ⅆ x$

(4.308)

$\begin{matrix} a_{V}^{'} (z, \bar{z}) = - \int_{0}^{ℓ} [E I {(z_{3,1} V)}_{, 11} {\bar{z}}_{3,11} + E I z_{3,11} {({\bar{z}}_{3,1} V)}_{, 11}] ⅆ x + \int_{0}^{ℓ} {(E I z_{3,11} {\bar{z}}_{3,11} V)}_{, 1} ⅆ x \\ = - 3 \int_{0}^{ℓ} (E I z_{3,11} {\bar{z}}_{3,11} V_{, 1}) ⅆ x - \int_{0}^{ℓ} [E I (z_{3,11} {\bar{z}}_{3,1} + z_{3,1} {\bar{z}}_{3,11}) V_{, 11}] ⅆ x \end{matrix}$

(4.309)

and

$\begin{matrix} ℓ_{V}^{'} (\bar{z}) = - \int_{Ω} f \nabla {\bar{z}}^{T} V Ω + \int_{Ω} div (f \bar{z} V) Ω = - \int_{0}^{ℓ} f {\bar{z}}_{3,1} V ⅆ x + \int_{0}^{ℓ} {(f {\bar{z}}_{3} V)}_{, 1} ⅆ x = \\ \int_{0}^{ℓ} (f_{, 1} V + f V_{, 1}) {\bar{z}}_{3} ⅆ x \end{matrix}$

(4.310)

Plugging these terms into Eq. 4.307, we have

$\begin{matrix} \int_{0}^{ℓ} E I {\dot{z}}_{3,11} {\bar{z}}_{3,11} ⅆ x = 3 \int_{0}^{ℓ} (E I z_{3,11} {\bar{z}}_{3,11} V_{, 1}) ⅆ x + \int_{0}^{ℓ} [E I (z_{3,11} {\bar{z}}_{3,1} + z_{3,1} {\bar{z}}_{3,11}) V_{, 11}] ⅆ x \\ + \int_{0}^{ℓ} (f_{, 1} V + f V_{, 1}) {\bar{z}}_{3} ⅆ x \end{matrix}$

(4.311)

Using finite element shape functions, such as those shown in Eq. 4.143, to discretize the sensitivity equation (Eq. 4.311), we have on the left-hand side

$\int_{0}^{ℓ} E I {\dot{z}}_{3,11} {\bar{z}}_{3,11} ⅆ x = {\bar{Z}}_{g}^{T} K_{g} {\dot{Z}}_{g}$

(4.312)

where

${\dot{Z}}_{g} = {[\begin{matrix} {\dot{z}}_{3_{i}} & {\dot{θ}}_{i} & {\dot{z}}_{3 j} & {\dot{θ}}_{j} \end{matrix}]}^{T}$

(4.313)

The right-hand side of Eq. 4.311 can be computed as

${\bar{Z}}^{T} F_{{fic}_{g}}$

, in which

$F_{{fic}_{g}}$

is the global fictitious load vector. By imposing the boundary conditions and removing the virtual displacement

$\bar{Z}$

from both sides, we have

$K \dot{Z} = F_{fic}$

(4.314)

which can be solved using the same decomposed stiffness matrix K as the original structure considering the fictitious load F_fic as additional load cases. Equation 4.311 represents the direct differentiation method of the continuum-discrete approach for shape sensitivity analysis. We illustrate a few more details in the following example.

EXAMPLE 4.25

Calculate shape sensitivity for the cantilever beam shown in Figure 4.13, assuming a linear design velocity of Eq. 4.294—that is,

$V = V^{A} = \frac{x}{ℓ}$

(set

$δℓ = 1$

)—using the continuum-discrete method.

Solutions

We first rewrite Eq. 4.311 for the cantilever beam with f = q (hence f_,1 = q_,1 = 0) and a linear velocity field (hence V_,11 = 0) as

$\int_{0}^{ℓ} E I {\dot{z}}_{3,11} {\bar{z}}_{3,11} ⅆ x = 3 \int_{0}^{ℓ} (E I z_{3,11} {\bar{z}}_{3,11} V_{, 1}) ⅆ x + \int_{0}^{ℓ} q V_{, 1} {\bar{z}}_{3} ⅆ x$

(4.315)

Using the shape functions of Eq. 4.144, we have

$z_{3} = N^{T} Z = [\begin{matrix} N_{3} & N_{4} \end{matrix}] [\begin{matrix} z_{3_{j}} \\ θ_{j} \end{matrix}] = [\frac{3 x^{2}}{ℓ^{2}} - \frac{2 x^{3}}{ℓ^{3}} - \frac{x^{2}}{ℓ} + \frac{x^{3}}{ℓ^{2}}] [\begin{matrix} \frac{q ℓ^{4}}{8 E I} \\ \frac{q ℓ^{3}}{6 E I} \end{matrix}] = \frac{q ℓ}{24 E I} (5 x^{2} ℓ - 2 x^{3})$

(4.316)

$z_{3,1} = {(N^{T} Z)}_{, 1} = [\begin{matrix} N_{3,1} & N_{4,1} \end{matrix}] [\begin{matrix} z_{3_{j}} \\ θ_{j} \end{matrix}] = [\frac{6 x}{ℓ^{2}} - \frac{6 x^{2}}{ℓ^{3}} - \frac{2 x}{ℓ} + \frac{3 x^{2}}{ℓ^{2}}] [\begin{matrix} \frac{q ℓ^{4}}{8 E I} \\ \frac{q ℓ^{3}}{6 E I} \end{matrix}] = \frac{q ℓ}{12 E I} (5 x ℓ - 3 x^{2})$

(4.317)

$z_{3,11} = {(N^{T} Z)}_{, 11} = [\begin{matrix} N_{3,11} & N_{4,11} \end{matrix}] [\begin{matrix} z_{3_{j}} \\ θ_{j} \end{matrix}] = [\frac{6}{ℓ^{2}} - \frac{12 x}{ℓ^{3}} - \frac{2}{ℓ} + \frac{6 x}{ℓ^{2}}] [\begin{matrix} \frac{q ℓ^{4}}{8 E I} \\ \frac{q ℓ^{3}}{6 E I} \end{matrix}] = \frac{q ℓ}{12 E I} (5 ℓ - 6 x)$

(4.318)

Bringing Eqs 4.316 to 4.318 to the right-hand side of Eq. 4.315, we have

$\int_{0}^{ℓ} E I z_{3,11} {\bar{z}}_{3,11} V_{, 1} ⅆ x = {\bar{Z}}^{T} \int_{0}^{ℓ} [\begin{matrix} \frac{6}{ℓ^{2}} - \frac{12 x}{ℓ^{3}} \\ - \frac{2}{ℓ} + \frac{6 x}{ℓ^{2}} \end{matrix}] [\frac{q ℓ}{12} (5 ℓ - 6 x)] \frac{1}{ℓ} ⅆ x = {\bar{Z}}^{T} \frac{q}{12} \int_{0}^{ℓ} [\begin{matrix} \frac{6}{ℓ^{2}} - \frac{12 x}{ℓ^{3}} \\ - \frac{2}{ℓ} + \frac{6 x}{ℓ^{2}} \end{matrix}] (5 ℓ - 6 x) ⅆ x = {\bar{Z}}^{T} [\begin{matrix} \frac{q}{2} \\ - \frac{q ℓ}{12} \end{matrix}]$

(4.319)

and

$\int_{0}^{ℓ} q {\bar{z}}_{3} V_{, 1} ⅆ x = {\bar{Z}}^{T} \int_{0}^{ℓ} q [\begin{matrix} \frac{3 x^{2}}{ℓ^{2}} - \frac{2 x^{3}}{ℓ^{3}} \\ - \frac{x^{2}}{ℓ} + \frac{x^{3}}{ℓ^{2}} \end{matrix}] \frac{1}{ℓ} ⅆ x = {\bar{Z}}^{T} \frac{q}{ℓ} \int_{0}^{ℓ} [\begin{matrix} \frac{3 x^{2}}{ℓ^{2}} - \frac{2 x^{3}}{ℓ^{3}} \\ - \frac{x^{2}}{ℓ} + \frac{x^{3}}{ℓ^{2}} \end{matrix}] ⅆ x = {\bar{Z}}^{T} [\begin{matrix} \frac{q}{2} \\ - \frac{q ℓ}{12} \end{matrix}]$

(4.320)

Bringing Eqs 4.319 and 4.320 to 4.315, we have

${\bar{Z}}^{T} {3 [\begin{matrix} \frac{q}{2} \\ - \frac{q ℓ}{12} \end{matrix}] + [\begin{matrix} \frac{q}{2} \\ - \frac{q ℓ}{12} \end{matrix}]} = {\bar{Z}}^{T} [\begin{matrix} 2 q \\ - \frac{q ℓ}{3} \end{matrix}] = {\bar{Z}}^{T} F_{fic}$

(4.321)

By solving the sensitivity vector in the same way as solving Z of the structure (see 4.149), we have

$[\begin{matrix} {\dot{z}}_{3 j} \\ {\dot{θ}}_{j} \end{matrix}] = K^{- 1} F_{fic} = \frac{ℓ}{12 E I} [\begin{matrix} 4 ℓ^{2} & 6 ℓ \\ 6 ℓ & 12 \end{matrix}] [\begin{matrix} 2 q \\ - \frac{q ℓ}{3} \end{matrix}] = [\begin{matrix} \frac{q ℓ^{3}}{2 E I} \\ \frac{2 q ℓ^{2}}{3 E I} \end{matrix}]$

(4.322)

4.6. Topology Optimization

Sizing optimization varies the sizes of the structural elements, such as the diameter of a bar or beam, as shown in Figure 4.31a, or the thickness of a sheet metal. In sizing optimization, the shape of the structure is known and unchanged. For shape optimization, the topology of geometric features in the structure, such as the number of holes, is known. The optimal shape is confined to the topology of the initial structural geometry, as illustrated in Figure 4.31b. No additional holes can be created during the shape optimization process. For both sizing and shape optimization, the topology of the structural geometry is unchanged in the design process.

Another important type of optimization problem is topology optimization, which solves the basic engineering problem of distributing a prescribed amount of material in a design space. Topology optimization is sometimes referred to as layout optimization. In general, the goal of topology optimization is to find the best use of material for a structural body that is subject to either a single load or multiple load distributions. The best use of material in the case of topology optimization represents a maximum stiffness (or minimum compliance), maximum buckling load, or maximum first vibration frequency design. With topology optimization, the resulting shape or topology is not known; the number of holes, structural members (for frame structures), etc. are not decided upon, as illustrated in Figure 4.31c.

In topology optimization, we start from a given design domain and proceed to find the optimum distribution of material and voids. In general, a design domain is discretized by using the FEM into discrete mesh. Individual finite elements in the discretized design domain are then evaluated to determine if they are staying or removed from the structure using optimization methods. This results in a so-called 0–1 problem—the elements either exist or do not.

FIGURE 4.31 Different types of structural optimization. (a) Sizing optimization. (b) Domain shape optimization. (c) Topology optimization.

The two main solution strategies for solving the topology optimization problem are the density method and the homogenization method. In this section, we briefly discuss the density method. Those interested in learning the homogenization method are referred to Bendsøe and Sigmund (2003).

4.6.1. Basic Concept and Problem Formulation

In general, the objective of a topology optimization problem is either to minimize the compliance, which is equivalent to maximizing the stiffness, maximize the lowest frequency, or maximize the lowest buckling load. A volume constraint is imposed to limit material usage. The optimization problem can be formulated as

$Minimize : ϕ (c (x))$

(4.323a)

$Subject to : \int_{Ω} c (x) ρ_{0} ⅆ Ω \leq m_{0}$

(4.323b)

$ψ_{i} - ψ_{i}^{u} \leq 0, i = 1, m$

(4.323c)

$0 \leq c (x) \leq 1$

(4.323d)

where ϕ is a structural performance measure to be minimized (or maximized), such as compliance; ψ_i is ith structural performance constraint, such as displacement; m is the total number of constraints; Ω is the structural domain; m₀ is the mass limit; c(x) is the design variable; and ρ₀ is material mass density. In implementation, the design variable c(x) is assigned to individual finite elements; therefore, c(x) = c_i, i = 1, NE (number of elements). In each element, the density becomes ρ_i = c_i ρ₀, and Eq. 4.323b becomes

$\int_{Ω} c (x) ρ_{0} ⅆ Ω = \sum_{i = 1}^{NE} c_{i} ρ_{0} V_{i} \leq m_{0}$

(4.324)

where V_i is the volume (or area for planar problems) of the ith finite element. Also, Eq. 4.323d becomes

$0 \leq c_{i} \leq 1, i = 1, NE$

(4.325)

It is desired that the solution of topology optimization only consists of solid or empty elements—that is, c_i = 0 or 1. One popular method to suppress or penalize intermediate densities is by letting the stiffness of the material be expressed as

$E = c_{i}^{p} E_{0}, i = 1, NE$

(4.326)

where E₀ is the modulus of elasticity of the material and p is a penalization factor that is greater than zero, typically 2 to 5 in implementation. For p > 1, local stiffness for values of c_i < 1 is lowered, as illustrated in Figure 4.32, thus making it “uneconomical” to have intermediate densities in the optimal design. In the literature, the density method together with this penalization is often called the SIMP (solid isotropic microstructures with penalization) method (Rozvany 2000).

Although the density method is straightforward to implement for topology optimization, one must be aware of numerical instabilities. These can manifest as checkerboard, mesh dependence, and local minimum. More detailed discussion and remedies to treat these instabilities can be found in Sigmund and Petersson (1998).

Topology optimization often converges to a design that contains a checkerboard pattern—that is, the alternately solid and void elements, as shown in Figure 4.33b. This is a typical result in topology optimization using finite element methods.

FIGURE 4.32 Relative stiffness as a function of variable c_i with different penalization factors p (Eschenauer and Olhoff 2001).

FIGURE 4.33 Numerical instabilities in topology optimization. (a) Design problem. (b) Example of checkerboards. (c) Solution for 600-element discretization. (d) Solution for 5400-element discretization (Sigmund and Petersson 1998).

The quality of topology optimization result is dependent on the discretization of the finite element model, the so-called mesh-dependence problem. Figure 4.33c shows the topology optimization result for the discretization using 600 elements, and Figure 4.33d shows the result using 5400 elements for the same physical problem. The result in Figure 4.33d is much more detailed than that in Figure 4.33c; however, the two topologies are different in nature.

Different solutions to the same discretized problem are observed when choosing different parameters, such as finite element type (triangular vs quadratic elements), optimization algorithms, convergence criteria, and so forth. These numerical instabilities are all related to whether the continuous constrained optimization problems are well posed or not—that is, whether they have a solution or not.

4.6.2. Two-Dimensional Cantilever Beam Example

A cantilever beam with a fixed left end and a vertical load applied at the midpoint of the free end, as shown in Figure 4.34a, is used to illustrate the density method for topology optimization. The material properties are modulus of elasticity E = 2.07 × 10⁵ psi and Poisson's ratio ν = 0.3.

The finite element model contains 32 × 20 mesh with four-node quad elements. There are 640 elements in the model and the density of each element is selected as the design variable. The design problem is to minimize the compliance with the area constraint of 25% imposed on the domain. Figure 4.34b shows the material distribution using the modified feasible direction after 10 iterations. Figure 4.34c shows the material distribution using the sequential linear programming after 36 iterations. Figure 4.34d shows the material distribution using the sequential quadratic programming after 28 iterations. They converge to slightly different topologies.

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Table of Contents for 4.6. Topology Optimization

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4.5.4.3. Domain velocity computation

4.5.4.3.1. Boundary displacement method

4.5.4.3.2. Isoparametric mapping method

4.5.5. Shape Sensitivity Analysis Using Finite Difference or Semianalytical Method

4.5.6. Material Derivatives

4.5.7. Shape Sensitivity Analysis Using the Continuum Approach

4.5.7.1. Shape sensitivity analysis for a cantilever beam

4.6. Topology Optimization

4.6.1. Basic Concept and Problem Formulation

4.6.2. Two-Dimensional Cantilever Beam Example

Table of Contents for
4.6. Topology Optimization