An efficient 3D topology optimization code written in Matlab

A topology optimization problem can be defined as a binary programming problem in which the objective is to find the distribution of material in a prescribed area or volume referred to as the design domain. A classical formulation, referred to as the binary compliance problem, is to find the “black and white” layout (i.e., solids and voids) that minimizes the work done by external forces (or compliance) subject to a volume constraint.

The binary compliance problem is known to be ill-posed (Kohn and Strang 1986a, b, c). In particular, it is possible to obtain a non-convergent sequence of feasible black-and-white designs that monotonically reduce the structure’s compliance. As an illustration, assume that a design has one single hole. Then, it is possible to find an improved solution with the same mass and lower compliance when this hole is replaced by two smaller holes. Improved solutions can be successively found by increasing the number of holes and reducing their size. The design will progress towards a chattering design within infinite number of holes of infinitesimal size. That makes the compliance problem unbounded and, therefore, ill-posed.

One alternative to make the compliance problem well-posed is to control the perimeter of the structure (Haber and Jog 1996; Jog 2002). This method effectively avoids chattering configurations, but its implementation is not free of complications. It has been reported that the addition of a perimeter constraint creates fluctuations during the iterative optimization process so internal loops need to be incorporated (Duysinx 1997) Op. cit. (Bendsøe and Sigmund 2003). Also, small variations in the parameters of the algorithm lead to dramatic changes in the final layout (Jog 2002).

Another alternative is to relax the binary condition and include intermediate material densities in the problem formulation. In this way, the chattering configurations become part of the problem statement by assuming a periodically perforated microstructure. The mechanical properties of the material are determined using the homogenization theory. This method is referred to as the homogenization method for topology optimization (Bendsøe 1995; Allaire 2001). The main drawback of this approach is that the optimal microstructure, which is required in the derivation of the relaxed problem, is not always known. This can be alleviated by restricting the method to a subclass of microstructures, possibly suboptimal but fully explicit. This approach, referred to as partial relaxation, has been utilized by many authors including Bendsøe and Kikuchi (1988), Allaire and Kohn (1993), Allaire et al. (2004), and references therein.

An additional problem with the homogenization methods is the manufacturability of the optimized structure. The “gray” areas found in the final designs contain microscopic length-scale holes that are difficult or impossible to fabricate. However, this problem can be mitigated with penalization strategies. One approach is to post-process the partially relaxed optimum and force the intermediate densities to take black or white values (Allaire et al. 1996). This a posteriori procedure results in binary designs, but it is purely numerical and mesh dependent. Other approach is to impose a priori restrictions on the microstructure that implicitly lead to black-and-white designs (Bendsøe 1995). Even though penalization methods have shown to be effective in avoiding or mitigating intermediate densities, they revert the problem back to the original ill-possedness with respect to mesh refinement.

An alternative that avoids the application of homogenization theory is to relax the binary problem using a continuous density value with no microstructure. In this method, referred to as the density-based approach, the material distribution problem is parametrized by the material density distribution. In a discretized design domain, the mechanical properties of the material element, i.e., the stiffness tensor, are determined using a power-law interpolation function between void and solid (Bendsøe 1989; Mlejnek 1992). The power law may implicitly penalize intermediate density values driving the structure towards a black-and-white configuration. This penalization procedure is usually referred to as the Solid Isotropic Material with Penalization (SIMP) method (Zhou and Rozvany 1991). The SIMP method does not solve the problem’s ill-possedness, but it is simpler than other penalization methods.

The SIMP method is based on a heuristic relation between (relative) element density x _i and element Young’s modulus E _i given by where E₀ is the elastic modulus of the solid material and p is the penalization power (p > 1). A modified SIMP approach is given by where E_min is the elastic modulus of the void material, which is non-zero to avoid singularity of the finite element stiffness matrix. The modified SIMP approach, as (2), offers a number of advantages over the classical SIMP formulation, as shown in (1), including the independency between the minimum value of the material’s elastic modulus and the penalization power (Sigmund 2007).

However, topology optimization methods are likely to encounter numerical difficulties such as mesh-dependency, checkerboard patterns, and local minima (Bendsøe and Sigmund 2003). In order to mitigate such issues, researchers have proposed the use of regularization techniques (Sigmund and Peterson 1998). One of the most common approaches is the use of density filters (Bruns and Tortorelli 2001). A basic filter density function is defined as where N _i is the neighborhood of an element x _i with volume v _i , and H _{i j} is a weight factor. The neighborhood is defined as where the operator dist(i, j) is the distance between the center of element i and the center of element j, and R is the size of the neighborhood or filter size. The weight factor H _{i j} may be defined as a function of the distance between neighboring elements, for example where j ∈ N _i . The filtered density \(\tilde {x_{i}}\) defines a modified (physical) density field that is now incorporated in the topology optimization formulation and the SIMP model as The regularized SIMP interpolation formula defined by (6) used in this work.

Following the regularized SIMP method given by (6) and the generalized Hooke’s law, the three-dimensional constitutive matrix for an isotropic element i is interpolated from void to solid as where \(\mathbf {C}^{0}_{i}\) is the constitutive matrix with unit Young’s modulus. The unit constitutive matrix is given by where ν is the Poisson’s ratio of the isotropic material. Using the finite element method, the elastic solid element stiffness matrix is the volume integral of the elements constitutive matrix C _i (x͂ _i ) and the strain–displacement matrix B in the form of where ξ _e (e = 1,…,3) are the natural coordinates as shown in Fig. 1, and the hexahedron coordinates of the corners are shown in Table 1. The strain–displacement matrix B relates the strain 𝜖 and the nodal displacement u, 𝜖 = Bu. Using the SIMP method, the element stiffness matrix is interpolated as where

Replacing values in (11), the 24 × 24 element stiffness matrix \( \mathbf {k}^{0}_{i}\) for an eight-node hexahedral element is where k _m (m = 1,…,6) are 6 × 6 symmetric matrices (see Appendix A). One can also verify that \(\mathbf{k}^{0}_{i}\) is positive definite. The global stiffness matrix K is obtained by the assembly of element-level counterparts k _i , where n is the total number of elements. Using the global versions of the element stiffness matrices K _i and \( \mathbf {k}^{0}_{i}\), (13) is expressed as where \(\mathbf {k}^{0}_{i}\) is a constant matrix. Using the interpolation function defined in (6), one finally observes that

Finally, the nodal displacements vector U(x͂) is the solution of the equilibrium equation where F is the vector of nodal forces and it is independent of the physical densities x͂. For brevity of notation, we omitted the dependence of physical densities x͂ on the design variables x, x͂ = x͂(x).

Consider the discretized prismatic structure in Fig. 2 composed of eight eight-noded cubic elements. The nodes identified with a number (node ID) ordered column-wise up-to-bottom, left-to-right, and back-to-front. The position of each node is defined with respect to Cartesian coordinate system with origin at the left-bottom-back corner.

Within each element, the eight nodes N ₁,…, N ₈ are ordered in counter-clockwise direction as shown in Fig. 3. Note that the “local” node number (N _i ) does not follow the same rule as the “global” node ID (NID _i ) system in Fig. 2. Given the size of the volume (nelx × nely × nelz) and the global coordinates of node N ₁ (x ₁, y ₁, z ₁), one can identify the global node coordinates and node IDs of the other seven nodes in that element by the mapping the relationships as summarized in Table 2.

Each node in the structure has three degrees of freedom (DOFs) corresponding to linear displacements in x-y-z directions (one element has 24 DOFs). The degrees of freedom are organized in the nodal displacement vector U as where n is the number of elements in the structure. The location of the DOFs in U, and consequently K and F, can be determined from the node ID as shown in Table 2.

The node IDs for each element are organized in a connectivity matrix edofMat with following Matlab lines: where nele is the total number of elements, nodegrd contains the node ID of the first grid of nodes in the x-y plane (for z = 0), the column vector edofVec contains the node IDs of the first node at each element, and the connectivity matrix edofMat of size nele × 24 containing the node IDs for each element. For the volume in Fig. 2, nelx=4, nely=1, and nelz=2, which results in

The element connectivity matrix edofMat is used to assemble the global stiffness matrix K as follows:

The element stiffness matrix KE (size 24 × 24) is obtained from the lk_H8 subroutine (lines 99-146). Matrices iK (size 24 nele × 24) and jK (size nele × 24²), reshaped as column vectors, contain the rows and columns identifying the 24 × 24 × nele DOFs in the structure. The three-dimensional array xPhys (size nely × nelx × nelz) corresponds to the physical densities. The matrix sK (size 24² × nele) contains all element stiffness matrices. The assembly procedure of the (sparse symmetric) global stiffness matrix K (line 71) avoids the use of nested for loops.

Finally, the nodal displacement vector U is obtained from the solution of the equilibrium equation (16) by pre-multiplying the inverse of the stiffness matrix K and the vector of nodal forces F, where the indices freedofs indicate the unconstrained DOFs. For the cantilevered structure in Fig. 2, the constrained DOFs where jf, and kf are the coordinate of the fixed nodes, fixednid are the node IDs, and fixeddof are the location of the DOFs. The free DOFs, are then defined as where ndof is the total number of DOFs. By default, the code constraints the left face of the prismatic structure and assigns a vertical load to the structure’s free-lower edge as depicted in Fig. 2. The user can define different load and support DOFs by changing the corresponding node coordinates (lines 12 and 16). Several examples are presented in Section 6.

The objective of the minimum compliance problem is to find the material density distribution \(\tilde {\mathrm {x}}\) that minimizes the structure’s deformation under the prescribed support and loading condition. The structure’s compliance, which provides a global measure of deformation, is defined as where F is the vector of nodal forces and U (x͂) is the vector of nodal displacements. Incorporating a volume constraint, the minimum compliance optimization problem is where the physical densities x͂ = x͂(x͂) are defined by (3), n is the number of elements used to discretize the design domain, v = [v ₁,…,v _n ]^T is a vector of element volume, and \(\bar {v}\) is the prescribed volume limit of the design domain. The nodal force vector F is independent of the design variables and the nodal displacement vector U(x͂) is the solution of K(x͂)U(x͂) = F.

The derivative of the volume constraint v(x͂) in (18) with respect to the design variable x _e is given where and The code uses a mesh with equally sized cubic elements of unit volume, then v _i =v _j =v _e =1.

The derivative of the compliance is where \(\partial \tilde {x_{i}} / \partial x_{e}\) is given by (21) and

The derivative of (16) with respect to \(\tilde {\mathrm {x}}_{i}\) is which yields Using (15), Using (25) and (26), (23) results in Since \( \mathbf {k}^{0}_{i}\) is the global version of an element matrix, (27) may be transformed from the global level to the element level, obtaining where u _i is the element vector of nodal displacements. Since \(\mathbf {k}^{0}_{i}\) is positive definite, ∂c(x͂)/∂x͂ _i < 0.

The numerical implementation of minimum compliance problem can be done as the following:

The objective function in (18) is calculated in Line 75. The sensitivities of the objective function and volume fraction constraint with respect to the physical density are given be lines 76-77. Finally, the chain rule as stated in (22) is deployed in lines 79-80.

A compliant mechanism is a morphing structure that undergoes elastic deformation to transform force, displacement, or energy (Bruns and Tortorelli 2001). A typical goal for a compliant mechanism design is to maximize the output port displacement. The optimization problem is where L is a unit length vector with zeros at all degrees of freedom except at the output point where it is one, and U (x͂) = K (x͂) ⁻¹ F.

To obtain the sensitivity of the new cost function c(x͂) in (29), let us define a global adjoint vector U _d(x͂) from the solution of the adjoint problem Using (30) in (29), the objective function can be expressed as

The derivative of c(x͂) with respect to the design variable x _e is again obtained by the chain rule,

where \({\partial \tilde x_{i}}/{\partial x_{e}}\) is described by (21), and ∂c(x͂)/∂x͂ _i can be obtained using direct differentiation. The use of the interpolation given by (6) yields an expression similar to the one obtained in (28), where \( \mathbf u_{di}({\mathbf {k}}_{i}^{0})\) is the part of the adjoint vector associated with element i. In this case, \({\partial c({\mathbf {k}}_{i}^{0})}/{\partial \tilde {\mathrm {x}}_{i}} \) may be positive or negative.

The numerical implementation of the objective function (31) and sensitivity (32) are

Vector Ud (Line 74a) is the dummy load displacement field and vector U (line 74b) is the input load displacement. The codes for the implementation of chain rule are not shown above since they are same as lines 79-80.

Heat in physics is defined as energy transferred between a system and its surrounding. The direct microscopic exchange of kinetic energy of particles through the boundary between two systems is called diffusion or heat conduction. When a body is at a different temperature from its surrounding, heat flows so that the body and the surroundings reach the same temperature. This condition is known as thermal equilibrium. The equilibrium condition for heat transfer in finite element formulation is described by where \( \mathbf U(\mathbf {k}_{i}^{0})\) now donates the finite element global nodal temperature vector, F donates the global thermal load vector, and \(\mathbf K({\mathbf {k}}_{i}^{0}) \) donates the global thermal conductivity matrix. For a material with isotropic properties, conductivity is the same in all directions.

The optimization problem for heat conduction is where U(x͂) = K (x͂)⁻¹ F, and K(x͂) is obtained by the assembly of element thermal conductivity matrices \(\mathbf {k}_{i}(\tilde {\mathrm {x}}_{i})\). Following the interpolation function in (6), the element conductivity matrix is expressed as where k _min and k ₀ represent the limits of the material’s thermal conductivity coefficient and \(\mathbf {k}_{i}^{0}\) donates the element conductivity matrix. Note that (34) may be considered as the distribution of two material phases: a good thermal conduction (k ₀) and the other a poor conductor (k _min).

The sensitivity analysis of the cost function in (33) is given by

where \({\partial \tilde x_{i}}/{\partial x_{e}}\) is described by (21) and

The numerical implementation only requires an optional change in the material property name: where k0 and kmin are the limits of the material’s thermal conductivity. The chain rule is applied same as before.

A QP problem has a quadratic objective function and linear constraints (Nocedal and Wright 2006). Given the current design x ^(k) and all corresponding active constraints, the QP approximation of the minimum compliance problem in (18) can be expressed as where c ^(k) is the value of the objective function evaluated at x ^(k) and d = x − x ^(k), and A is the matrix of active constraints. The optimality and feasibility conditions of (36) yield This SQP approach, referred to as the active set algorithm (Nocedal and Wright 2006), allows to determine a step \(\tilde {d}^{(k)}\) from the solution of the of system of linear equations in (37) expressed as The updated design is given by where the step size parameter α ^(k) is determined by a line search procedure. The Hessian ∇² c ^(k) can be numerically approximated but, for the problems considered in this paper, one can determined the closed-form expression (see Appendix B), which is given by This line-search, active-set SQP algorithm is summarized in Algorithm 1.

The implementation of (40) in the program can be done in just two lines, since the term \(\mathbf {u}_{i}^{\text {T}}+\mathbf {k}_i^0 +\mathbf u_{i}\) has already been calculated, namely matrix ce in line 74:

Finally, Matlab has built-in constrained NLP solver fmincon. The implementation of using fmincon as an optimizer in our top3d program is quite easy, but need some reconstructions of the program (one needs to divide program into different subfunctions, e.g., objective function, constraint function, Hessian function). To further assist on the implementation of an SQP strategy, the reader can find a step-by-step tutorial on our website http://​top3dapp.​com.

The MMA algorithm (Svanberg 1987) was proposed to adjust the curvature of the convex linearization (CONLIN) method introduced by Fleury (1989). Give the current design x ^(k) the MMA approximation of the minimum compliance problem in (18) yields to the following linear programming problem: where The lower and upper asymptotes \(L_{i}^{\left (k\right )}\) and \(U_{i}^{\left (k\right )}\) are iteratively updated to mitigate oscillation or improve convergence rate. The heuristic rule proposed by Svanberg (1987) is as follows: For k = 1 and k = 2, For \(k\geqslant 3\), where Note from (45) that the signs of three successive iterations are stored. If the signs are opposite, meaning x _i oscillates, the two asymptotes are brought closer to \(x_{i}^{\left (k\right )}\) to have a more conservative MMA approximation. On the other hand, if the signs are same, the two asymptotes are extended away from \(x_{i}^{\left (k\right )}\) in order to speed up the convergence. The MMA algorithm is explained in Algorithm 2.

The MMA algorithm is available for Matlab (mmasub). The reader may obtain a copy by contacting Prof. Krister Svanberg (http://​www.​math.​kth.​se/​%7Ekrille/​Welcome.​html) from KTH in Stockholm Sweden. Although mmasub has total of 29 input and output variables, its implementation for top3d is straightforward. The details can be found at http://​top3dapp.​com.

A classical approach to structural optimization problems is the Optimality Criteria (OC) method. The OC method is historically older than sequential approximation methods such as Sequential Linear Programming (SLP) or SQP. The OC method is formulated on the grounds that if constraint 0 ⩽ x ⩽ 1 is inactive, then convergence is achieved when the KKT condition is satisfied for k = 1,…, n, where λ is the Lagrange multiplier associated with the constraint v(x͂). This optimality condition can be expressed as B _e =1, where

The code implements the OC updating scheme proposed by (Bendsøe 1995) to update design variables: where m is a positive move-limit, and η is a numerical damping coefficient. The choice of m = 0.2 and η = 0.5 is recommended for minimum compliance problems (Bendsøe 1995; Sigmund 2001). For compliant mechanisms, η = 0.3 improves the convergence of the algorithm. The only unknown in (48) is the value of the Lagrange multiplier λ, which satisfies that Numerically, λ is found by a root-finding algorithm such as the bisection method. Finally, the termination criteria are satisfied when a maximum number of iterations is reached or where the tolerance 𝜖 is a relatively small value, for example 𝜖 = 0.01.

The numerical implementation begins with the initialization of design and physical variables, where volfrac represents the volume fraction limit. Initially, the physical densities are assigned a constant uniform value, which is iteratively updated following the OC updating scheme (Algorithm 3).

By default, the code solves a minimum compliance problem for the cantilevered beam in Fig. 4. The prismatic design domain is fully constrained in one end and a unit distributed vertical load is applied downwards on the lower free edge. Figure 4 shows the topology optimization results for solving minimum compliance problem with the following Matlab input lines:

A compliant mechanism problem involves loading cases: input loading case and dummy loading case. The code also needs to implement a new objective function and its corresponding sensitivity analysis. To demonstrate this implementation, let us consider a three-dimensional force inverter problem as shown in Fig. 9. With an input load defined in the positive direction, the design goal is to maximize the negative horizontal output displacement. Both the top face and the side force are imposed with symmetric constraints; i.e., nodes can only move within the plane.

The new loading conditions as well as input and output points are defined as follows:

and the boundary conditions are defined as below:

The external springs with stiffness 0.1 are added at input and output points after line 71.

The expressions of the objective function (31) and sensitivity (32) are modified in lines 74-76.

The convergence criteria for the bi-sectioning algorithm (lines 82-83) is improved by the following lines:

To improve the convergence stability, the damping factor of OC-method changes from 0.5 to 0.3 and also takes the positive sensitivities into account, then line 85 is changed to:

The final design shown in Fig. 10 is promoted by the line in the Matlab:

The implementation of heat conduction problems is not more complex than the one for compliant mechanism synthesis since the number of DOF per node is one rather than three. Following the implementation of heat conduction problems in two dimensions (Bendsøe and Sigmund 2003), the implementation for three dimension problems is suggested in the following steps.

First, the elastic material properties (lines 8-10) are changed to the thermal conductivities of materials Furthermore, the boundary conditions for the heat condition problem, i.e., a rectangular plate with a heat sink on the middle of top face and all nodes are given a thermal load as shown in Fig. 11, are changed corresponding (lines 10-18).

Also, since there is only one DOF per node in heat condition problems, some variables need to change correspondingly, such as ndof, edofMat.

Change the total number of DOFs and the force vector in lines 21–22

The element conductivity matrix is called in line 25 by

and it is defined in lines 99–145

The finite element connectivity matrix edofMat is changed in lines 30–35

The global conductivity matrix is assembled in a different way, hence line 70 need to change as

The evulation of the objective function and analysis of the sensitivity are given in lines 75–76

The optimized topology is derived as shown in Fig. 12 by promoting the following line in the Matlab: