Topology optimization considering the Drucker–Prager criterion with a surrogate nonlinear elastic constitutive model

Abstract

We address material nonlinear topology optimization problems considering the Drucker–Prager strength criterion by means of a surrogate nonlinear elastic model. The nonlinear material model is based on a generalized J₂ deformation theory of plasticity. From an algorithmic viewpoint, we consider the topology optimization problem subjected to prescribed energy, which leads to robust convergence in nonlinear problems. The objective function of the optimization problem consists of maximizing the strain energy of the system in equilibrium subjected to a volume constraint. The sensitivity analysis is quite effective and efficient in the sense that there is no extra adjoint equation. In addition, the nonlinear structural equilibrium problem is solved through direct minimization of the structural strain energy using Newton’s method with an inexact line search strategy. Four numerical examples demonstrate features of the proposed nonlinear topology optimization framework considering the Drucker–Prager strength criterion.

Appendices

Appendix A. Illustration of the energy control approach

Let us consider a ground structure-based elastic formulation (Bendsøe and Sigmund 2003; Christensen and Klarbring 2009; Sanders et al. 2017), in which (43) can be written as:

$$ {\displaystyle \begin{array}{c}\underset{\boldsymbol{x}}{\max}\kern0.50em U\left(\boldsymbol{x},\boldsymbol{u}\left(\boldsymbol{x}\right)\right)\\ {}\mathrm{s}.\mathrm{t}.\begin{array}{c}\sum \limits_{e=1}^n{L}_e^T{x}_e-{V}_{\mathrm{max}}\le 0\\ {}{x}_e^{\mathrm{min}}\le {x}_e\le {x}_e^{\mathrm{max}},e=1,\dots, n\end{array}\\ {}\mathrm{with}\left\{\begin{array}{c}\boldsymbol{T}\left(\boldsymbol{x},\boldsymbol{u}\left(\boldsymbol{x}\right)\right)=\chi \left(\boldsymbol{x},\boldsymbol{u}\left(\boldsymbol{x}\right)\right){\boldsymbol{f}}_0\\ {}{\boldsymbol{f}}_0^T\boldsymbol{u}\left(\boldsymbol{x}\right)=2{C}_0\end{array}\right.\end{array}} $$

(56)

The vector x is a vector of design variables, with component x_e being the cross-sectional area of truss member e—it is subjected to lower bound ${x}_e^{\mathrm{min}}$ and upper bound ${x}_e^{\mathrm{max}}$. In addition, n is the number of truss members in the ground structure, L_e is the length of truss member e, V_max is the upper bound on the total volume, and u(x) is the displacement vector. For illustrative purpose, we assume the particular case of linear elasticity. In the following, we solve a simple three-bar truss example to explain how to estimate a proper value for the prescribed energy C₀ and demonstrate that C₀ remains constant at each design iteration during the entire optimization process.

Fig. 18

The three-bar truss example. a Design domain and boundary conditions. b, c, and d are the topologies at optimization iteration #1, #35, and #72, respectively. Blue bars are in tension and red bars are in compression

The three-bar example shown in Fig. 18(a) is made of a linear elastic material with the Young’s modulus E = 200 GPa. The structure has two degree of freedoms (dofs), and two reference forces are applied at each dof, respectively. The magnitudes of the two reference forces are f₀₁ = 40 N and f₀₂ = 80 N. The displacements at each of the two dofs are u₁ and u₂.

Let us assume that the designer suggests that the magnitudes of initial displacements at each of dofs are the same, i.e., u₁ = u₂ = 10⁻⁵ m. Based on this assumption, we estimate C₀ as:

$$ {C}_0={C_0}_1+{C_0}_2=\frac{1}{2}{f_0}_1{u}_1+\frac{1}{2}{f_0}_2{u}_2=6\times {10}^{-4}\ \mathrm{N}\cdotp \mathrm{m} $$

(57)

This three-bar optimization problem converges with 72 iterations. As an example, we plot the topologies at optimization iteration #1, #35, and #72 in Fig. 18(b), (c), and (d), respectively. Table 11 shows the prescribed energy C₀ at each optimization design iteration, which is composed of the energy C₀₁ and C₀₂ at each dof, respectively. As expected, C₀ remains a constant in each optimization iteration. The data in Table 11 can be visualized in Fig. 19(a) and (b), which illustrate the energy control approach during the optimization process.

Table 11 Calculation of the prescribed energy at each optimization iteration

Fig. 19

Displacement versus reaction force diagrams at (a) dof #1 and (b) dof #2, respectively. The shaded area represents prescribed energy at optimization iteration #0 (yellow), #1 (green), #35 (red), and #72 (blue), i.e., optimal design

Appendix B. Estimating the limit value of the prescribed energy ${\boldsymbol{C}}_{\mathbf{0}}$

Here, we provide a rational approach to estimate the limit value of the prescribed energy C₀ for a design optimization problem given a fixed volume constraint. This approach includes two phases as follows:

• Phase #1: Calculate an initial guess of the ${C}_0^0$ based on an approximated displacement vector u

$$ {C}_0^0=\frac{1}{2}{\boldsymbol{f}}_0^T\boldsymbol{u} $$

(58)

• Phase #2: Estimate the limit value of the prescribed energy ${\left({C}_0^i\right)}_{\mathrm{lim}}$ for the given optimized topology corresponding to the prescribed energy ${C}_0^i$.

  • At step i (i = 0, 1, 2…), perform FEM analysis of the optimized topology obtained with ${C}_0^i$, and plot the curve that represents the relationship between the prescribed energy C₀ and the reaction load factor χ.

  • Select two points on the curve (i.e., C₀ versus χ). One of the two points is $\left[{C}_0^i,\kern0.5em {\chi}^i\right]$, and the other point $\left[{\left({C}_0^i\right)}_k,\kern0.5em {\left({\chi}^i\right)}_k\right]$ is obtained iteratively such that:

$$ \frac{{\left({\chi}^i\right)}_k-{\left({\chi}^i\right)}_{k-1}}{{\left({C}_0^i\right)}_k-{\left({C}_0^i\right)}_{k-1}}\le \beta {K}_0 $$

(59)

where K₀ is the slope of the curve as C₀ is close to zero, and β is a small ratio (e.g., 4 × 10⁻² as appropriate).

• Based on the two selected points $\left[{C}_0^i,\kern0.5em {\chi}^i\right]$ and $\left[{\left({C}_0^i\right)}_k,\kern0.5em {\left({\chi}^i\right)}_k\right]$, build an asymptotic function with the functional format as follows:

$$ \chi \left({C}_0\right)=\frac{a{C}_0}{b+{C}_0} $$

(60)

• Calculate the limit value of the prescribed energy ${\left({C}_0^i\right)}_{\mathrm{lim}}$ at the current step i as follows:

$$ {\left({C}_0^i\right)}_{\mathrm{lim}}=b\ \left(\frac{1}{\sqrt{\alpha }}-1\right) $$

(61)

where α is a small ratio (e.g., 2 × 10⁻² as appropriate).

• Proceed to step i + 1. Stop, if the following criterion is satisfied

$$ \frac{\left|{\left({C}_0^{i+1}\right)}_{\mathrm{lim}}-{\left({C}_0^i\right)}_{\mathrm{lim}}\right|}{{\left({C}_0^i\right)}_{\mathrm{lim}}}\le 5\% $$

(62)

• Then, ${\left({C}_0^{i+1}\right)}_{\mathrm{lim}}$ is the estimated limit value of the prescribed energy for the given a fixed volume constraint.

For example, we investigate the limit value of the prescribed energy for the clamped design optimization problem in Section 4.2 using the approach mentioned above. In Fig. 20(a), the red curve represents the structural response of the optimized topology obtained with ${C}_0^0=0.013\ \mathrm{MJ}$ (see Fig. 20(b)), and the black curve is an asymptotic approximation based on (60). From (61), we can obtain the limit prescribed energy for this topology (Fig. 20(b)) as ${\left({C}_0^0\right)}_{\mathrm{lim}}=0.05\ \mathrm{MJ}$. With this value of the prescribed energy, the corresponding optimized topology is shown in Fig. 20(c). Similarly, we estimate the limit value of the prescribed energy for this topology as ${\left({C}_0^1\right)}_{\mathrm{lim}}=0.13\ \mathrm{MJ}$. We repeat the procedure until (62) is satisfied, and then, the limit value of the prescribed energy for this clamped problem is obtained as ${\left({C}_0^2\right)}_{\mathrm{lim}}=0.134\ \mathrm{MJ}$. As the prescribed energy increases, the corresponding topologies shown in Fig. 20(b), (c), and (d) are not changing. Instead, we note that the shape of those topologies is different.

Fig. 20

(a) Structural response and its approximation for the optimized topology in (b) with ${C}_0^0=0.013\ \mathrm{MJ}$. (c) Optimized topology with ${C}_0^1=0.05\ \mathrm{MJ}$. (d) Optimized topology with ${C}_0^2=0.13\ \mathrm{MJ}$

Appendix C. Relationship between the increment of principal stress on the strength surface and the increment of principal strain

A reference principal strain tensor of the deformation at a material point can be written as:

$$ \left[{\varepsilon}^{\mathrm{ref}}\right]=\left[\begin{array}{ccc}{\varepsilon}_1^{\mathrm{ref}}& 0& 0\\ {}0& {\varepsilon}_2^{\mathrm{ref}}& 0\\ {}0& 0& {\varepsilon}_3^{\mathrm{ref}}\end{array}\right] $$

(63)

and the principal strain tensor controlled by a positive scaling factor ξ is denoted by:

$$ \left[\varepsilon \right]=\xi \left[{\varepsilon}^{\mathrm{ref}}\right] $$

(64)

Then, the first invariant of the principal strain tensor is:

$$ {J}_1=\xi {J}_1^{\mathrm{ref}} $$

(65)

By making reference to (1), we can calculate the principal stress components as:

$$ {\sigma}_i=\xi \left(\lambda\ {J}_1^{\mathrm{ref}}+2\ \mu\ {\varepsilon}_i^{\mathrm{ref}}\right),\kern0.5em i=1,2,3 $$

(66)

where λ and μ can be obtained from (4), (9), and (10) considering the Drucker–Prager criterion.

Next, by taking the derivatives of the principal stresses components with respect to the scaling factor ξ, we obtain that:

$$ \frac{d{\sigma}_i}{d\xi}=\left(\lambda +\xi \frac{d\lambda}{d\xi}\right){J}_1^{\mathrm{ref}}+2\ \left(\mu +\xi \frac{d\mu}{d\xi}\right){\varepsilon}_i^{\mathrm{ref}} $$

(67)

We then check that the terms in parentheses in (67) are independent of the scaling factor ξ, i.e.,

$$ \lambda +\xi \frac{d\lambda}{d\xi}=\frac{\left(1-6{\beta}^2\right){J}_1^{\mathrm{ref}}\sqrt{J_2^{\mathrm{ref}}}-\beta \left(6{J}_2^{\mathrm{ref}}-{J_1^{\mathrm{ref}}}^2\right)}{3{J}_1^{\mathrm{ref}}\sqrt{J_2^{\mathrm{ref}}}\left[6{\beta}^2\left(1+\upsilon \right)+1-2\upsilon \right]}E $$

(68)

and

$$ \mu +\xi \frac{d\mu}{d\xi}=\frac{6\upbeta \sqrt{J_2^{\mathrm{ref}}}-{J}_1^{\mathrm{ref}}}{2\sqrt{J_2^{\mathrm{ref}}}\left[6{\beta}^2\left(1+\upsilon \right)+1-2\upsilon \right]}\beta E $$

(69)

Therefore, we conclude that the increment of principal stress on the Drucker–Prager strength surface is constant for each reference strain tensor with respect to the increment of the principal strain.

Appendix D. Solving the nonlinear state equations: Newton’s method with line search

We solve (44) using Newton’s method with a backtracking line search strategy. We start with the Lagrangian function.

$$ \mathcal{L}\left(\boldsymbol{u},\chi \right)=U\left(\boldsymbol{u}\right)+\overset{\sim }{\chi}\left(2{C}_0-{\boldsymbol{f}}_0^T\boldsymbol{u}\ \right) $$

(70)

where $\overset{\sim }{\chi }$ is the Lagrangian multiplier, which is also the reaction load factor in (43). According to the KKT optimality conditions, we readily obtain:

$$ \left\{\begin{array}{c}\frac{\partial \mathcal{L}}{\partial \boldsymbol{u}}\left({\boldsymbol{u}}^{\ast },\backslash {\mathrm{chi}}^{\ast}\right)=\mathbf{\nabla}U\left({\boldsymbol{u}}^{\ast}\right)-\backslash {\mathrm{chi}}^{\ast }{\boldsymbol{f}}_0=0\\ {}\frac{\partial \mathcal{L}}{\partial \chi}\left({\boldsymbol{u}}^{\ast },\backslash {\mathrm{chi}}^{\ast}\right)=2{C}_0-{\boldsymbol{f}}_0^T{\boldsymbol{u}}^{\ast }=0\end{array}\right. $$

(71)

At iteration k, we interpret the Newton step ∆u_k, and the associated multiplier χ_k + 1, as the solutions of a linearized approximation of the optimality conditions in (71). We substitute u_k + ∆u_k for u^∗ and χ_k + 1 for χ^∗, and replace the gradient by its linearized approximation near u_k, to obtain the equations:

$$ \left\{\begin{array}{c}\mathbf{\nabla}U\left({\boldsymbol{u}}_k+\Delta {\boldsymbol{u}}_k\right)-{\chi}_{k+1}{\boldsymbol{f}}_0\approx \mathbf{\nabla}U\left({\boldsymbol{u}}_k\right)+\mathbf{\nabla}{\mathbf{\nabla}}^TU\left({\boldsymbol{u}}_k\right)\Delta {\boldsymbol{u}}_k-{\chi}_{k+1}{\boldsymbol{f}}_0=0\\ {}2{C}_0-{\boldsymbol{f}}_0^T\ \left({\boldsymbol{u}}_k+\Delta {\boldsymbol{u}}_k\right)=0\end{array}\right. $$

(72)

Since ∇∇^TU(u_k) = K(u_k) and ∇U(u_k) = T(u_k), then, (72) becomes:

$$ \left\{\begin{array}{c}\boldsymbol{T}\left({\boldsymbol{u}}_k\right)+\boldsymbol{K}\left({\boldsymbol{u}}_k\right)\Delta {\boldsymbol{u}}_k-{\chi}_{k+1}{\boldsymbol{f}}_0=0\\ {}2{C}_0-{\boldsymbol{f}}_0^T\ \left({\boldsymbol{u}}_k+\Delta {\boldsymbol{u}}_k\right)=0\end{array}\right. $$

(73)

Solving for ∆u_k using the first equation in the system (73), we obtain:

$$ \Delta {\boldsymbol{u}}_k={\boldsymbol{K}}^{-1}\left({\boldsymbol{u}}_k\right)\left[-\boldsymbol{T}\left({\boldsymbol{u}}_k\right)+{\chi}_{k+1}{\boldsymbol{f}}_0\right] $$

(74)

By means of the equality ${\boldsymbol{f}}_0^T{\boldsymbol{u}}_k=2{C}_0$, the second equation in the system (73) becomes:

$$ {\boldsymbol{f}}_0^T\ \Delta {\boldsymbol{u}}_k=0 $$

(75)

Substituting (74) into (75), and solving for χ_k + 1, we obtain:

$$ {\chi}_{k+1}=\frac{{\boldsymbol{f}}_0^T\Delta {\boldsymbol{u}}_k^{\prime }}{{\mathbf{f}}_0^T\Delta {\boldsymbol{u}}_k^{\prime \prime }}\kern0.5em ,\Delta {\boldsymbol{u}}_k^{\prime }={\boldsymbol{K}}^{-1}\left({\boldsymbol{u}}_k\right)\boldsymbol{T}\left({\boldsymbol{u}}_k\right)\kern0.5em ,\Delta {\boldsymbol{u}}_k^{\prime \prime }={\boldsymbol{K}}^{-1}\left({\boldsymbol{u}}_k\right){\boldsymbol{f}}_0 $$

(76)

By substituting the expression of χ_k + 1 in (76) into (74), we finally obtain the expression of the Newton step ∆u_k as:

$$ \Delta {\boldsymbol{u}}_k=-\Delta {\boldsymbol{u}}_k^{\prime }+{\chi}_{k+1}\Delta {\boldsymbol{u}}_k^{\prime \prime } $$

(77)

For the sake of completeness, the detailed algorithm for Newton’s method, as employed in the present work, is provided in Table 12. The stiffness matrix might become singular near the limit state which can cause numerical difficulties. To prevent the possibility of a singular stiffness matrix, we add a Tikhonov regularization (Tikhonov and Arsenin 1977; Ramos Jr and Paulino 2016) parameter t_TK into the tangent stiffness matrix as shown in lines 5 and 6 of Table 12. Through the testing of the numerical examples, we verify that the Tikhonov regularization technique is effective.

Table 12 Newton’s algorithm for solving nonlinear state equations

Appendix E. ABAQUS® UMAT subroutine (ESM)

The ABAQUS® user subroutine UMAT for the surrogate nonlinear elastic constitutive model considering the Drucker–Prager criterion is provided as ESM (Electronic Supplementary Material). A representative example of the supplementary material is presented here. For the Abaqus/CAE usage, please follow this sequence: Property module → Material Editor → General → User Material → Mechanical Constants → Input the user-defined material properties in the sequence of Young’s modulus, Poisson’s ratio, plastic Poisson’s ratio, and uniaxial strength stress. Run analysis with the present UMAT subroutine: Analysis → Edit job → General → User subroutine file.

Appendix F. Nomenclature

σ stress tensor

ɛ strain tensor

s deviatoric stress tensor

ɛ_d deviatoric strain tensor

I second-order identity tensor

σ_i principal stress components

ε_i principal strain components

λ, μ Lame’s parameters

ϕ₁, ϕ₂ functions representing the hardening behavior of the material

J₁(ɛ) first invariant of the strain tensor

J₂(ɛ_d) second invariant of the deviatoric strain tensor

J₁(σ) first invariant of the stress tensor

J₂(s) second invariant of the deviatoric stress tensor

σ^L linear elastic limit of the stress tensor

ɛ^L linear elastic limit of the strain tensor

σ^N nonlinear elastic limit of the stress tensor

ɛ^N nonlinear elastic limit of the strain tensor

E Young’s modulus

υ Poisson’s ratio

υ^p plastic Poisson’s ratio

σ_y uniaxial strength stress

β, k_spositive constants for an elastic-perfectly-plastic material

φ strain energy density

φ^L linear elastic strain energy density

c Cohesion

ψ friction angle

η, ζ parameters defined based on the approximation to the Mohr–Coulomb criterion

σ_c material compressive strength

σ_t material tensile strength

U structural strain energy

C₀ prescribed energy

f₀ vector of given applied forces

u nodal displacement vector

ρ vector of element density variables

p constant penalization parameter

Vol Fracvolume fraction

R linear density filter radius

n number of elements discretizing the design domain

v_e volume of element e

V_max maximum material volume

T internal force vector

χ reaction load factor

$\overset{\sim }{\chi }$Lagrangian multiplier

K_T tangent stiffness matrix

∆u Newton step

α step size by backtracking line search

J objective function

$\mathcal{L}$Lagrangian function

w magnitude of distributed load

x vector of the cross-sectional area for truss members

L_e length of truss member e

κ a factor used in the inexact line search approach