Nonlinear Topology Optimization Research Articles

Improved fracture resistance of 3D-printed elastoplastic structures with respect to their topology and orientation of deposited layers

3D-printed structures may be characterized by anisotropic fracture behavior because of their layered nature. Depending on the orientation of the sample during the layer deposition, a completely different mechanical response can be obtained, ranging from quasi-brittle to elastoplastic, and with large variations in the maximum stress to failure. In this study, an optimization framework is proposed for 3D-printed samples to maximize their resistance to fracture with respect to both the orientation of the deposited layers during the process and the topology of the sample. To achieve this, a phase-field anisotropic elastoplastic fracture model is combined with a Bidirectional Evolutionary Structural Optimization topology optimization. The model makes it possible to predict the response of the structure until failure with respect to the orientation of the deposited layers in the 3D-printing process and then optimize this orientation to maximize the mechanical response. A large increase in fracture resistance can be obtained by optimizing the orientation, and a significant increase in fracture resistance can be achieved using the present nonlinear anisotropic topology optimization compared with the use of linear topology optimization.

Nonlinear eigenvalue topology optimization for structures with frequency-dependent material properties

Eigenvalue topology optimization problem has been a hot topic in recent years for its wide applications in many engineering areas. In the previous studies, the applied materials are usually assumed as elastic, and the resulting structural eigenfrequencies are obtained by solving a linear eigenvalue problem. However, many engineering materials, such as viscoelastic materials, have frequency-dependent modulus, which results in a more complicated nonlinear eigenvalue problem. This paper presents a systematic study on the nonlinear eigenvalue topology optimization problem with frequency-dependent material properties. The nonlinear eigenvalue problem is solved by a continuous asymptotic numerical method based on the homotopy algorithm and perturbation expansion technique, which involves higher-order differentiation of the nonlinear term and shows a fast convergence. Several schemes are proposed to improve the computational accuracy, applicability, and robustness of the method for the application in topology optimization, including Faà di Bruno's theorem, bisection method, and inverse iteration based eigenvector modification method. Three optimization problems are solved to demonstrate the effectiveness of the developed methods, including the maximization of the fundamental frequency, the eigenfrequency separation interval between two adjacent eigenfrequencies of given orders, and the eigenfrequency separation interval at a given frequency. Numerical examples show the large influence of the frequency-dependent material properties on the optimized results and validate the effectiveness of the developed method.

Eigenvectors-guided topology optimization to control the mode shape and suppress the vibration of the multi-material plate

Although the structural vibration can be suppressed by adjusting eigenvalues outside the excitation frequency band, it becomes increasingly difficult to suppress vibration as the width of the excitation frequency band increases. Besides, for some high-precision equipment, e.g. space telescope mirror substrate and rocket motor casings, their complex and diverse modals directly affect the performance. Thus, a novel optimization algorithm was proposed, which applies to achieve eigenvector-based modal control and vibration suppression. The eigenvectors were defined as the objective function, and Nelson’s method without truncation error was used to calculate the sensitivity information. Due to the introduction of dynamic equations, the non-linearity of the objective function is prominent. Then, an improved solver that can handle this non-linear topology optimization problem was proposed. The optimization was performed under a multi-material framework, and the extended multi-material interpolation scheme was proposed to readily realize the optimization with three or more materials. In addition, the consistent mass matrix constructed by multi-material interpolation was used to describe the mass matrix without concentrated mass in the dynamic equation. Moreover, the modal assurance criterion was used to track jumping modals. Finally, the modal controllability was achieved through several numerical examples, which verify the effectiveness of the proposed method.

Multi-resolution nonlinear topology optimization with enhanced computational efficiency and convergence

Multi-resolution nonlinear topology optimization with enhanced computational efficiency and convergence

Failure of existing structures with semi-brittle mechanical properties on meso scale and reduction of computational cost using non-linear topology optimization

The current research proposes a novel technique to reduce the computational cost and memory usage of numerical modeling of existing structures with semi-brittle mechanical properties on a meso scale. Macro and Meso scales are utilized to investigate the behavior of existing structures. Therefore, rather than modeling the whole numerical sample on meso scale via non-linear topology optimization analysis, certain parts of the numerical model are modeled on meso scale, and the remaining are modeled on the macro scale. Accordingly, without reducing the accuracy of the numerical modeling results, computational complexity and memory use are decreased. In the paper, non-linear Topology optimization is developed using the Drucker-Prager yield criterion and added to the ANSYS finite element software as an extension. Finally, the application of this method is analyzed by proposing some examples.

Elasto-Plastic limit analysis of reliability based geometrically nonlinear bi-directional evolutionary topology optimization

This paper presents elasto-plastic limit analysis of reliability-based geometrically nonlinear topology optimization. For this purpose, by reason of uncertainties the volume fraction is considered randomly during optimization. Thus reliability-based design has been considered for solving the problems. To perform reliability-based topology optimization design, the Monte-Carlo simulation method has been applied to calculate the probability of failure, thus the reliability index. Besides, bi-directional evolutionary structural optimization (BESO) method is used to consider the effect of geometrically nonlinear design for elasto-plastic analysis. Plastic behavior is controlled by applying a bound on the plastic limit load multipliers using limit analysis. The adequacy of the proposed method is exhibited by three 2D benchmark problems. 2D models of L-shape beam and U-shaped plate are considered for reliability-based design and geometrically nonlinear analysis topology optimization in case of elastic material. Additionally, 2D and 3D elasto-plastic material models have been considered to demonstrate that the proposed method can find the optimal topology of elasto-plastic models for reliability-based design and geometrically nonlinear analysis.

An Approach for Geometrically Nonlinear Topology Optimization Using Moving Wide-Bézier Components With Constrained Ends

Abstract Standard moving morphable component (MMC)-based topology optimization methods use free components with explicitly geometrical parameters as design units to obtain the optimal structural topology by moving, deforming, and covering such components. In this study, we intend to present a method for geometrically nonlinear explicit topology optimization using moving wide-Bézier components with constrained ends. Not only can the method efficiently avoid the convergence issues associated with nonlinear structural response analysis, but it can also alleviate the component disconnection issues associated with the standard MMC-based topology optimization methods. The numerical investigations proposed in this work indicate that the proposed method allows us to obtain results in accordance with the current literature with a more stable optimization process. In addition, the proposed method can easily achieve minimum length scale control without adding constraints.

Nonlinear dynamic topology optimization with explicit and smooth geometric outline via moving morphable components method

Nonlinear dynamic topology optimization with explicit and smooth geometric outline via moving morphable components method

Geometrically Nonlinear Topology Optimization of Continuum Structures Based on an Independent Continuous Mapping Method

A geometrically nonlinear topology optimization method for continuum structures is proposed based on the independent continuous mapping method. The stress constraint problem is studied due to the importance of structural strength in engineering applications. First, a topology optimization model is established for a lightweight structure with element stress as constraints. Second, the stress globalization method is adopted to convert local stress constraints into strain energy constraints, which overcomes the difficulties caused by local stress constraints, such as model establishment, sensitivity analysis, and massive solution calculations. Third, the sensitivity of the objective function and constraint function is analyzed, and the method of moving asymptotes is employed to solve the optimization model. In addition, the additive hyperelasticity technique is utilized to solve the numerical instability induced by structures undergoing large deformation. Numerical examples are given to validate the feasibility of the proposed method. The method provides a significant reference for geometrically nonlinear optimization design.

An efficient 137-line MATLAB code for geometrically nonlinear topology optimization using bi-directional evolutionary structural optimization method

Topology optimization, as a powerful conceptual design method, has been widely adopted in both academic research and industrial applications. To further promote the development of topology optimization, many computer programs have been published for educational purposes over the past decades. However, most of the computer programs are constructed based on a linear assumption. On the basis of bi-directional evolutionary structural optimization (BESO) method, the paper presents a MATLAB implementation of the geometrically nonlinear topology optimization code for compliance minimization of statically loaded structures. Excluding 19 lines which are used for explanation, only 118 lines are needed for the initialization of the design parameters, nonlinear finite element analysis, sensitivity calculation, sensitivity filtration, and topological design variables update. Different design problems can be solved by modifying several lines in the proposed program. The complete 137-line code is included as an Appendix and is intended for educational purposes only.

Topology Optimization of Nonlinear Material Structures Based on Parameterized Level Set and Substructure Methods

In order to overcome the problems of complicated calculation process and lower computational efficiency of the traditional level set method (LSM), for nonlinear structure topology optimization, a parameterized level set method (PLSM) was introduced. Through interpolation of the initial level set function with the globally supported radial basis function (GSRBF), a nonlinear material structure topology optimization model was established with the interpolation coefficient as the design variable, the minimum strain energy of the structure as the objective function, and the material amount as the constraint condition. The equilibrium equation for the nonlinear material structure was established by finite element analysis, and solved with the iterative method. In addition, the substructure method (i.e. the domain decomposition method) was used to divide the design area into several sub-areas, and the solution to the full degree of freedom equilibrium equation was decomposed into a set of solutions of reduced equilibrium equations and solutions of multiple substructures’ internal displacements, which could reduce the computation cost. Examples show that, this method can ensure the numerical stability, improve the computational efficiency, and obtain the topology optimization configuration with clear boundaries and reasonable structures.

Linear versus nonlinear reliability-based topology optimization with application to bridge structures

In general, two types of topology optimization models can be found in literature. The first type is called Deterministic Topology Optimization (DTO) leading to a single layout when considering a given design space. The second type is called Reliability-Based Topology Optimization (RBTO) leading to various solutions. In our previous work, two strategies based on the Inverse Optimum Safety Factors (IOSF) were established and applied simply to the normal (Gaussian or linear) distribution law. In this work, linear and nonlinear (normal and lognormal) distribution RBTO cases are compared. A bridge structure is considered a numerical application to perform this comparison where different layouts at the same level of reliability can be found. The numerical results show that regarding the distribution laws, certain output parameters and safety factor values are affected. This change can affect the resulting topology layouts as well as the output parameter such as the compliance. When rising the values of the reliability index, the values of the compliance become larger and the volume values become lesser for the lognormal distribution when comparing to the normal one.

Adaptive Plan Using Sigmoid Function for Nonlinear Topology Optimization

Adaptive Plan Using Sigmoid Function for Nonlinear Topology Optimization

Bi-directional evolutionary stress-based topology optimization of material nonlinear structures

Stress-based topology optimization and nonlinear structural topology optimization is gaining increasing attention in order to make topology optimization more realistic. Thus, this paper extends current concepts of topology optimization to the design of structures made of nonlinear materials. An extended bi-directional evolutionary structural optimization (BESO) method for stress minimization topology optimization of material nonlinear structures is proposed in this work. BESO method based on discrete variables can effectively avoid the well-known singularity problem in density-based methods with low-density elements. The maximum von Mises stress is approximated by the p-norm global stress. The sensitivity information for designing variable updates is derived in detail by adjoint method. As for the highly nonlinear stress behavior, the updated scheme takes advantages from two filters respectively of the sensitivity and topological variables to improve convergence. Moreover, the filtered sensitivity numbers are combined with their historical sensitivity information to further stabilize the optimization process. The effectiveness of the proposed method is demonstrated by several 2D benchmark design problems.

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

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.

An 89-line code for geometrically nonlinear topology optimization written in FreeFEM

Topology optimization has emerged as a powerful tool for structural configuration design. To further promote the development of topology optimization, many computer programs have been published for educational purposes over the past decades. However, most of the computer programs are constructed based on a linear assumption. This paper presents an 89-line code for nonlinear topology optimization written in FreeFEM based on the popular SIMP (solid isotropic material with penalization) method. Excluding thirteen lines which are used for explanation, only 76 lines are needed for the initialization of the design parameters, nonlinear finite element analysis, sensitivity calculation, and updated design variables. Different design problems can be solved by modifying several lines in the proposed program. The complete program is given in the Appendix and is intended for educational purposes only.

Topology optimization of hyperelastic structures using a modified evolutionary topology optimization method

Soft materials are finding widespread implementation in a variety of applications, and it is necessary for the structural design of such soft materials to consider the large nonlinear deformations and hyperelastic material models to accurately predict their mechanical behavior. In this paper, we present an effective modified evolutionary topology optimization (M-ETO) method for the design of hyperelastic structures that undergo large deformations. The proposed M-ETO method is implemented by introducing the projection scheme into the evolutionary topology optimization (ETO) method. This improvement allows nonlinear topology optimization problems to be solved with a relatively big evolution rate, which significantly enhances the robustness. The minimal length scale is achieved as well. Numerical examples show that the proposed M-ETO method can stably obtain a series of optimized structures under different volume fractions with smooth boundaries. Moreover, compared with other smooth boundary methods, another merit of M-ETO is that the problem of the dependency on initial layout can be eliminated naturally due to the inherent characteristic of ETO.

A new geometrically nonlinear topology optimization formulation for controlling maximum displacement

This article presents a novel formulation for geometric nonlinear topology optimization problems. In practical engineering, maximum deflection is frequently used to quantify the stiffness of continuum structures, yet not applied generally as the optimization constraint in geometrically nonlinear topology optimization problems. In this study, the maximum nodal displacement is formulated as a sole constraint. The p-mean aggregation function is adopted to efficiently treat a mass of local displacement constraints imposed on nodes in the user-specified region. The sensitivities of the objective and constraint functions with respect to relative densities are derived. The effect of the aggregate parameter on the final design is further investigated through numerical examples. By comparison with final designs from the traditional formulation, i.e. minimization end compliance with the volume fraction constraint, or minimization of total volume subject to multiple nodal displacement constraints, the optimized results clearly demonstrate the necessity for and efficiency of the present approach.

On the co-rotational method for geometrically nonlinear topology optimization

This paper investigates the application of the co-rotational method to solve geometrically nonlinear topology optimization problems. The main benefit of this approach is that the tangent stiffness matrix is naturally positive definite, which avoids some numerical issues encountered when using other approaches. Three different methods for constructing the tangent stiffness matrix are investigated: a simplified method, where the linear elastic stiffness matrix is simply rotated; the consistent method, where the tangent stiffness is derived by differentiating residual forces by displacements; and a symmetrized method, where the consistent tangent stiffness is approximated by a symmetric matrix. The co-rotational method is implemented for 2D plane quadrilateral elements and 3-node shell elements. Matlab code is given in the appendix to modify an existing, freely available, density-based topology optimization code so it can solve 2D problems with geometric nonlinear analysis using the co-rotational method. The approach is used to solve four benchmark problems from the literature, including optimizing for stiffness, compliant mechanism design, and a plate problem. The solutions are comparable with those obtained with other methods, demonstrating the potential of the co-rotational method as an alternative approach for geometrically nonlinear topology optimization. However, there are differences between the methods in terms of implementation effort, computational cost, final design, and objective value. In summary, schemes involving the symmetrized tangent stiffness did not outperform the other schemes. For problems where the optimal design has relatively small displacements, then the simplified method is suitable. Otherwise, it is recommended to use the consistent method, as it is the most accurate.

Topology optimization using material-field series expansion and Kriging-based algorithm: An effective non-gradient method

Topology optimization is now a very effective and important tool for designing the layouts of various structural and multidisciplinary problems, but most existing methods require information about the sensitivity of the performance function with respect to an enormous number of design variables. This paper presents an efficient non-gradient approach to the topology optimization of structures when no information is available about design sensitivity. Based on the material-field series expansion (MFSE), the problem of topology optimization is constructed as a constrained minimization model with the series expansion coefficients as the design variables, thereby involving a considerable reduction of design variables. The Kriging-based optimization algorithm incorporating two infill criteria is used to solve the optimization problem. A special strategy of (i) using a self-adjusting design domain and (ii) remodeling the surrogate function is proposed to improve the searching efficiency of the Kriging-based algorithm. Several examples are given in the form of linear, nonlinear, and fluid topology optimization problems to demonstrate the effectiveness and applicability of the proposed Kriging-based MFSE method.