Topology Optimization Formulation Research Articles

Topology optimization with a finite strain nonlocal damage model using the continuous adjoint method

This study presents a unified formulation of topology optimization with a finite strain nonlocal damage model using the continuous adjoint method. For the primal problem to describe the material response including deterioration, we consider the standard Neo–Hookean constitutive model and incorporate crack phase-field theory for brittle fracture within the finite strain framework. For the optimization problem, the objective function is set to accommodate multiple objectives by weighting each sub-function, and the continuous adjoint method is employed to derive the sensitivity. Thus, both the governing equations for primal and adjoint problems are written as strong forms and hold at any moment and at any location in the continuum body or on its boundary. Accordingly, the proposed formulation is independent of the requirements from numerical implementation, such as element type or discretization method. In addition, the reaction–diffusion equation is used to update the design variable in an optimizing process, by which the continuous distribution of the design variable, as well as material properties, are realized. After the basic performance of the proposed formulation is demonstrated with a simple numerical setup, two-material (matrix and inclusion materials) and single-material (material and null) topology optimizations are presented, and discussions are made.

Anatomically and mechanically conforming patient-specific spinal fusion cages designed by full-scale topology optimization

Cage subsidence after instrumented lumbar spinal fusion surgery remains a significant cause of treatment failure, specifically for posterior or transforaminal lumbar interbody fusion. Recent advancements in computational techniques and additive manufacturing, have enabled the development of patient-specific implants and implant optimization to specific functional targets. This study aimed to introduce a novel full-scale topology optimization formulation that takes the structural response of the adjacent bone structures into account in the optimization process. The formulation includes maximum and minimum principal strain constraints that lower strain concentrations in the adjacent vertebrae. This optimization approach resulted in anatomically and mechanically conforming spinal fusion cages. Subsidence risk was quantified in a commercial finite element solver for off-the-shelf, anatomically conforming and the optimized cages, in two representative patients. We demonstrated that the anatomically and mechanically conforming cages reduced subsidence risk by 91% compared to an off-the-shelf implant with the same footprint for a patient with normal bone quality and 54% for a patient with osteopenia. Prototypes of the optimized cage were additively manufactured and mechanically tested to evaluate the manufacturability and integrity of the design and to validate the finite element model.

An improved peridynamics topology optimization formulation for compliance minimization

This work proposes an improved peridynamics density-based topology optimization framework for compliance minimization. One of the main advantages of using a peridynamics discretization relies in the fact that it provides a consistent regularization of classical continuum mechanics into a nonlocal continuum, thus containing an inherent length scale called the horizon. Furthermore, this reformulation allows for discontinuities and is highly suitable for treating fracture and crack propagation. Partial differential equations are rewritten as integrodifferential equations and its numerical implementation can be straightforwardly done using meshfree collocation, inheriting its advantages. In the optimization formulation, Solid Isotropic Material with Penalization (SIMP) is used as interpolation for the design variables. To improve the peridynamic formulation and to evaluate the objective function in a energetically consistent manner, surface correction is implemented. Moreover, a detailed sensitivity analysis reveals an analytical expression for the objective function derivatives, different from an expression commonly used in the literature, providing an important basis for gradient-based topology optimization with peridynamics. The proposed implementation is studied with two examples illustrating different characteristics of this framework. The analytical expression for the sensitivities is validated against a reference solution, providing an improvement over the referred expression in the literature. Also, the effect of using the surface correction is evidenced. An extensive analysis of the horizon size and sensitivity filter radius indicates that the current method is mesh-independent, i.e. a sensitivity filter is redundant since peridynamics intrinsically filters length scales with the horizon. Different optimization methods are also tested for uncracked and cracked structures, demonstrating the capabilities and robustness of the proposed framework.

Multi‐material topology optimization considering arbitrary strength and yield criteria constraints with single‐variable interpolation

AbstractMaterial heterogeneity gives composite constructions unique mechanical and physical qualities. Combining multiple materials takes full use of these features in stress‐constrained topology optimization. Traditional research in this field often assumes a consistent yield criterion for all possible materials but adapts their stiffness and strengths accordingly. To cope with this challenge, an innovative single‐variable interpolation approach is proposed to enable the simultaneous inclusion of distinct yield criteria and material strengths. A stress‐constrained topology optimization formulation is presented based on this yield function interpolation method, which can independently support various materials with different elastic characteristics, material strengths, and yield criteria. Then, the large‐scale problem of local stress constraints can be effectively solved by the Augmented Lagrangian (AL) method. Several two‐dimensional (2D) and three‐dimensional (3D) design scenarios are investigated to reduce the overall mass of the structure while considering stress constraints. The optimal composite designs exhibit several crucial benefits resulting from material heterogeneity, including the enlargement of the design possibilities, the dispersion of stress, and the utilization of asymmetry in tension‐compression strength.

Topology optimization of finite strain elastoplastic materials using continuous adjoint method: Formulation, implementation, and applications

This study presents a unified formulation of topology optimization for finite strain elastoplastic materials. As the primal problem to describe the elastoplastic behavior, we consider the standard J2-plasticity model incorporated into Neo-Hookean elasticity within the finite strain framework. For the optimization problem, the objective function is set to accommodate both single and multiple objectives, the latter of which is realized by weighting each sub-function. The continuous adjoint method is employed to derive the sensitivity, which is a general form that accepts any kind of discretization method. Then, the governing equations of the adjoint problem are derived as a format that holds at any moment and at any location in the continuum body or on its boundary. Accordingly, the proposed formulation is independent of any requirements in numerical implementation. In addition, the reaction–diffusion equation is used to update the design variable in an optimizing process, for which the continuous distribution of the design variable as well as material properties are maintained. Two specific optimization problems, stiffness maximization and plastic hardening maximization, for two and three-dimensional structures are presented to demonstrate the ability of the proposed formulation.

Porous structures inspired by porcupine quill: multiscale design optimization approach

This paper presents a novel approach for designing a freeform bending-resistant structure from the combination of explicit discrete component-based topology optimization (TO) and the porcupine quill-inspired features. To embed the porcupine quill’s features into the TO formulations, the method involves constructing discrete components at various scales to imitate features including solid shell, stochastically distributed pores, and graded stiffeners. The components are iteratively updated, and the optimization process allows for the grading of quill-inspired features while achieving optimal structural compliance under bending loads. The proposed approach is demonstrated to be effective through the resolution of Messershmitt–Bolkow–Blohm (MBB) beam designs, parameterized studies of geometric parameters, and numerical validation of long-span and short-span quill-inspired beam designs. By examining the von Mises stress distribution, the study highlights the mitigation of material yielding at the shell region brought by the geometric features of porcupine quills, leading to the potential theory support for the bending resistance. The optimized MBB beams are manufactured using the material extrusion technique, and three-point bending tests are conducted to explore the failure mitigation capability of the quill-inspired beam under large deformation. Consequently, the study concludes that the proposed quill-inspired component-based TO approach can design a structure with excellent bending resistance according to the improved energy absorption as well as increased deformation after reaching 75% peak load.

Isogeometric topology optimization of structures using the overweight approach

In this paper, a 2D isogeometric formulation of the material distribution for structural topology optimization considering minimum weight and local stress constraints using the overweight approach is proposed. The aim of this isogeometric formulation is to provide solutions with high spatial definition using a lower number of design variables in comparison with the formulations previously developed to define the material layout. Despite of this, an important number of local stress constraints has to be considered in the solution of the problem. For this purpose, an Overweight Constraint is used to consider all of them. The structural analysis is performed by means of the Isogeometric Analysis (IGA) and the distribution of material is modeled by means of quadratic B-splines. Moreover, the optimization is addressed by means of the Sequential Linear Programming algorithm (SLP) that is driven by the information provided by a full first-order sensitivity analysis extension of the IGA formulation. Finally, the proposed formulation is tested by means of some benchmark problems, and the results show that the isogeometric formulation provides solutions with high spatial definition. A comparison with a Finite Element Method (FEM) topology optimization formulation is included.

Toward static and transient stress-constrained topology optimization for shell-infill structures

This paper contributes to a stress-constrained topology optimization approach for the design of shell-infill structures. Based on the density method, we utilize the two-step filtering and projection scheme and local volume constraint for generating the shell and non-uniform infills, respectively. The topology optimization formulation is defined as a Robust Minimum Compliance problem with Volume and Stress constraints (RMCVS), where the robust method is used to eliminate undesired topological characteristics. We globalize the static and transient stress constraints through the p-norm function, and tailor material interpolation and stress relaxation schemes for both cases, respectively, to avoid numerical difficulties. Sensitivity analysis is provided, and the derivatives of the objective function and constraints with respect to the design variable fields are derived. We then validate the suggested method through several 2D and 3D benchmarks. The results illustrate that the method is robust to generate various shell-infill structures while limiting the maximum static and transient stress.

Connectivity Constraints Ensuring Continuous Electrodes in Topology Optimization of Electroactive Polymer

Abstract Electroactive polymers (EAPs) deform when subject to an electric field, which is generated by two or more electrodes. To ensure proper function of the EAP, these electrodes are connected to a source and they are therefore required to be continuous such that no isolated islands exist. Increasing an EAP’s performance using topology optimization while ensuring electrode connectivity is the goal of this work. A topology optimization formulation is introduced where electrode connectivity is ensured using the virtual temperature method. Numerical experiments demonstrate that this is an efficient method to guarantee connectivity.

Topology optimization for flow machine rotor design considering resonance and low mass density flows

When designing structures that take part in rotating fluid flow machines, one important effect that may be taken into account is resonance, which is evaluated through the modal analysis. Under rotating conditions, the formulation and behavior of the structure changes, requiring a new formulation. Thus, this work formulates the fluid flow topology optimization for rotating resonance, while also considering a low mass density (compressible or incompressible) fluid flow. Therefore, three different effects are considered in the fluid flow-based design: from the modal analysis, from the solid structure, and from the low mass density fluid flow. The topology optimization problem is formulated for a simplified 2D model of a rotating fluid flow device, and a nodal design variable. The topology optimization is performed through an integer linear programming algorithm. Numerical examples are presented for the developed topology optimization formulation.

Topology Optimization of Shape Memory Alloy Actuators for Prescribed Two-Way Transforming Shapes

This paper proposes a new topology optimization formulation for obtaining shape memory alloy actuators which are designed with prescribed two-way transforming shapes. The actuation behaviors of shape memory alloy structures are governed by austenite-martensite phase transformations effected by thermal-mechanical loading processes; therefore, to realize the precise geometric shape variations of shape memory alloy actuators, traditional methods involve iteration processes including heuristic structural design, numerical predictions and experimental validation. Although advanced structural optimization methods such as topology optimization have been used to design three-dimensional (3D) shape memory alloy actuators, the maximization/minimization of quantities such as structural compliance or inaccurate stroke distances has usually been selected as the optimization objective to obtain feasible solutions. To bridge the gap between precise shape-morphing requirements and efficient shape memory alloy actuator designs, this paper formulates optimization criteria with quantitatively desired geometric shapes, and investigates the automatic designs of two-way prescribed shape morphing shape memory alloy structures based on the proposed topology optimization method. The super element method and adjoint method are used to derive the analytical sensitivities of the objective functions with respect to the design variables. Numerical examples demonstrate that the proposed method can obtain 3D actuator designs that have the desired two-way transforming shapes.

A compatible boundary condition-based topology optimization paradigm for static mechanical cloak design

We propose a novel compatible boundary condition-based topology optimization formulation for realizing static mechanical cloaking of linear elastic structures. The key idea is to make the actual boundary conditions along the exterior of the cloaking region consistent with the reference boundary conditions, including the displacement and the nodal force, by designing the cloak structure solely. This compatible boundary condition-based optimization paradigm is not only applicable to various shaped voids (e.g., circle, ellipse, star, Mickey, square, “DUT” logo) and diverse loading conditions (e.g., pressure-sliding, biaxial-stretching, linear-stretching, random-stretching, stretching-sliding) but also demonstrates insensitivity to element quantities and slight variations of external load, ensuring consistent and robust elastostatic cloaking performance. Additionally, adopting the Moving Morphable Voids (MMV) method guarantees the optimized structures have the merits of explicit geometric boundaries and easy fabrication. Numerical examples and experimental validation demonstrate the effectiveness of the proposed approach. The present approach can be easily extended for cloak design in three dimensions and multiphysics.

Adaptive fail-safe topology optimization using a hierarchical parallelization scheme

This work presents an efficient, flexible, and scalable strategy to implement density-based topology optimization formulation for fail-safe structural design. Such optimized designs can operate after faults modeled as loss of stiffness. However, the need for assessing the physical behavior in all the fault cases can make this problem unfeasible computationally. We combine many ingredients to exploit the different levels of parallelism of the formulation to deal successfully with this problem. We use a non-overlapping domain decomposition method to solve the failure cases using multi-core computing with distributed memory computation. These subdomains use adaptive mesh refinement (AMR) techniques to focus computing efforts on the regions of interest. A key point to evaluate the fault cases is the organization of the computing threads, especially in clusters of computers. We group the computing threads by physical computing nodes to reduce the inter-node communications, maximizing intra-node communications to mitigate bandwidth problems. Another ingredient of paramount importance is the use of computing buffers to adapt the evaluation of the fault cases to the computing resources. We test the scalability of the computational framework using a large cluster of computers.

Topology optimization of stability‐constrained structures with simple/multiple eigenvalues

AbstractThis work focuses on topology optimization formulations with linear buckling constraints wherein eigenvalues of arbitrary multiplicities can be canonically considered. The non‐differentiability of multiple eigenvalues is addressed by a mean value function which is a symmetric polynomial of the repeated eigenvalues in each cluster. This construction offers accurate control over each cluster of eigenvalues as compared to the aggregation functions such as ‐norm and Kreisselmeier–Steinhauser (K–S) function where only approximate maximum/minimum value is available. This also avoids the two‐loop optimization procedure required by the use of directional derivatives (Seyranian et al. Struct Optim. 1994;8(4):207‐227.). The spurious buckling modes issue is handled by two approaches—one with different interpolations on the initial stiffness and geometric stiffness and another with a pseudo‐mass matrix. Using the pseudo‐mass matrix, two new optimization formulations are proposed for incorporating buckling constraints together with the standard approach employing initial stiffness and geometric stiffness as two ingredients within generalized eigenvalue frameworks. Numerical results show that all three formulations can help to improve the stability of the optimized design. In addition, post‐nonlinear stability analysis on the optimized designs reveals that a higher linear buckling threshold might not lead to a higher nonlinear critical load, especially in cases when the pre‐critical response is nonlinear.

EMsFEM based concurrent topology optimization method for hierarchical structure with multiple substructures

An efficient concurrent topology optimization method based on the Extended Multiscale Finite Element Method (EMsFEM) is proposed to design the hierarchical structure with multiple substructures in this paper. Firstly, the mathematical description model for the topology of hierarchical structures and the interpolation model for material are established at both the macro and substructural levels. Then the concurrent topology optimization formulation for hierarchical structures is proposed based on EMsFEM. After that, the sensitivity analysis scheme of the objective function and constraint functions is given, based on which the Method of Moving Asymptotes (MMA) is employed to update design variables. The assumption of separation of length scales is not required in the proposed method, which avoids the disconnection between adjacent substructures. Besides, without limiting the topology and spatial distribution of substructures in advance, the concurrent topology optimization for hierarchical structures with multiple substructures can be realized using this method. The provided numerical examples verify the correctness and effectiveness of the proposed sensitivity analysis method as well as the feasibility, effectiveness and scalability of the proposed concurrent topology optimization method for hierarchical structures. Finally, the influence of the initial substructure configurations and the number of substructure types on the final configuration and mechanical performance of hierarchical structures are explored with a series of numerical examples.

Topology optimization of differentiable microstructures

Recent years have seen a growing interest in topology optimization of functionally graded microstructures, characterized by an array of microstructures with varying volume fractions. However, microstructures optimized at slightly different volume fractions do not necessarily connect well when placed adjacently. Furthermore, optimization is commonly performed on a finite set of volume fractions, limiting the number of microstructure configurations.In this paper, we introduce the concept of differentiable microstructures, which are parameterized microstructures that exhibit continuous variations in both geometry and mechanical properties. To construct such microstructures, we propose a novel formulation for topology optimization. In this approach, a series of 2-dimensional microstructures is represented using a height field, and the objective is to maximize the bulk modulus of the entire series. Through this optimization process, an initial microstructure with a small volume fraction undergoes non-uniform transformations, generating a series of microstructures with progressively increasing volume fractions. Notably, when compared to traditional uniform morphing methods, our proposed optimization approach yields a series of microstructures with bulk moduli that closely approach the theoretical limit.

A new design method for stiffened plate based on topology optimization with min-max length-scale control

Stiffened plates are widely used in engineering due to their excellent manufacturing and mechanical properties. This paper introduces a novel method for designing stiffener plates that combines the H-DGTP formulation, robust topology optimization formulation, and maximum length-scale control. In comparison to existing methods, the proposed approach not only provides a clear layout of stiffeners but also optimizes their height. Sensitivity analysis of all design variables is derived for utilization with gradient-based optimizers. The study demonstrates that the implementation of the robust filter approach enables precise control of both structural features and gap widths, effectively avoiding sharp angles. Moreover, as the maximum length approaches the minimum length, the stiffeners assume uniform thickness, which better meets engineering requirements. Numerical examples are presented to validate the effectiveness of the proposed method.

A fatigue-resistance topology optimization formulation for continua subject to general loads using rainflow counting

A fatigue-resistance topology optimization formulation for continua subject to general loads using rainflow counting

Tuning Buckling Behaviors in Magnetically Active Structures: Topology Optimization and Experimental Validation

Abstract Buckling, a phenomenon historically considered undesirable, has recently been harnessed to enable innovative functionalities in materials and structures. While approaches to achieve specific buckling behaviors are widely studied, tuning these behaviors in fabricated structures without altering their geometry remains a major challenge. Here, we introduce an inverse design approach to tune buckling behavior in magnetically active structures through the variation of applied magnetic stimuli. Our proposed magneto-mechanical topology optimization formulation not only generates the geometry and magnetization distribution of these structures but also informs how the external magnetic fields should be applied to control their buckling behaviors. By utilizing the proposed strategy, we discover magnetically active structures showcasing a broad spectrum of tunable buckling mechanisms, including programmable peak forces and buckling displacements, as well as controllable mechano- and magneto-induced bistability. Furthermore, we experimentally demonstrate that multiple unit designs can be assembled into architectures, resulting in tunable multistability and programmable buckling sequences under distinct applied magnetic fields. By employing a hybrid fabrication method, we manufacture and experimentally validate the generated designs and architectures, confirming their ability to exhibit precisely programmed and tunable buckling behaviors. This research contributes to the advancement of multifunctional materials and structures that harness buckling phenomena, unlocking transformative potential for various applications, including robotics, energy harvesting, and deployable and reconfigurable devices.

Conceptual design of compliant bone scaffolds by full-scale topology optimization

A promising new treatment for large and complex bone defects is to implant specifically designed and additively manufactured synthetic bone scaffolds. Optimizing the scaffold design can potentially improve bone in-growth and prevent under- and over-loading of the adjacent tissue. This study aims to optimize synthetic bone scaffolds over multiple-length scales using the full-scale topology optimization approach, and to assess the effectiveness of this approach as an alternative to the currently used mono- and multi-scale optimization approaches for orthopaedic applications. We present a topology optimization formulation, which is matching the scaffold's mechanical properties to the surrounding tissue in compression. The scaffold's porous structure is tuneable to achieve the desired morphological properties to enhance bone in-growth. The proposed approach is demonstrated in-silico, using PEEK, cortical bone and titanium material properties in a 2D parameter study and on 3D designs. Full-scale topology optimization indicates a design improvement of 81% compared to the multi-scale approach. Furthermore, 3D designs for PEEK and titanium are additively manufactured to test the applicability of the method. With further development, the full-scale topology optimization approach is anticipated to offer a more effective alternative for optimizing orthopaedic structures compared to the currently used multi-scale methods.