Structural Topology Optimization Problems Research Articles

An enhanced proportional topology optimization method with new density filtering weight function for the minimum compliance problem

This article proposes an enhanced proportional topology optimization (EPTO) method to solve the structural topology optimization problem of minimizing compliance under material volume constraints. In the proposed method, a new filtering weight function based on the improved Heaviside threshold function is adopted to filter element density during the optimization process. The optimization process of the EPTO method consists of an inner loop and an outer loop. In the inner loop, density distribution is modified in combination with a new filtering weight function. By locally averaging the weighted element compliance, an improved density distribution function in the inner loop is put forward to make the topology configuration of the optimized structure more reasonable. In addition, the projection method is used to reduce the number of intermediate density elements, thereby obtaining optimization results with clear boundaries. In the outer loop, a new termination criterion is adopted, which terminates the optimization process by determining that the relative error of the objective function in several consecutive iterations is less than the specified value. The effectiveness and efficiency of the proposed method are demonstrated through several numerical examples involving two-dimensional (2D) and three-dimensional (3D) topology optimization problems. The results of numerical examples show that the proposed method can not only accelerate the convergence of optimization iterations, but also obtain optimized structures with smaller objective function values and better topology configurations. HIGHLIGHTS A new density filtering weight function is proposed based on an improved Heaviside threshold function. An improved inner loop density distribution formula is proposed by combining a new filtering weight function. Using projection based methods to suppress the appearance of intermediate density elements. Verify the effectiveness of the proposed algorithm by comparing the optimization results with existing algorithms such as top88 and PTO.

Data-driven and physics-based methods to optimize structures against delamination

Composite materials and multi-material components often fail at their internal interfaces/adhesive joints, and hence special attention should be given to such catastrophic delamination events to guarantee the system’s functional requirements. So far, however, the majority of structural topology optimization problems have focused on optimal distribution of the bulk materials by considering interfaces as perfectly bonded. This motivates the introduction of optimization methods that explicitly take into account the role of the material interfaces to optimize structures against delamination. In this work, we propose a data-driven heuristic optimization approach for the identification of optimal cohesive interfaces with linearly graded fracture properties to increase the ability of the composite structure to withstand peeling. Moreover, for given cohesive interface properties, we investigate the applicability of the physics-based Solid Isotropic Material with Penalty (SIMP) topology optimization approach to optimize the internal structure of a substrate in problems where the stress field is affected by interface delamination.

Multi-material topology optimization with a direct meshless local Petrov–Galerkin method applied to a two-dimensional boundary value problem

ABSTRACT This work presents a novel approach for the solution of multi-material topology optimization problems of elastic structures coupling the Direct Meshless Local Petrov–Galerkin (DMLPG) method with the Gradual Bi-direction Evolutionary Structural Optimization (gBESO) method. In recent years, meshless methods have been used in several engineering fields, showing some advantages compared to mesh-based methods. DMLPG is a truly meshless method, which achieves results with good accuracy and computational efficiency, avoiding the numerical integration of complicated shape functions by adopting low-degree polynomials. In the proposed algorithm, DMLPG is used to obtain smooth nodal displacements, strains and stresses, and gBESO updates the structural geometry based on design sensitivity values of the compliance while it gradually increases the Young's modulus of the material. The optimization problem minimizes the compliance objective function subject to volume constraints. Numerical examples demonstrate the applicability and validity of the technique.

Mapping structural topology optimization problems to quantum annealing

Mapping structural topology optimization problems to quantum annealing

A polygonal topology optimization method based on the alternating active-phase algorithm

<abstract> <p>We propose a polygonal topology optimization method combined with the alternating active-phase algorithm to address the multi-material problems. During the process of topology optimization, the polygonal elements generated by signed distance functions are utilized to discretize the structural design domain. The volume fraction of each material is considered as a design variable and mapped to its corresponding element variable through a filtering matrix. This method is used to solve a multi-material structural topology optimization problem of minimizing compliance, in which a descriptive model is established by using the alternating active-phase algorithm and the solid isotropic microstructure with penalty theory. This method can accomplish the topology optimization of multi-material structures with complex curve boundaries, eliminate the phenomena of checkerboard patterns and a one-node connection, and avoid sensitivity filtering. In addition, this method possesses fine numerical stability and high calculation accuracy compared to the topology optimization methods that use quadrilateral elements or triangle elements. The effectiveness and feasibility of this method are demonstrated through several commonly used and representative numerical examples.</p> </abstract>

A topology optimization method for problems with design-dependent loads based on non-uniform rational basis spline entities

In this paper, topology optimization (TO) problems of structures subjected to design-dependent loads are formulated in the context of a special density-based TO approach wherein a Non Uniform Rational Basis Spline (NURBS) hyper-surface is used to represent the topological descriptor. Unlike classical density-based TO approaches, the NURBS-density-based method allows representing the pseudo-density field through a purely geometric computer-aided design compatible entity. In this context, the problem is formulated in the most general case of inhomogeneous Neumann-Dirichlet boundary conditions. Moreover, a thorough study of the penalty function of the design-dependent loads is carried out to investigate its effect on the optimized topologies and overcome the singularity effect related to the zones characterized by low values of the pseudo-density field. Finally, a wide campaign of sensitivity analyses is conducted to investigate the influence of the integer parameters of the NURBS entity, of the combination of design-depedent loads and inhomogeneous Neumann-Dirichlet boundary conditions, and of the concentrated load on the optimized topology. The effectiveness of the approach is tested on 2D and 3D benchmark problems.

Improved Sine-cosine Algorithm for the Optimization Design of Truss Structures

The sine cosine algorithm (SCA) is a new meta-heuristic algorithm. It has been proven to be very competitive compared with other meta-heuristic algorithms and has attracted significant attention from researchers in different fields. At present, there is no research on the application of SCA in the truss structure. In order to solve the truss optimization problem, the following improvements are made to SCA in this paper. Firstly, a non-linear conversion parameter is set to replace the linear conversion parameter, so that the exploration area is better in line with the characteristics of the actual optimization situation. Secondly, the Lévy flight is employed to enhance the algorithm's global search ability. Thirdly, a novel elite guidance strategy taking advantage of memory data is proposed to strengthen the local search ability. Finally, a greedy selection strategy is adopted for the selection of the candidate. Combining the finite element method, we apply ISCA to optimize the size, shape and topology of truss structures. The examples show that ISCA can use less calculation time, get better results, and also better stability. So, the ISCA is effective to solve the size, shape and topology optimization problems of truss structures.

A novel numerical manifold method and its application in parameterized LSM-based structural topology optimization

In this paper, a high-efficiency structural topology optimization framework based on combination of numerical manifold method (NMM) and parameterized level set method (PLSM) is established. The NMM uses two cover systems to discretize the model, which can accurately represent the complex boundary of the design domain. A new numerical manifold element has been derived for the application of NMM in PLSM-based optimization. To obtain enhanced accuracy for structural analysis, a multilevel subdivision technique for numerical manifold element generation is presented, and the related numerical integration scheme coupled with the interpolation point selection strategy is provided. For the proposed topology optimization, the new update method of element stiffness matrix is exclusively formulated as well as the calculation method of volume fraction. Some representative structural optimization problems demonstrate that the proposed structural topology optimization method is very effective for two-dimensional (2D) and three-dimensional (3D) structural topology optimization problems.

部材の密集度を考慮した骨組構造物の大域的トポロジー最適化

It is known that the topology optimization (TopOpt) problem of frame structures can be formulated as a mixed-integer second-order cone programming problem (MISOCP), when the compliance is minimized and the cross-section of each member is selected from a set of predetermined candidates. However, when considering the constructability of real-world building structures, it is insufficient to only minimize compliance. In this paper, in order to control the density of members, the constraints of the number of members connected to each node are additionally considered into TopOpt by MISOCP treated in the previous study.

GPU-based mesh reduction strategy utilizing active nodes for structural topology optimization

This paper presents a novel mesh reduction strategy on graphics processing unit (GPU) that aims to reduce the computational complexity of conventional structural topology optimization. In the proposed strategy, the effective number of design variables is reduced by using the concept of active nodes and active elements in the finite element mesh. A novel mesh numbering scheme is also introduced to facilitate parallel identification of active nodes using a proposed GPU-based algorithm. The preconditioned conjugate gradient (PCG) solver is further developed using the proposed strategy and the numbering scheme. The proposed strategy is tested on three structural topology optimization problems using solid isotropic material with penalization method. Results of the proposed strategy demonstrate up to 8× speedup over the standard GPU implementation considering the entire mesh.

Crash topology optimization for front-end safety parts of battery electric vehicle using an improved equivalent static loads method

Crash topology optimization is a typical nonlinear dynamic response structural topology optimization problem, which is one of the most difficult problem in the structural design field. The equivalent static load method (ESLM) provides a well-defined pattern to solve such difficult problems, which can convert a nonlinear dynamic response optimization into multi-load steps optimization problem with the equivalent static loads (ESLs). However, due to the large deformation in the crash condition and nodal characteristics of the ESLs, it is hard to solve the crash topology optimization directly using the standard ESLM. To expand the application scope of the ESLM, an improved ESLs calculation method is proposed by using the model order reduction method and energy principle, which only acting on some nodes and can be scaled adaptively. Correspondingly, to enrich the connotation of topology optimization, a crash topology optimization method is proposed by using the improved ESLs, which can solve the numerical problems in the design domain by guaranteeing the topology optimization with the improved ESLs perform in the linear rang. First, the principle of the standard ESLM is introduced, and corresponding problems and deficiencies in solving the crash topology optimization are summarized. Then, to solve the above problems, the improved ESLs calculation method is proposed. Meanwhile, the corresponding crash topology strategy is proposed based on the improved ESLs. Finally, to verify the effectiveness and engineering application value of the proposed crash topology optimization method, a test-verified frontal crash simulation model of the BEV front-end is established, and their safety parts are redesigned by using the proposed method. The results show that, the proposed method can effectively solve the crash topology optimization of thin-walled structures under large deformation crash condition. This method also provides a new idea and practical method for the crashworthiness and lightweight design of automobile structures.

Overhang Penalization in Additive Manufacturing via Phase Field Structural Topology Optimization with Anisotropic Energies

A phase field approach for structural topology optimization with application to additive manufacturing is analyzed. The main novelty is the penalization of overhangs (regions of the design that require underlying support structures during construction) with anisotropic energy functionals. Convex and non-convex examples are provided, with the latter showcasing oscillatory behavior along the object boundary termed the dripping effect in the literature. We provide a rigorous mathematical analysis for the structural topology optimization problem with convex and non-continuously-differentiable anisotropies, deriving the first order necessary optimality condition using subdifferential calculus. Via formally matched asymptotic expansions we connect our approach with previous works in the literature based on a sharp interface shape optimization description. Finally, we present several numerical results to demonstrate the advantages of our proposed approach in penalizing overhang developments.

Chaotic marine predators algorithm for global optimization of real-world engineering problems

A novel metaheuristic called Chaotic Marine Predators Algorithm (CMPA) is proposed and investigated for the optimization of engineering problems. CMPA integrates the exploration merits of the recently proposed Marine Predators Algorithm (MPA) with the chaotic maps exploitation capabilities. Several chaotic maps were applied in the proposed CMPA to govern MPA parameters that eventually led to controlled exploration and exploitation of search. This study makes an initial attempt to explore and employ CMPA in decoding complex and challenging design and manufacturing problems. For performance evaluation of the proposed algorithm, CEC 2020 numerical problems having different dimensions and five widely adopted constrained design problems were solved. For all problems, both qualitative and qualitative results are examined and discussed. Moreover, two case studies of multi-pass turning were examined by the proposed CMPA algorithm to optimize the cutting operation with a minimum cost of production per unit objective. Furthermore, the suggested CMPA algorithm has been investigated for solving a real-world structural topology optimization problem. Statistical analysis is performed, and the results of CMPA are compared with twelve distinguished algorithms. Outcomes of the proposed variant algorithm on the benchmarks demonstrate its significantly improved performance relative to other optimizers including a variant of MPA and two state-of-the-art IEEE CEC competitions winners algorithms. Findings from the manufacturing process exhibit CMPA proficiency in solving arduous real-world design problems.

Aeroelastic Topology Optimization of Wing Structure Based on Moving Boundary Meshfree Method

The increasing structural flexibility of large aircraft leads to significant aeroelastic effects. More efficient topology optimization techniques are required for the design to further take advantage of aeroelasticity and obtain lightweight structures. This paper proposes a moving boundary meshfree topology optimization that combines the Galerkin method of weighted residuals and non-uniform rational B-splines (NURBS). The solution domain is described by the control points of NURBS and its property is calculated adaptively with an integration subtraction technique. The minimal compliance is searched for using the globally convergent method of moving asymptotes (GCMMA) by designing the locations of control points as subject to volume and flux constraints. The method is first applied to a typical two-dimensional design example with symmetric boundary conditions. The results show that the shape constraints can be conveniently applied, and smoother boundaries are obtained with fewer parameters. Then, a three-dimensional wing structure with asymmetric boundary conditions is optimized. A three-dimensional flight load that combines the high-order-panel and meshfree methods is employed to calculate the elastic loads and update asymmetric external loads during the optimization process. The designed wing satisfies engineering requirements and the presented method can solve the practical topology optimization problems of three-dimensional structures.

Topology optimization design of three-dimensional multi-material and multi-body structure based on irregular cellular hybrid cellular automata method

In recent years, Hybrid Cellular Automata Method (HCAM) has been successfully applied to solve structural topology optimization problems. However, there was no report on HCAM research of three-dimensional composite structure composed of multiple materials and multiple bodies. Therefore, in this paper, three-dimensional non-cube cells of irregular size (such as tetrahedral cells with adaptive changes inside length) and Finite Element Method (FEM) are introduced to extend HCAM, which is better and more flexibly to fit complex geometric shapes. Furthermore, a better structure configuration of multi-material and multi-body structure is obtained. The typical example study showed that the proposed topology optimization method could effectively remove the redundant materials of multi-material and multi-body structure, and the optimized structure configuration could still meet the requirements of the original condition after geometric reconstructed. Thus it provided a reference for the intelligent design of other products.

Topology optimization of structures subject to self-weight loading under stress constraints

PurposeThe purpose of this paper is to present an approach for structural weight minimization under von Mises stress constraints and self-weight loading based on the topological derivative method. Although self-weight loading topology has been the subject of intense research, mainly compliance minimization has been addressed.Design/methodology/approachThe resulting minimization problem is solved with the help of the topological derivative method, which allows the development of efficient and robust topology optimization algorithms. Then, the derived result is used together with a level-set domain representation method to devise a topology design algorithm.FindingsNumerical examples are presented, showing the effectiveness of the proposed approach in solving a structural topology optimization problem under self-weight loading and stress constraint. When the self-weight loading is dominant, the presence of the regularizing term in the formulation is crucial for the design process.Originality/valueThe novelty of this research work lies in the use of a regularized formulation to deal with the presence of the self-weight loading combined with a penalization function to treat the von Mises stress constraint.

A NEW ELEMENT-BASED METHOD FOR SOLVING STRUCTURAL TOPOLOGY OPTIMIZATION PROBLEMS WITH NON-UNIFORM MESHES

This study proposes a new element-based method to solve structural topology optimization problems with non-uniform meshes. The objective function is to minimize the compliance of a structure, subject to a volume constraint. For a structure of a fixed volume, the method is intended to find a topology that could almost conform to the compliance minimum. The method is refined from the evolutionary switching method, whose policy of exchanging elements is improved by replacing some empirical decisions with ones according to optimization theories. The method has the evolutionary stage and the element exchange stage to conduct topology optimization. The evolutionary stage uses the evolutionary structural optimization method to remove inefficient elements until the volume constraint is satisfied. The element exchange stage performs a procedure refined from the element exchange method. Notably, the procedures of both stages are refined to conduct non-uniform finite element meshes. The proposed method was implemented to use the Abaqus Python scripting interface to call the services of Abaqus such as running analysis and retrieving the output database of an analysis. Numerical examples demonstrate that the proposed optimization method could determine the optimal topology of a structure that is subject to a volume constraint and whose mesh is non-uniform.

Robust topology optimization of multi-material structures under load uncertainty using the alternating active-phase method

The synergy between different constituent materials can drastically improve the performance of composite structures. The optimal design of such structures for practical applications is complicated by the often-encountered non-deterministic loading conditions. This paper proposes an efficient method for robust multi-material topology optimization problems of continuum structures under load uncertainty. Specifically, the weighted sum of the mean and standard deviation of structural compliance is minimized under volume constraints for each material phase. Based on the theory of linear elasticity and using the orthogonal diagonalization of real symmetric matrices, the Monte Carlo sampling is separated from the topology optimization procedure and an efficient procedure for sensitivity analysis is established. By employing an alternating active-phase algorithm of the Gauss-Seidel version, the multi-material topology optimization problem is split into a series of binary topology optimization sub-problems, which can be easily solved using the modified SIMP model. Several 2D examples are presented to demonstrate the effectiveness of the proposed method.

A gradient-based optimization method with functional principal component analysis for efficient structural topology optimization

Structural topology optimization (STO) is usually treated as a constrained minimization problem, which is iteratively addressed by solving the equilibrium equations for the problem under consideration. To reduce the computational effort, several reduced basis approaches that solve the equilibrium equations in a reduced space have been proposed. In this work, we apply functional principal component analysis (FPCA) to generate the reduced basis, and we couple FPCA with a gradient-based optimization method for the first time in the literature. The proposed algorithm has been tested on a large STO problem with 4.8 million degrees of freedom. Results show that the proposed algorithm achieves significant computational time savings with negligible loss of accuracy. Indeed, the density maps obtained with the proposed algorithm capture the larger features of maps obtained without reduced basis, but in significantly lower computational times, and are associated with similar values of the minimized compliance.

Internal contact modeling for finite strain topology optimization

The present work proposes an extension of the third medium contact method for solving structural topology optimization problems that involve and exploit self-contact. A new regularization of the void region, which acts as the contact medium, makes the method suitable for cases with very large deformations. The proposed contact method is implemented in a second order topology optimization framework, which employs a coupled simultaneous solution of the mechanical, design update, and adjoint problems. All three problems are derived and presented in weak form, and discretized with finite elements of suitable order. The capabilities and accuracy of the developed method are demonstrated in a topology optimization problem for achieving a desired non-linear force-displacement path.