Stress-based Topology Optimization Research Articles

Topology Optimization based Integrated Design of Structure-Foundation System

This study introduces a novel topology optimization method with the goal of achieving an integrated design for offshore infrastructure. A key feature of this method is its capability to eliminate artificial boundaries between the design of the above-ground support structure and underlying soil foundation, thus facilitating a more efficient and effective design solution. In contrast to traditional stress-based topology optimization methods, the proposed approach combines limit analysis with density-based topology optimization techniques. Formulated as a volume minimization task, it targets the structure’s bearing capacity, ensuring that the external force applied is precisely matches the bearing capacity of the system obtained in the design. Furthermore, it is demonstrated that the proposed method exhibits favorable convergence properties within a prescribed number of iterations, indicating a steady progression towards an optimized solution.

Stress-based topology optimization approach using binary variables and geometry trimming

Stress-based topology optimization approach using binary variables and geometry trimming

Topology optimization for infinite fatigue life of cyclic symmetric structures subjected to non-proportional loading

This paper presents a density based topology optimization method for infinite fatigue life constraints of non-proportional load cases, with a specific focus on parts with cyclic symmetry. Considering non-proportional loads in topology optimization significantly broadens the types of design problems that can be handled. The method estimates the local variation in Signed von Mises stress using a smooth min/max function and constrains the resulting stress amplitude using established stress based topology optimization methods. Accounting for non-proportionality of loading significantly increases the computation cost with respect to existing proportional methods, as the time-varying stress field needs to be computed. Inertia effects are neglected in the structural analysis. Therefore, a quasi-static analysis is used to obtain the stress history. To reduce the computational cost, advantage is taken of cyclic symmetric properties to reduce the number of necessary time steps to evaluate. This reduces the computational cost roughly proportional to the number of unique load time steps present in the repeated segments as opposed to a standard implementation. The method is tested on numerical examples in 2D and 3D for both proportional and non-proportional loads and was found to be locally accurate up to the accuracy of the constraint aggregation.

Stress-based evolutionary topology optimization via XIGA with explicit geometric boundaries

This paper presents a novel approach to stress-based topology optimization that employs NURBS representations of explicit geometric boundaries to address stress minimization and stress-constrained problems. The proposed methodology is compatible with CAD systems and utilizes Extended IsoGeometric Analysis (XIGA) to ensure an accurate representation of stress field in the evolving structure. A p-norm aggregation scheme is used to measure global stress levels, and the aggregated stress constraint is augmented to the objective (compliance) via a Lagrange multiplier in stress-constrained problems. The proposed approach introduces a novel and efficient Lagrange multiplier trail strategy aimed at reducing computational costs. Additionally, a normalization scheme and a partial differential equation filter are presented to stabilize the highly nonlinear stress-based optimization. Validation studies demonstrate the effectiveness and characteristics of the proposed approach. Moreover, control points of the structural design can be directly edited to further improve strength performance. Comparative studies show clear advantages of the proposed methodology in terms of computational efficiency compared to other Finite element analysis (FEA)-based optimization methods. Overall, this paper presents a practical and novel approach that addresses the limitations of existing FEA-based optimization methods in terms of accuracy, flexibility, and degree of freedom (DOF) cost, and makes a significant contribution to the field of stress-based topology optimization.

Stress-based topology optimization for fiber composites with improved stiffness and strength: Integrating anisotropic and isotropic materials

Fiber-reinforced materials offer high stiffness- and strength-to-mass ratios to lightweight composite structures. To design stiff, strong, and lightweight fiber-reinforced composite structures, we propose a multimaterial anisotropic stress-constrained topology optimization framework that simultaneously optimizes geometry, distribution of anisotropic (i.e., orthotropic) and isotropic material phases, and local orientations of fiber reinforcements in the anisotropic phase. To achieve high performance in both stiffness and strength, we discover that both isotropic and anisotropic materials are needed: anisotropic materials are preferred in uniaxial members to increase stiffness, while isotropic materials are crucial at multi-axially stressed joints to enhance strength. We introduce multimaterial interpolation schemes to characterize both the stiffness and strength of composites made up of anisotropic and isotropic materials. The characterization of strength is enabled by a novel load factor-based yield function interpolation that consistently integrates anisotropic Tsai–Wu and isotropic von Mises yield criteria. We optimize stress-sensitive domains considering materials with various levels of stiffness and strength anisotropy as well as multiple load cases. The anisotropic stress constraints in the proposed framework effectively inform geometries to reduce stress concentration in fiber composites. The proposed framework provides a rational design paradigm for composite structures, capitalizing on dissimilar stiffness and strength properties of anisotropic and isotropic materials, to potentially benefit various engineering applications.

Stress-based topology optimization of thermoelastic structures considering self-support constraints

A thermoelastic topology optimization method considering stress and manufacturing constraints is developed for engineering structures under thermo-mechanical coupled field in this paper. A design dependent non-uniform temperature field is considered for the heat conduction and the thermoelastic analysis during the topology optimization. Based on the solid isotropic material with penalization (SIMP), a topology optimization model is formulated to minimize the structural compliance under stress and overhang angle constraints. The overhang angle constraint to achieve the self-support property in additive manufacturing is established by connecting the elemental density values between the printing element and the corresponding support elements in the lower layer. The stress aggregation function is built to approximate the maximum stress value and an improved adaptive normalization scheme is applied to deal with the highly nonlinearity of the optimization. The sensitivity of the objective function and constraints with respect to the design variables are derived by the adjoint method, and the design variables are updated by the gradient-based optimization method. Moreover, a robustness scheme is proposed for the thermoelastic topology optimization to eliminate possible manufacturing issues such as “single node connection” and “thin rod” phenomena. Several typical numerical examples are provided to demonstrate the effectiveness of the proposed method in stress value reduction and overhang angle control. Meanwhile, the topology optimization with robustness scheme is also proved to result in more robust designs for manufacturing.

Real-time stress-based topology optimization via deep learning

In this paper, an iteration-free method based on the convolutional neural network (CNN) is proposed to increase the computational efficiency of stress-based topology optimization (SBTO). The p-norm stress aggregation scheme is adopted to measure the global stress level, and the method of moving asymptotes (MMA) is utilized to generate a dataset consisting of various load conditions and volume fractions. A deep CNN based on U-Net architecture is constructed, and based on the generated dataset, different input tensors are designed to train the network to find the optimal input mode for the SBTO problem. With a well-trained network, the optimized structures can be generated in real-time without any iterations with little sacrifice on the performance of the design scheme. Finally, the deep learning-based SBTO method is extended to a 3D case, and the results show that the proposed method can realize near optimal 3D SBTO prediction using negligible calculation cost.

Topology-optimized patient-specific osteosynthesis plates

Abstract Patient-specific osteosynthesis plates can be used to reduce complications related to bone fracture treatment, such as infection, malocclusion and fatigue fractures of plates and screws. However, the implant design process is tedious. We propose a semi-automatic workflow to computationally design patient-specific titanium osteosynthesis plates for mandibular angle fractures. In this process, the plate stiffness is maximized while the mass is reduced. Two plate designs with different numbers of screw holes (implant #1 with four holes, implant #2 with eight holes) were generated with identical topology optimization settings and compared in a finite element model simulating various biomechanical masticatory loads. Differences in von Mises stresses in the implants and screws were observed. The load case of clenching the jaw on the opposite side of the fracture showed the highest stress distribution in implant #1 and higher peak stresses in implant #2. Stress concentrations were observed in sharp corners of the implant and could be reduced using local stress-based topology optimization. We conclude that the design process is an effective method to generate patientspecific implants.

Stress-based topology optimization of continuum structures under harmonic force excitation

In this paper, a new optimization method for the maximum von Mises stress minimization design of continuum structures subjected to harmonic force excitation is proposed. By using an extended Bi-directional Evolutionary Structural Optimization (BESO) method, the intermediate density elements with high stress state are avoided. To reduce the computational cost, the global von Mises stress is approximately measured by using a global stress measure based on the p-norm. Both sensitivities and topological variables are filtered to overcome the highly nonlinear stress behavior and stabilize the optimization procedure. A series of comparisons have been conducted to validate the effectiveness and practicability of the method on two-dimensional and three-dimensional benchmark design problems. The influence of varying excitation frequencies, damping and boundary conditions on the optimized results and the mode shapes are considered. The optimized results indicate that the proposed approach has good convergence and can achieve a reasonable design that effectively reduces the stress concentration effect at the critical stress areas. Strength of engineering structure under harmonic load is improved.

Stress-based topology optimization with the parameterized level-set method based on radial basis functions

Stress-based topology optimization with the parameterized level-set method based on radial basis functions

Correction to: An efficient 146-line 3D sensitivity analysis code of stress-based topology optimization written in MATLAB

Correction to: An efficient 146-line 3D sensitivity analysis code of stress-based topology optimization written in MATLAB

A finite element level-set method for stress-based topology optimization of plate structures

In this paper, we present a stress-based topology optimization of plate structures by using a finite element level-set (LS) method. Since stress constraints are independent with LS-based design variables, the singularity phenomenon arising in topology optimization under stress constraints can be resolved without using any relaxation technique. The finite element method is used for solving LS-based equations of design variables and the state and adjoint equations of plate structures with arbitrary complex geometries and boundaries of the design domain. To obtain stable finite element solutions without any oscillation, both unsteady LS convection-diffusion equation and LS re-initialization convection-reaction equation are solved using a stabilized least-squares finite element method.

Stress-based topology optimization of continuum structures for the elastic contact problems with friction.

Structural problems have various nonlinearities in the real world and these nonlinearities should be accommodated in structural topology optimization. This work proposes a topology optimization method for minimizing the maximum von Mises stress of elastic continuum structures with frictional contact under material usage constraint, using an extended Bi-directional Evolutionary Structural Optimization (BESO) method. Stresses are treated as global performance (objective) function, the global von Mises stress is measured by the p-norm stress aggregation approach, and the friction behavior is governed by the Coulomb friction law regularized in analogy with the perfect elasto-plastic theory. BESO method based on discrete variables which can avoid the well-known stress singularity and the numerical instability issue in frictional contact problems. The adjoint sensitivity analysis method is adopted to derive the sensitivity numbers. The effectiveness of the proposed method is validated through a series of comparison studies including elastic-rigid and elastic-elastic contact problems. The influence of varying friction coefficient on the optimized results and the stress distributions are investigated in comparison with the maximum stiffness design. The effect of different parameters including p-norm, volume fraction and mesh density on the optimized results are discussed. The optimized results, for elastic-rigid contact, indicate that the maximum stress can be reduced compared with elastic-elastic contact. The optimized stress decreases as the friction coefficient increases because the friction behavior resists the tangential deformation at the contact interface. The results also show that the proposed approach can achieve a reasonable design that effectively controls the stress level and reduces the stress concentration effect at the critical stress areas.

Stress-based structural topology optimization for design-dependent self-weight loads problems using the BESO method

This article aims to propose an approach to the stress-based topology optimization of continuous elastic bi-dimensional structures subjected to design-dependent self-weight loads using the Bi-directional Evolutionary Structural Optimization (BESO) method. Topology optimization is developed through the minimization of P-norm von Mises stress while satisfying a volume constraint. To implement the algorithm, a consistent sensitivity analysis including design-dependent loads has been developed by the adjoint method. A series of tests has been performed to explore and validate the method through three numerical examples: an L-bracket; a doubly supported beam with one pre-existing crack notch; and a cantilever beam. Comparison between traditional compliance minimization and stress minimization analyses, including design-dependent self-weight loads, shows that the method is an effective way to reduce the maximum stress.

Stress-Based topology optimization under the stress relaxation effect

One of the effects of creep phenomena in structures is stress relaxation. Owing to this effect, the stiffness of each point is inversely related to the stress at that point, which causes a stress redistribution, a reduction of the maximum stress, and an increment in the displacement field. By contrast, for structural optimization, it is a practical and challenging problem to consider the maximum stress as a failure criterion or objective function. In the present study, the stress relaxation effect was investigated in stress-based topology optimization. The so-called solid isotropic material with penalization method and a nonlinear finite element analysis are utilized for this purpose. The maximum von Mises stress and the total volume of the structure are selected as the objective function and constraint, respectively. The main novelty of this research is to find the optimized topology with the least maximum stress by developing and implementing a time-dependent adjoint method for sensitivity analysis. Implementing the calculated sensitivity for stress-based problems in a gradient-based optimization method, such as the optimality criteria, is quite challenging, and some numerical instabilities in the optimization process are addressed. Several examples are solved with and without considering the stress relaxation effect. The results reveal that the influence of stress relaxation in the reduction of stress concentration makes the optimized layout more similar to the optimized layout obtained by minimizing the compliance for the linear structures.

An efficient 146-line 3D sensitivity analysis code of stress-based topology optimization written in MATLAB

This paper presents an efficient and compact MATLAB code for three-dimensional stress-based sensitivity analysis. The 146 lines code includes the finite element analysis and p-norm stress sensitivity analysis based on the adjoint method. The 3D sensitivity analysis for p-norm global stress measure is derived and explained in detail accompanied by corresponding MATLAB code. The correctness of the analytical sensitivity is verified by comparison with finite difference approximation. The nonlinear optimization solver is chosen as the Method of moving asymptotes (MMA). Three typical volume-constrained stress minimization problems are presented to verify the effectiveness of sensitivity analysis code. The MATLAB code presented in this paper can be extended to resolve different stress related 3D topology optimization problems. The complete program for sensitivity analysis is given in the Appendix and is intended for educational purposes. MATLAB code is additionally provided in electronic supplementary material for a simple cantilever beam optimization.

A new stress-based topology optimization approach for finding flexible structures

This paper presents a new FE-based stress-related topology optimization approach for finding bending governed flexible designs. Thereby, the knowledge about an output displacement or force as well as the detailed mounting position is not necessary for the application. The newly developed objective function makes use of the varying stress distribution in the cross section of flexible structures. Hence, each element of the design space must be evaluated with respect to its stress state. Therefore, the method prefers elements experiencing a bending or shear load over elements which are mainly subjected to membrane stresses. In order to determine the stress state of the elements, we use the principal stresses at the Gauss points. For demonstrating the feasibility of the new topology optimization approach, three academic examples are presented and discussed. As a result, the developed sensitivity-based algorithm is able to find usable flexible design concepts with a nearly discrete 0 − 1 density distribution for these examples.

Stress-based topology optimization through non-uniform rational basis spline hyper-surfaces

In this work, the Huber-Hencky-Von Mises criterion for isotropic materials is integrated into a special density-based algorithm for topology optimization (TO). The algorithm makes use of (a) Non-Uniform Rational Basis Spline (NURBS) hyper-surfaces to represent the pseudo-density field describing the topology of the continuum and (b) the well-known Solid Isotropic Material with Penalization approach. The local behavior and the singularity of stresses are efficiently handled thanks to the NURBS blending functions properties and a suitable aggregation function. To this end, a dedicated strategy is proposed to properly update the parameters governing the behavior of the aggregation function during the iterations of the optimization process. Moreover, the gradient of the criterion is derived in closed form (in the most general case when both displacements and forces are applied as boundary conditions) by exploiting the local support property of NURBS entities. A sensitivity analysis of the optimized topology to the integer parameters of the NURBS hyper-surface is carried out. Furthermore, a manufacturing requirement related to the minimum allowable size is also integrated into the problem formulation. The effectiveness of the approach is proven on 2 D and 3 D benchmark problems taken from the literature.

Stress Relaxation and Sensitivity Weight for Bi-Directional Evolutionary Structural Optimization to Improve the Computational Efficiency and Stabilization on Stress-Based Topology Optimization

Stress-based topology optimization is one of the most concerns of structural optimization and receives much attention in a wide range of engineering designs. To solve the inherent issues of stress-based topology optimization, many schemes are added to the conventional bi-directional evolutionary structural optimization (BESO) method in the previous studies. However, these schemes degrade the generality of BESO and increase the computational cost. This study proposes an improved topology optimization method for the continuum structures considering stress minimization in the framework of the conventional BESO method. A global stress measure constructed by p-norm function is treated as the objective function. To stabilize the optimization process, both qp-relaxation and sensitivity weight scheme are introduced. Design variables are updated by the conventional BESO method. Several 2D and 3D examples are used to demonstrate the validity of the proposed method. The results show that the optimization process can be stabilized by qp-relaxation. The value of q and p are crucial to reasonable solutions. The proposed sensitivity weight scheme further stabilizes the optimization process and evenly distributes the stress field. The computational efficiency of the proposed method is higher than the previous methods because it keeps the generality of BESO and does not need additional schemes.

基于LiToSim平台的海上风机过渡段优化软件开发

The software development of stress-based topology optimization was addressed for the offshore turbine transition structure based on the self-developed LiToSim software platform. Firstly, the multi-scale model for the mixed beam and bulk elements was applied to the structural analysis of the wind turbine transition structure. Secondly, the extreme loads of the wind-wave coupling were taken into account. Thirdly, the stress-based topology optimization method was used for the design of the transition structure. Based on the self-developed LiToSim software platform, a customized software TUR/TOPT for the topology optimization of the transition structures of offshore wind turbines was formed. With TUR/TOPT, the traditional compliance optimization and stress-based optimization results of the transition structure were compared to show the advantages of stress-based optimization in reducing the structural stress during the material reduction design process and effectively avoiding stress concentration. The software TUR/TOPT provides important guiding value for the selection process of wind turbine construction.