Stress-constrained Topology Optimization Research Articles

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.

Reformulation for stress topology optimization of continuum structures by floating projection

Stress topology optimization of continuum structures is an important topic for structural design and has been widely investigated. However, stress topology optimization itself is ill-conditioned and the resulting optimized designs highly depend on the used parameters, mesh size, and so on. In this paper, stress minimization and stress-constrained topology optimization problems are reformulated by introducing a global measure of an optimized design, the average solid stress. The floating projection topology optimization method is established with ε-relaxation and the p-norm function. Due to the modified problem formulations, the optimized design becomes insensitive to the selection of p in the p-norm function once p is large enough, such as p=16, to ensure the total elimination of stress concentration. Numerical results show the uniform stress distribution of optimized designs, demonstrating the effectiveness of the proposed method. Some findings on the elimination of stress concentration, mesh-independent optimized topology, equivalent smooth design, and impractical design are presented and discussed.

Stress-constrained topology optimization for multi-material thermo-hyperelastic compliant mechanisms based on inverse motion

A stress-constrained topology optimization design for the multi-material thermo-hyperelastic compliant mechanism is investigated in this article, using inverse motion analysis. Through the inverse motion analysis approach, precise control over the geometry of the deformed structure is achieved. A topology optimization model is established for the multi-material thermal hyperelastic compliant mechanism, incorporating constraints on volume, cost and stress, while maximizing displacement as the objective function. Filtering is implemented using an improved partial differential equation, and the stress problem is addressed using the ϵ-relaxation method and the p-norm aggregation function. The sensitivity analysis of the displacement objective function and the stress constraint is conducted. Considering the multi-material thermally actuated actuator as an example, the influence of the stress constraint on the topology optimization design is studied. The results indicate that inverse motion analysis allows for effective control of the output end position of the deformed compliant mechanism.

Stress-Constrained Topology Optimization for Commercial Software: A Python Implementation for ABAQUS®

Topology optimization has evidenced its capacity to provide new optimal designs in many different disciplines. However, most novel methods are difficult to apply in commercial software, limiting their use in the academic field and hindering their application in the industry. This article presents a new open methodology for solving geometrically complex non-self-adjoint topology optimization problems, including stress-constrained and stress minimization formulations, using validated FEM commercial software. The methodology was validated by comparing the sensitivity analysis with the results obtained through finite differences and solving two benchmark problems with the following optimizers: Optimality Criteria, Method of Moving Asymptotes, Sequential Least-Squares Quadratic Programming (SLSQP), and Trust-constr optimization algorithms. The SLSQP and Trust-constr optimization algorithms obtained better results in stress-minimization problem statements than the methodology available in ABAQUS®. A Python implementation of this methodology is proposed, working in conjunction with the commercial software ABAQUS® 2023 to allow a straightforward application to new problems while benefiting from a graphic user interface and validated finite element solver.

An exact penalty function optimization method and its application in stress constrained topology optimization and scenario based reliability design problems

A smooth penalty function method which does not involve dual and slack implicit variables to formulate constrained optimization conditions is proposed. New active and loss functions are devised to independently and adaptively handle the violation of each constraint function. A single unconstrained minimization of the proposed penalty function produces a solution to the original optimization problem. The constrained optimization conditions depending only on the primal variable are derived and reduce to the canonical Karush-Kuhn-Tucker(KKT) conditions in a much more general framework. The derivative of the proposed loss functions with respect to constraint function can be interpreted as the Lagrange multiplier of the traditional Lagrangian function. The appearance of the first-order Hessian information is unveiled. This is in contrast to the existing works where the classical Lagrange Hessian contains only second order derivatives. Finally, some numerical examples including medium-scale stress constrained topology optimization and scenario based reliability design problems are presented to demonstrate the effectiveness of the proposed methodology.

Topology Optimization for Digital Light Projector Additive Manufacturing Addressing the In-Situ Structural Strength Issue.

A topology optimization approach is proposed for the design of self-supporting structures for digital light projector (DLP) 3D printing. This method accounts for the adhesion forces between the print part and the resin base during DLP printing to avoid failure of the part due to stress concentration and weak connections. Specifically, the effect of the process-related adhesion forces is first simulated by developing a design variable-interpolated finite element model to capture the intricate mechanical behavior during DLP 3D printing. Guided by the process model, a stress-constrained topology optimization algorithm is formulated with both the SIMP and RAMP interpolation schemes. The interpolations on the stress term and the design-dependent adhesion load are carefully investigated. A sensitivity result on the P-norm stress constraint is fully developed. Finally, the approach is applied to several 2D benchmark examples to validate its efficacy in controlling the process-caused peak P-norm stresses. The effects of alternating between the SIMP and RAMP interpolations and changing the stress upper limits are carefully explored during the numerical trials. Moreover, 3D printing tests are performed to validate the improvement in printability when involving the process-related P-norm stress constraint.

Three-dimensional plasticity-based topology optimization with smoothed finite element analysis

AbstractThis paper presents a novel plasticity-based formulation for three-dimensional (3D) topology optimization of continuum structures. The proposed formulation addresses the optimization problem by combining mixed rigid-plastic analysis with density-based topology optimization, resulting in a volume minimization approach. Unlike conventional stress-constrained topology optimization methods that rely on linear elastic structural analysis, our developed formulation focuses on enhancing the loading capacity of the designed structures based on the plastic limit theory, leading to more cost-effective designs. To improve computational efficiency, we employ the smoothed finite element technique in our proposed method, enabling the utilization of linear tetrahedral elements for 3D mesh refinement. Moreover, the final formulation of our developed method can be efficiently solved using the advanced primal–dual interior point method, eliminating the need for a separate nonlinear finite element structural analysis. Numerical examples are presented to demonstrate the effectiveness of the proposed approach in offering enhanced design possibilities for continuum structures.

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.

Optimal design of compliant displacement magnification mechanisms using stress-constrained topology optimization based on effective energy

In this paper, stress-constrained topology optimization is applied to the design of compliant displacement magnification mechanisms. By formulating the objective function based on the concept of effective energy, it is not necessary to place artificial spring components at the boundaries of the output and input ports as in previous methods. This makes it possible to design mechanisms that do not receive a reaction force at the output port, such as sensors. Furthermore, by imposing a constraint on the maximum stress evaluated in terms of the p-norm of the von Mises equivalent stress, problems such as stress concentration can be avoided. Several numerical examples of displacement magnification mechanisms are provided to demonstrate the effectiveness of the proposed method.

Transient stress-constrained topology optimization of impacted structures

Transient stress-constrained topology optimization of impacted structures

Topology optimization of turbine disk considering maximum stress prediction and constraints

For the stress-constrained topology optimization of a turbine disk under centrifugal loads, the jagged boundaries of the mesh and the gray densities on the solid/void interfaces could make the calculated stress field inconsistent with the actual value. It may result in overestimating the maximum stress and thus affect the effectiveness of stress constraints. This paper proposes a new method for predicting the maximum stress to overcome the difficulty. In the process, a predicted density is newly defined to obtain stable boundaries with thin layers of gray elements, a transition factor is innovatively proposed to evaluate the effects of intermediate-density elements, two different stiffness penalty schemes are flexibly used to calculate the elastic modulus of elements, and a linear stress penalty is further adopted to relax the stress field of the structure. The proposed approach for predicting the maximum stress value is verified by the analysis of a structure with smooth boundaries and the topology optimization of a turbine disk. An updating scheme of the stress constraint in the topology optimization is also developed using the predicted maximum stress. Some key ingredients affecting the optimization results are discussed in detail. The results prove the effectiveness and efficacy of the proposed maximum stress prediction and developed stress constraint methods.

Stress constraint topology optimization of coupled thermo-mechanical problems using the temperature dependence of allowable stress

Engine components in the aerospace industry are used in high-temperature environments to improve performance. The material strength of metals significantly reduces at high temperatures compared to room temperature. This phenomenon is important in design since the design allowance considers temperature dependence. This study proposed a new stress-constrained topology optimization methodology for coupled thermo-mechanical problems. As for stress constraints, global constraints are applied, which can reduce computational cost, and constraints are imposed on the maximum stress in the body. Assuming an actual design, the proposed methodology defines allowable stress as a function of temperature and considers temperature dependence of the allowable stress. The proposed methodology simplifies the optimization problem by allowing the user to constrain temperature and stress with a single constraint function. The temperature constraint is automatically determined based on the degree of stress generated, making it easier to obtain valid optimization conditions and optimize results for the design. Further, the analytical sensitivity needed to obtain an optimized structure is developed. Finally, numerical calculations are performed to show the effects of temperature dependence on the allowable stress.

Topology optimization with local stress constraints and continuously varying load direction and magnitude: towards practical applications

Topology optimization problems typically consider a single load case or a small, discrete number of load cases; however, practical structures are often subjected to infinitely many load cases that may vary in intensity, location and/or direction (e.g. moving/rotating loads or uncertain fixed loads). The variability of these loads significantly influences the stress distribution in a structure and should be considered during the design. We propose a locally stress-constrained topology optimization formulation that considers loads with continuously varying direction to ensure structural integrity under more realistic loading conditions. The problem is solved using an Augmented Lagrangian method, and the continuous range of load directions is incorporated through a series of analytic expressions that enables the computation of the worst-case maximum stress over all possible load directions. Variable load intensity is also handled by controlling the magnitude of load basis vectors used to derive the worst-case load. Several two- and three-dimensional examples demonstrate that topology-optimized designs are extremely sensitive to loads that vary in direction. The designs generated by this formulation are safer, more reliable, and more suitable for real applications, because they consider realistic loading conditions.

Stress constrained topology optimization of energy storage flywheels using a specific energy formulation

A variable density, stress-constrained topology optimization approach is used, along with the solid isotropic material with penalization (SIMP) power law and a P-norm aggregated global stress measure to optimize the rotor of a flywheel energy storage systems (FESS). A new specific energy maximization optimization formulation is proposed which eliminates the need to impose an arbitrary volume fraction constraint to the optimization problem as it is done in traditional kinetic energy maximization approaches. The FESS rotors obtained with the specific energy formulation achieved a 15.8% increase in the specific energy compared to the kinetic energy based approach which improved the specific energy by 12.8%. Factors such as the operating speed, maximum stress, rotational symmetry and rotor material also influence the moment of inertia and stress distribution in the flywheel, and the effects of these parameters on the optimal topology and its energy capacity are investigated. At low speeds, the topology is seen to have elongated holes, which become rounded and move away from central shaft as the speed is increased. The reverse effect is seen when the upper limit on the stress constraint is increased. While the choice of rotor material affects the kinetic energy and mass of the flywheel, its optimal topology and specific energy is seen to be nearly identical for rotor materials with the same Eρ ratio, and slightly different for other cases. The size of the circular sections used as the topology design domain is seen to affect the number of spokes in the rotor, but the overall specific energy is unaffected, so manufacturing considerations can be used to choose the rotational symmetry of the rotor. Due to the frequent charge–discharge cycles associated with short term storage, both centrifugal and acceleration forces might play an important role in determining the peak stresses in the flywheel. Therefore, the influence of acceleration loads on the optimal rotor topology is also studied and shown to only have an impact for very large instantaneous accelerations of 5235.9 rad/s2 or higher.

A Topology Optimization Method Based on the Edge-Based Smoothed Finite Element Method

In this paper, we combined the edge-based smoothed finite element method (ES-FEM) with topology optimization. The edge-based gradient smoothing operation was introduced to overcome the accuracy-loss of the classical finite element method raised by the coarse mesh and “overly stiff” phenomenon. By employing the ES-FEM, design variables can be related to the smoothed edge, thus more design variables can be adaptively obtained without additional remeshing. Two classical topology optimization problems were considered, namely compliance minimization and stress-constrained topology optimization. We presented several numerical examples, among which the compliance minimization examples illustrated the potential of the proposed method, and the advantages of applying such a numerical method in topology optimization were demonstrated through the stress-constrained topology optimization.

A maximum-rectifier-function approach to stress-constrained topology optimization

A maximum-rectifier-function approach to stress-constrained topology optimization

Stress-constrained topology optimization using approximate reanalysis with on-the-fly reduced order modeling

Most of the methods used today for handling local stress constraints in topology optimization, fail to directly address the non-self-adjointness of the stress-constrained topology optimization problem. This in turn could drastically raise the computational cost for an already large-scale problem. These problems involve both the equilibrium equations resulting from finite element analysis (FEA) in each iteration, as well as the adjoint equations from the sensitivity analysis of the stress constraints. In this work, we present a paradigm for large-scale stress-constrained topology optimization problems, where we build a multi-grid approach using an on-the-fly Reduced Order Model (ROM) and the p-norm aggregation function, in which the discrete reduced-order basis functions (modes) are adaptively constructed for adjoint problems. In addition to reducing the computational savings due to the ROM, we also address the computational cost of the ROM learning and updating phases. Both reduced-order bases are enriched according to the residual threshold of the corresponding linear systems, and the grid resolution is adaptively selected based on the relative error in approximating the objective function and constraint values during the iteration. The tests on 2D and 3D benchmark problems demonstrate improved performance with acceptable objective and constraint violation errors. Finally, we thoroughly investigate the influence of relevant stress constraint parameters such as the p norm factor, stress penalty factor, and the allowable stress value.

Stress-constrained topology optimization of lattice-like structures using component-wise reduced order models

Lattice-like structures can provide a combination of high stiffness with light weight that is useful in many applications, but a resolved finite element mesh of such structures results in a computationally expensive discretization. This computational expense may be particularly burdensome in many-query applications, such as optimization. We develop a stress-constrained topology optimization method for lattice-like structures that uses component-wise reduced order models as a cheap surrogate, providing accurate computation of stress fields while greatly reducing run time relative to a full order model. We demonstrate the ability of our method to produce large reductions in mass while respecting a constraint on the maximum stress in a pair of test problems. The ROM methodology provides a speedup of about 150x in forward solves compared to full order static condensation and provides a relative error of less than 5% in the relaxed stress.

Direction-oriented stress-constrained topology optimization of orthotropic materials

Efficient optimization of topology and raster angle has shown unprecedented enhancements in the mechanical properties of 3D printed materials. Topology optimization helps reduce the waste of raw material in the fabrication of 3D printed parts, thus decreasing production costs associated with manufacturing lighter structures. Fiber orientation plays an important role in increasing the stiffness of a structure. This paper develops and tests a new method for handling stress constraints in topology and fiber orientation optimization of 3D printed orthotropic structures. The stress constraints are coupled with an objective function that maximizes stiffness. This is accomplished by using the modified solid isotropic material with penalization method with the method of moving asymptotes as the mathematical optimizer. Each element has a fictitious density and an angle as the main design variables. To reduce the number of stress constraints and thus the computational cost, a new clustering strategy is employed in which the highest stresses in the principal material coordinates are grouped separately into two clusters using an adjusted P-norm. A detailed description of the formulation and sensitivity analysis is discussed. While we present an analysis of 2D structures in the numerical examples section, the method can also be used for 3D structures, as the formulation is generic. Our results show that this method can produce efficient structures suitable for 3D printing while thresholding the stresses.

Multimaterial stress-constrained topology optimization with multiple distinct yield criteria

Composite structures offer unique mechanical and physical properties enabled by material heterogeneity. To harness these properties in stress-constrained topology optimization, the incorporation of multiple materials is necessary. Established studies in the field typically assume the same yield criterion for all the candidate materials while vary their stiffness and strengths. To open up the full design capability for composite structures, we propose a novel yield function interpolation scheme that allows for the simultaneous incorporation of distinct yield criteria and material strengths. Built upon this yield function interpolation scheme, we introduce a stress-constrained topology optimization formulation that handles multiple materials with distinct elastic properties, material strengths, and yield criteria simultaneously. We investigate several two-dimensional and three-dimensional design cases with the objective of minimizing the total volume subjected to stress constraints. The optimized composite designs reveal several fundamental advantages enabled by material heterogeneity, including design space enlargement, stress deconcentration effect, and exploitation of tension–compression strength asymmetry. These advantages lead to composite designs with 10−40% reduced minimized volumes as compared to single-material designs and provide new insights for the discovery of more efficient composite structures.