Local Stress Constraints Research Articles

Isogeometric Shape Optimization for Design Dependent Loads

Abstract This paper presents a new isogeometric formulation for shape optimization of structures subjected to design dependent loads. This work considers two types of design dependent loads, namely, surface loads like pressure where the direction and/or magnitude of force changes with the variation of boundary shape, and body forces that depend on the material layout. These problems have been mostly solved by topology optimization methods which are prone to difficulties in determination of the loading surface for pressure loads and problems associated with non-monotonous behavior of compliance and low density regions for body forces. This work uses an isogeometric shape optimization approach where the geometry is defined using non-uniform rational B-spline and the control point coordinates and control weights of the boundary are chosen as design variables. This approach accommodates the design dependent loads easily, in addition to its other advantages like exact geometry representation, local control, fewer design variables, excellent shape sensitivity, efficient mesh refinement strategies, and smooth results that can be integrated with computer aided design. Two classes of optimization problems have been discussed; they are minimum compliance problems subjected to volume constraint and minimum weight problems subjected to local stress constraints. These problems are solved using convex optimization programs. Hence, expressions for full sensitivities are derived which are new for structural shape optimization problems with design dependent loads. Some representative engineering examples are solved and compared with existing literature to demonstrate the application of the proposed method.

Fail-safe optimization of tubular frame structures under stress and eigenfrequency requirements

In conventional fail-safe optimization of frame structures the damage is usually modelled as complete removal of one or more members. We propose and incorporate two additional types of damage models into the fail-safe design problem. The first describes thickness degradation caused e.g. by corrosion. The second describes severe local damage by removal of a part of a member that causes a gap and free ends in the member. The latter damage model can cause undesirable local vibration modes. By combining the two damage models, local thickness degradation in a part of a member can be modelled. The considered design problem minimizes structural mass and includes local stress constraints and limits on eigenfrequencies. Besides the new damage models, a working-set algorithm is applied on the fail-safe optimization problem to reduce the computational cost. Numerical experiments on two-dimensional frame structures illustrate that the working-set algorithm can effectively handle the relatively large number of constraints and damage scenarios in fail-safe optimization.

Local stress constraints in topology optimization of structures subjected to arbitrary dynamic loads: a stress aggregation-free approach

Local stress constraints in topology optimization of structures subjected to arbitrary dynamic loads: a stress aggregation-free approach

Local versus global stress constraint strategies in topology optimization: A comparative study

AbstractStress‐constrained topology optimization requires techniques for handling thousands to millions of stress constraints. This work presents a comprehensive numerical study comparing local and global stress constraint strategies in topology optimization. Four local and four global solution strategies are presented and investigated. The local strategies are based on either the augmented Lagrangian or the pure exterior penalty method, whereas the global strategies are based on the P‐mean aggregation function. Extensive parametric studies are carried out on the L‐shaped design problem to identify the most promising parameters for each solution strategy. It is found that (1) the local strategies are less sensitive to the continuation procedure employed in standard density‐based topology optimization, allowing achievement of better quality results using less iterations when compared with the global strategies; (2) the global strategies become competitive when P values larger than 100 are employed, but for this to be possible a very slow continuation procedure should be used; (3) the local strategies based on the augmented Lagrangian method provide the best compromise between computational cost and performance, being able to achieve optimized topologies at the level of a pure P‐continuation global strategy with , but using less iterations.

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

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

PolyStress: a Matlab implementation for local stress-constrained topology optimization using the augmented Lagrangian method

We present PolyStress, a Matlab implementation for topology optimization with local stress constraints considering linear and material nonlinear problems. The implementation of PolyStress is built upon PolyTop, an educational code for compliance minimization on unstructured polygonal finite elements. To solve the nonlinear elasticity problem, we implement a Newton-Raphson scheme, which can handle nonlinear material models with a given strain energy density function. To solve the stress-constrained problem, we adopt a scheme based on the augmented Lagrangian method, which treats the problem consistently with the local definition of stress without employing traditional constraint aggregation techniques. The paper discusses several theoretical aspects of the stress-constrained problem, including details of the augmented Lagrangian-based approach implemented herein. In addition, the paper presents details of the Matlab implementation of PolyStress, which is provided as electronic supplementary material. We present several numerical examples to demonstrate the capabilities of PolyStress to solve stress-constrained topology optimization problems and to illustrate its modularity to accommodate any nonlinear material model. Six appendices supplement the paper. In particular, the first appendix presents a library of benchmark examples, which are described in detail and can be explored beyond the scope of the present work.

Fail-Safe Topology Optimization of Continuum Structures with Multiple Constraints Based on ICM Method

Traditional topology optimization methods may lead to a great reduction in the redundancy of the optimized structure due to unexpected material removal at the critical components. The local failure in critical components can instantly cause the overall failure in the structure. More and more scholars have taken the fail-safe design into consideration when conducting topology optimization. A lot of good designs have been obtained in their research, though limited regarding minimizing structural compliance (maximizing stiffness) with given amount of material. In terms of practical engineering applications considering fail-safe design, it is more meaningful to seek for the lightweight structure with enough stiffness to resist various component failures and/or to meet multiple design requirements, than the stiffest structure only. Thus, this paper presents a fail-safe topology optimization model for minimizing structural weight with respect to stress and displacement constraints. The optimization problem is solved by utilizing the independent continuous mapping (ICM) method combined with the dual sequence quadratic programming (DSQP) algorithm. Special treatments are applied to the constraints, including converting local stress constraints into a global structural strain energy constraint and expressing the displacement constraint explicitly with approximations. All of the constraints are nondimensionalized to avoid numerical instability caused by great differences in constraint magnitudes. The optimized results exhibit more complex topological configurations and higher redundancy to resist local failures than the traditional optimization designs. This paper also shows how to find the worst failure region, which can be a good reference for designers in engineering.

Three‐dimensional manufacturing tolerant topology optimization with hundreds of millions of local stress constraints

AbstractIn topology optimization, the treatment of stress constraints for very large scale problems (more than 100 million elements and more than 600 million stress constraints) has so far not been tractable due to the failure of robust agglomeration methods, that is, their inability to accurately handle the locality of the stress constraints. This article presents a three‐dimensional design methodology that alleviates this shortcoming using both deterministic and robust problem formulations. The robust formulation, based on the three‐field density projection approach, is extended and proved necessary to handle manufacturing uncertainty in three‐dimensional stress‐constrained problems. Several numerical examples are solved and further postprocessed with body‐fitted meshes using commercial software. The numerical investigations demonstrate that: (1) the employed solution approach based on the augmented Lagrangian method is able to handle very large problems, with hundreds of millions of stress constraints; (2) three‐dimensional stress‐based results are extremely sensitive to slight manufacturing variations; (3) if appropriate interpolation parameters are adopted, voxel‐based (fixed grid) models can be used to compute von Mises stresses with excellent accuracy; and (4) in order to ensure manufacturing tolerance in three‐dimensional stress‐constrained topology optimization, a combination of double filtering and more than three density field realizations may be required.

Topology optimization with local stress constraints: a stress aggregation-free approach

This paper presents a consistent topology optimization formulation for mass minimization with local stress constraints by means of the augmented Lagrangian method. To solve problems with a large number of constraints in an effective way, we modify both the penalty and objective function terms of the augmented Lagrangian function. The modification of the penalty term leads to consistent solutions under mesh refinement and that of the objective function term drives the mass minimization towards black and white solutions. In addition, we introduce a piecewise vanishing constraint, which leads to results that outperform those obtained using relaxed stress constraints. Although maintaining the local nature of stress requires a large number of stress constraints, the formulation presented here requires only one adjoint vector, which results in an efficient sensitivity evaluation. Several 2D and 3D topology optimization problems, each with a large number of local stress constraints, are provided.

Controlling the maximum first principal stress in topology optimization

Previous studies on topology optimization subject to stress constraints usually considered von Mises or Drucker–Prager criterion. In some engineering applications, e.g., the design of concrete structures, the maximum first principal stress (FPS) must be controlled in order to prevent concrete from cracking under tensile stress. This paper presents an effective approach to dealing with this issue. The approach is integrated with the bi-directional evolutionary structural optimization (BESO) technique. The p-norm function is adopted to relax the local stress constraint into a global one. Numerical examples of compliance minimization problems are used to demonstrate the effectiveness of the proposed algorithm. The results show that the optimized design obtained by the method has slightly higher compliance but significantly lower stress level than the solution without considering the FPS constraint. The present methodology will be useful for designing concrete structures.

A novel stress influence function (SIF) methodology for stress-constrained continuum topology optimization

This study presents a new stress influence function (SIF) methodology for continuum topology optimization under consideration of local strength failure. Firstly, the qp-relaxation criterion is involved to circumvent the stress singularity. To deal with the large-scale stress constraints in topology optimization, the local stress constraint is reflected in the objective along with the material volume by multiplication, and the weight of stress is characterized by stress influence function. Meanwhile, three types of stress influence functions are proposed for comparison. By means of the study on the characteristic of high-stress elements, the rationality of the SIF methodology is illustrated, in which the proposed method may achieve the full-stress state of high-stress element. Numerical examples are given to demonstrate the applicability and validity of the proposed methodology. It is shown that the proposed methodology can obtain reasonable results. Consequently, the proposed SIF methodology provides a novel strategy with high computational efficiency for topology optimization considering local strength failure.

A unified approach for topology optimization with local stress constraints considering various failure criteria: von Mises, Drucker-Prager, Tresca, Mohr-Coulomb, Bresler- Pister and Willam-Warnke.

An interesting, yet challenging problem in topology optimization consists of finding the lightest structure that is able to withstand a given set of applied loads without experiencing local material failure. Most studies consider material failure via the von Mises criterion, which is designed for ductile materials. To extend the range of applications to structures made of a variety of different materials, we introduce a unified yield function that is able to represent several classical failure criteria including von Mises, Drucker-Prager, Tresca, Mohr-Coulomb, Bresler-Pister and Willam-Warnke, and use it to solve topology optimization problems with local stress constraints. The unified yield function not only represents the classical criteria, but also provides a smooth representation of the Tresca and the Mohr-Coulomb criteria-an attribute that is desired when using gradient-based optimization algorithms. The present framework has been built so that it can be extended to failure criteria other than the ones addressed in this investigation. We present numerical examples to illustrate how the unified yield function can be used to obtain different designs, under prescribed loading or design-dependent loading (e.g. self-weight), depending on the chosen failure criterion.

Stress-constrained level set topology optimization for compliant mechanisms

This work presents a level set-based topology optimization procedure to design compliant mechanisms subject to local stress constraints. A classical technical difficulty related to the design of compliant mechanisms is the emergence of non-realistic flexible joints (hinges) subject to unfeasible strain and stresses. A natural way to overcome this occurrence is the inclusion of stress constraints within the formulation. This approach brings, on the other hand, particular difficulties associated with the local nature of stress constraints. To deal with this issue, stress constraints are included in the present work using an augmented Lagrangian mathematical programming technique. A level set approach defines the design of the mechanical part and a reaction–diffusion equation is used to update the level set function towards the optimized state. A natural extension of the velocity field provided for the reaction–diffusion equation allows for hole nucleation during the design updating. The joining of an augmented Lagrangian approach and level set implicit boundaries to handle local stress constraints in order to obtain hinge-free compliant mechanisms is the novelty of this paper. Several two-dimensional numerical examples show the efficacy of the proposed procedure. In particular, an example analyzes the influence of the weak material in the design of compliant mechanisms. Level set designs enable that final solutions are validated in commercial finite element codes without any post-processing procedure.

Uncertainty-oriented topology optimization of interval parametric structures with local stress and displacement reliability constraints

This paper presents the interval reliability-based topology optimization (IRBTO) framework and an effective solution procedure for interval parametric structures to achieve optimal material configurations under consideration of local stiffness and strength failure. Firstly, ε-relaxed stress criterion and global stress aggregation approach are involved to circumvent the stress singularity and multi-constrained problems. Combined the orthogonal polynomial expansion with the set allocation theorem, an interval dimension-by-dimension method (IDDM) is proposed to determine feasible bounds of structural responses under unknown-but-bounded load and material uncertainties. The interval reliability (IR) is then applied to handle the limited reliability constraints of concerned displacements and the global stress measure. Meanwhile, the adjoint-vector based sensitivity analysis of presented IR indexes to design variables is further discussed to avoid expensive computational cost from the large-scale nature of variable updating. The usage, rationality, and superiority of the developed methodology are demonstrated by several case applications.

Free material optimization with local stress constraints for laminated structures

ABSTRACTIn Free Material Optimization (FMO), the design variable is the elastic material tensor, which is allowed to change its values freely over the design domain resulting in both optimal material properties and optimal material distribution. Models are mostly available for two- and three-dimensional continuum structures. Recently, these models have been extended to laminated plates and shells. The goal of this article is to introduce constraints on local stresses to these new models. The associated optimization problems are highly nonlinear semidefinite programming problems that are known to be challenging for numerical tractability, and are solved by a primal–dual interior point method previously proposed for FMO problems without stress constraints. The algorithm utilizes the FMO problem structure, wherein the matrix inequality constraints involve small many block matrices. Several numerical experiments demonstrate the stress constrained models and the capability of the method to obtain solutions to these problems within a modest number of iterations.

Non-probabilistic robust continuum topology optimization with stress constraints

This paper proposes a non-probabilistic robust design approach, based on optimization with anti-optimization, to handle unknown-but-bounded loading uncertainties in stress-constrained topology optimization. The objective of the proposed topology optimization problem is to find the lightest structure that respects the worst possible scenario of local stress constraints, given predefined bounds on magnitudes and directions of applied loads. A solution procedure based on the augmented Lagrangian method is proposed, where worst-case local stress constraints are handled without employing aggregation techniques. Results are post-processed, demonstrating that maximum stress of robust solutions is almost insensitive with respect to changes in loading scenarios. Numerical examples also demonstrate that obtained robust solutions satisfy the stress failure criterion for any load condition inside the predefined range of unknown-but-bounded uncertainties in applied loads.

Stress-constrained level set topology optimization for design-dependent pressure load problems

This work presents a level set framework to solve topology optimization problems subject to local stress constraints considering design-dependent pressure loads. Two technical difficulties are related to this problem. The first one is the local nature of stresses. To deal with this issue, stress constraints are included to the problem by means of an Augmented Lagrangian scheme. The second is the adequate association between the moving boundary and the pressure acting on it. This difficulty is easily overcome by the level set method that allows for a clear tracking of the boundary along the optimization process. In the present approach, a reaction–diffusion equation substitutes the classical Hamilton–Jacobi equation to control the level set evolution. This choice has the advantage of allowing the nucleation of holes inside the domain and the elimination of the undesirable level set reinitializations. In addition, the optimization algorithm allows the rupture of loading boundaries, that is, the crossing of the pressured (wet) boundary with the traction free boundary is not avoided. This gives more freedom to the algorithm for topological changes. In order to validate the proposed scheme against stress concentrations, all numerical examples are performed on constrained domains containing singularities. Moreover, optimized designs obtained from stress-constrained and compliance problems are compared.

STRESS-CONSTRAINED TOPOLOGY OPTIMIZATION BASED ON IMPROVED BI-DIRECTIONAL EVOLUTIONARY OPTIMIZATION METHOD

It is necessary to limit maximum nominal stress for engineering structural design generally, so as to avoid that the failure of fracture or fatigue occurs. Topology optimization approach is a feasible strategy. The conventional bi-evolutionary structural optimization (BESO) method cannot effectively address the topology optimization problem with stress constraint. To overcome this limitation, this paper suggests an improved BESO method for stress-constrained topology optimization, in which the minimum compliance design problem with volume and stress constraints is considered. A global stress measure based on the K-S function is introduced to reduce the computational cost associated with the local stress constraint. Meanwhile, the stress constraint function is added to the objective function by using the Lagrange multiplier method. Moreover, the appropriate value of the Lagrangian multiplier is then determined by a bisection method so that the stress constraint is satisfied. The model of BESO method for solving stress-constrained topology optimization and its sensitivity analysis are detailed. Finally, three typical topology optimization examples are performed to demonstrate the validity of the present method, in which the stress constrained designs are compared with the traditional stiffness based designs to illustrate the merit of considering stress constraint. The optimized results indicate that the improved BESO method, as a robust algorithm with stable iterative history, achieves an ambiguous topology with clearly defined boundaries, and realizes a design that effectively reduces stress concentration effect at the critical stress areas.

Stress-constrained topology optimization based on maximum stress measures

This paper proposes two effective constraint schemes to address the stress-constrained topology optimization of continuum structures. By considering the maximum stress measure in the global and local forms, respectively, the STM (stability transformation method)-based stress correction scheme and the violated set enhanced stress measure are developed to tackle the challenging issues from numerous local stress constraints and highly nonlinear stress behavior. Particularly, a stress aggregation function is involved in the design sensitivity analysis. Moreover, the nodal variable based SIMP method and adjoint sensitivity analysis are employed to solve the optimum topological design problems with two different optimization formulations. Finally, several representative examples demonstrate the validity of the present approach. It is also indicated that the numerical performance of the stress aggregation function is closely related to the problem formulation of topology optimization. The STM-based stress correction scheme is appropriate to the material volume minimization design, while the violated set enhanced stress measure is suitable for the mean compliance minimization design. Meanwhile, the proposed optimization approach can handle the stress-constrained topology optimization with easy implementation, low computational cost and stable convergence.

On the influence of local and global stress constraint and filtering radius on the design of hinge-free compliant mechanisms

Distributed compliant mechanisms are components that use elastic strain to obtain a desired kinematic behavior. Compliant mechanisms obtained via topology optimization using the standard approach of minimizing/maximizing the output displacement with a spring at the output port, representing the stiffness of the external medium, usually contain one-node connected hinges. Those hinges are undesired since an ideal compliant mechanism should be a continuous part. This work compares the use of two strategies for stress constrained problems: local and global stress constraints, and analyses their influence in eliminating the one-node connected hinges. Also, the influence of spatial filtering in eliminating the hinges is studied. An Augmented Lagrangian formulation is used to couple the objective function and constraints, and the resulting optimization problem is solved by using an algorithm based on the classical optimality criteria approach. Two compliant mechanisms problems are studied by varying the stress limit and filtering radius. It is observed that a proper combination of filtering radius and stress limit can eliminate one-node connected hinges.