Constrained Topology Optimization Research Articles

Nonlinear fatigue damage constrained topology optimization

In engineering applications, plenty of components are subjected to variable-amplitude cyclic loading, resulting in fatigue damage, which is one of the main forms of structural damage. While the linear damage rule has long served as a fundamental approach, its limitations necessitate advancements for more accurate fatigue life predictions. Hence, this paper introduces a pioneering method termed nonlinear fatigue damage constrained topology optimization (NFDCTO). This method integrates several key components: the rainflow counting method to evaluate the non-proportional cyclic load levels, Basquin's equation to describe the S-N curve, and Morrow's plastic work interaction rule to calculate the nonlinear cumulative damage of the structure. Consequently, we establish a mathematical model for the NFDCTO method based on these components. The incorporation of penalized damage aggregation with the P-Normal function is used to address significant constraints and singularity challenges. Furthermore, by employing the adjoint method, we derive sensitivity equations for both the objective function and fatigue constraint function concerning the design variables. Subsequently, the superiority of the NFDCTO method over traditional linear fatigue damage method were verified applying 2D l-shaped beam and simply supported structures as examples. Concurrently, bridge structures were employed to investigate the effect of the sensitivity index of the material to the history of variable amplitude stresses on the optimization results. In addition, the influence of the fatigue penalty factor on the topology optimization results was assessed using a 2D cantilever beam. Finaly, we verify the effectiveness and feasibility of the NFDCTO method in addressing fatigue optimization challenges for 3D structures utilizing two 3D examples.

A decoupled probabilistic constrained topology optimization method based on the constraint shift

AbstractTopology optimization (TO) has recently emerged as an advanced design method. To ensure practical reliability in the design process, it is imperative to incorporate considerations of uncertainty. Consequently, performing reliability analysis (RA) during the design phase becomes necessary. However, RA itself constitutes an optimization problem. Combining these two optimization problems can result in inefficiency. To address this challenge, we propose a decoupled approach that integrates deterministic topology optimization (DTO) and RA cycles. The reliability‐based stress‐constrained TO (RBSCTO) problem is considered in this paper. The DTO constraint is derived based on shifting vectors derived from the previous cycle's RA outcomes, enabling low‐reliability constraint shift towards the feasible direction. The DTO is solved based on solid‐isotropic‐material‐with‐penalization (SIMP) and augmented Lagrangian method. Meanwhile, the optimization problem in RA is addressed using finite differences and the interior point method. To reduce the errors resulting from linear approximation and optimization in RA when the target reliability is very low, an outlier handling method is employed. Meantime, we utilize a probabilistic neural network to enhance the efficiency of reliability assessment. Comparative studies against traditional methods across four RBSCTO tasks are demonstrated to validate its effectiveness. Monte Carlo simulations are used to validate the reliability of results.

Fatigue constrained topology optimization for the jacket support structure of offshore wind turbine under the dynamic load

This paper proposes a topology optimization method considering fatigue constraints for the jacket support structure of offshore wind turbine. Jacket support structure is an important component and supports the whole wind turbine weight under the dynamic environmental loads, which would induce serious fatigue problems. To obtain the optimal structural layout, the topology optimization method considering fatigue constrained is introduced for the jacket support structure under the dynamic load. The fatigue lifetime is constrained base on the Miner's cumulative damage rule. The design sensitivity of the objective and constraint functions are evaluated analytically, while the equivalent static load approach is applied. Some typical topology optimization problems are introduced to verify the effectiveness of the proposed method, and this method can obtain the appropriate layouts with stable convergence. A case study on the OC4 reference jacket indicates that the method can achieve a reasonable layout structure under the dynamic load, which meets the fatigue constraint. And the final design is reconstructed based on the topology optimization layout. Finally, a lighter jacket structure has good performance on the ultimate bearing capacity, eigenfrequency, and buckling, which is suited for long operation life.

Strength constrained topology optimization of hyperealstic structures with large deformation-induced frictionless contact

This paper developed a strength constrained topology optimization method for hyperelastic structures with large deformation-induced frictionless contact. The Neo-Hookean hyperelastic constitutive equation is adopted as the material model that incorporates both material and geometric nonlinearity. The large deforming contact is described by the node-to-segment algorithm. The topology optimization model is developed under the SIMP framework. Given the SIMP-related fictitious domain, we develop a nodal variable-based interpolation scheme to build smooth correlation between the contact force and nodal density variables, thus enabling the smooth transition and variation of contact conditions. Moreover, to guarantee the structure strength, local strain energy constraints are formulated and aggregated by the P-norm function. The details of sensitivity analysis are explicitly explained. At last, several benchmark examples are conducted that validate the proposed method.

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.

Environmental impact of process constrained topology optimization design on automotive component’ life

The components’ lightweighting has been pursued, especially in the transport industry, for greenhouse gas reduction. Topology optimization, being able to allocate the material within a provided design space, is a mathematical method that can support the design of lightweight components, preserving, at the same time, their mechanical performances. In this paper, a standard shape of a component, specifically an automotive bracket, was topology optimized by estimating the impacts of the new designs from an eco-friendly point of view. A subtractive, an additive and a casting manufacturing process were considered as possible manufacturing routes achieving an optimized geometry of the component for each of them. The topology optimizations were performed considering each processes’ peculiarities, introduced as constraints. Same strength for a given set of loads and boundary conditions was the target of each analysis. The component’s lightening can be considered environmentally friendly just after assessing the impacts associated with all the stages of the product’ life cycle. Indeed, each phase of the product’ life cycle can be affected, differently, by the performed topology optimization taking into account the peculiarities of the employed manufacturing process. The overall considerations on the most environmentally safe strategies can, therefore, change according to the specificities of the optimized shapes. The topology optimization showed its utmost potentiality, from a sustainable point of view, if applied to additive manufacturing techniques for the advantages arisen by the capability to manufacture complex shapes benefiting also of reduction time process owing to less material to be deposited.

Efficient buckling constrained topology optimization using reduced order modeling

We present an efficient computational approach to continuum topology optimization with linearized buckling constraints, using Reduced Order Models (ROM). A reanalysis technique is employed to generate basis vectors that subsequently are used to significantly reduce the size of the generalized eigenvalue problems. We demonstrate the efficacy of this approach by optimizing for stiffness with buckling constraints and show results for several test cases. Based on our findings, we conclude that the ROM can potentially save significant computational effort without compromising the quality of the results.

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 constrained topology optimization methodology based on budget constrained min-cut

PurposeThe purpose of this paper is to propose a novel methodology based on budget constrained Min-Cut theorem to solve constrained topology optimization (TO).Design/methodology/approachThis paper establishes a weighted network with budget, which is derived from the sensitivity with respect to the constraint function. The total budget carried by the topology evaluates the extent to which the constraint is satisfied. By finding the Min-Cut under budget constraint in each step, the proposed method is able to solve constrained TO problem.FindingsThe results obtained from a magnetic actuator including a yoke, a coil and an armature have demonstrated that the proposed method is effective to solve constrained TO problem.Originality/valueA novel methodology based on budget constrained Min-Cut is proposed to solve constrained TO problem.

Stress Constrained Structure Optimization for the High Power Bus Bar of the Electromagnetic Launcher

The optimization design of the breech bus bar, especially lightweight design, has great significance for the large-caliber electromagnetic launcher (EML). In this article, based on the global stress constrained topology optimization technology and hierarchical optimization technology, a lightweight design model of the bus bar is established. The optimization algorithm adopts the method of moving asymptotes (MMA). The calculation model is the half of 840 mm <inline-formula> <tex-math notation="LaTeX">$\times230$ </tex-math></inline-formula> mm <inline-formula> <tex-math notation="LaTeX">$\times20$ </tex-math></inline-formula> mm flat-plate bus bar fed through 16 cables. The results show that the area of the optimum structure can be reduced to about 59.2&#x0025; of the initial structure. It indicates that these methods can effectively deal with the bus bar lightweight design. Besides, the influence of the structure change on the electromagnetic field is analyzed. Finally, the optimum structures are given when the bus bar plate thicknesses are 25, 30, 35, and 40 mm. If pursuing the space utilization efficiency and safety, it is not recommended to use the optimum structure of the bus bar with a thickness beyond 30 mm. It is hoped that these can provide meaningful guidance for the design of the bus bar.

Stress constrained topology optimization simultaneously considering the uncertainty of load positions

AbstractThis study presents a topological optimization method simultaneously considering the stress constraint and the uncertainty of the load positions for practical applications. The phase field design method is used to derive the topologically optimal structure shape. The stress penalization function, which makes intermediate design variable values disproportionately expensive, is employed to ensure numerical stability and avoid the singularity during the optimization process. The adaptive mesh refinement and a modified P‐norm stress with correction factor are also employed to reduce the computational cost. As numerical examples, cantilever beam, L‐shaped, and MBB‐beam are presented to verify the proposed design method. The open‐source code FreeFEM++ is used for the finite element analysis and the design process.

Structural stability and artificial buckling modes in topology optimization

This paper demonstrates how a strain energy transition approach can be used to remove artificial buckling modes that often occur in stability constrained topology optimization problems. To simulate the structural response, a nonlinear large deformation hyperelastic simulation is performed, wherein the fundamental load path is traversed using Newton’s method and the critical buckling load levels are estimated by an eigenvalue analysis. The goal of the optimization is to minimize displacement, subject to constraints on the lowest critical buckling loads and maximum volume. The topology optimization problem is regularized via the Helmholtz PDE-filter and the method of moving asymptotes is used to update the design. The stability and sensitivity analyses are outlined in detail. The effectiveness of the energy transition scheme is demonstrated in numerical examples.

On material selection for topology optimized compliant mechanisms

Although compliant mechanism design is a thoroughly studied field, surprisingly little information can be found in literature regarding selection of optimal materials. This paper is intended to fill this gap. Density-based, geometrically robust, stress constrained topology optimization based on a total Lagrangian FEM formulation is used for investigation of a compliant inverter and a compliant gripper example. Changes in handling of the projection parameters and the stress constraints are proposed for improved algorithmic stability and accurate representation of the stresses and material stiffness in topology optimization. Large-scale optimization studies are carried out, varying elastic modulus and allowable stress in a wide range. A comparison with an Ashby chart shows the characteristics of the best suited material. It is shown, that an optimal Young’s modulus and a minimum required material strength (depending on the modulus) can be identified from the results. Altering critical optimization parameters, e.g. allowable volume fraction and the minimum length scale, their influence on the optimal material choice is investigated. Guidelines for compliant mechanism designers for efficient selection of suited materials are developed.

A discrete adjoint based level set topology optimization method for stress constraints

This paper proposes a new methodology for computing boundary sensitivities in level set topology optimization using the discrete adjoint method. The adjoint equations are constructed using the discretized governing field equations. The objective function is differentiated with respect to the boundary point movement for computing boundary sensitivities using the discrete adjoint equations. For this purpose, we present a novel approach where we perturb the boundary implicitly by locally modifying the level set function around a given boundary point. These local perturbations are combined with the derivatives of the objective function with respect to the volume fractions of individual elements to compute boundary sensitivities. This enables the circumvention of smoothing or interpolation methods typically used in level set topology optimization to compute sensitivities; and improves the accuracy of the sensitivities and the convergence characteristics. We demonstrate the effectiveness of our method in the context of stress minimization and stress constrained topology optimization problems for orthogonal bracket design under multiple load cases.

Topology optimization of elasto-plastic structures under reliability constraints: A first order approach

The objective of this study is to propose a relatively simple and efficient method for reliability based topology optimization for structures made of elasto-plastic material. The process of determining the optimal topology of elasto-perfectly plastic structures is associated with the removal of material from the structure. Such a process leads to weakening of structural strength and stiffness causing at the same time increase the likelihood of structural failure. An important aspect of engineering design is to track this probability during the optimization process and not allow the structure safety to exceed a certain level specified by the designer. The purpose of this work is to combine the previously developed yield-limited topology optimization method with reliability analysis using first order approach. Effectiveness of the proposed methodology is demonstrated on benchmark problems proposed by Rozvany and Maute, and the elasto-plastic topology design of L-shape structure which is frequently used in different approaches for stress constrained topology optimization.

Eigenfrequency constrained topology optimization of finite strain hyperelastic structures

This paper incorporates hyperelastic materials, nonlinear kinematics, and preloads in eigenfrequency constrained density–based topology optimization. The formulation allows for initial finite deformations and subsequent small harmonic oscillations. The optimization problem is solved by the method of moving asymptotes, and the gradients are calculated using the adjoint method. Both simple and degenerate eigenfrequencies are considered in the sensitivity analysis. A well-posed topology optimization problem is formulated by filtering the volume fraction field. Numerical issues associated with excessive distortion and spurious eigenmodes in void regions are reduced by removing low volume fraction elements. The optimization objective is to maximize stiffness subject to a lower bound on the fundamental eigenfrequency. Numerical examples show that the eigenfrequencies drastically change with the load magnitude, and that the optimization is able to produce designs with the desired fundamental eigenfrequency.

Piecewise length scale control for topology optimization with an irregular design domain

This paper presents a piecewise length scale control method for level set topology optimization. Different from the existing methods, where a unique lower limit or upper limit was applied to the entire design domain, this new method decomposes the topological design into pieces of strip-like components based on the connectivity condition, and then, the lower or upper limit for length scale control could be piecewise and dynamically defined based on each component’s real-time status (such as position, orientation, or dimension). Specifically, a sub-algorithm of structural skeleton identification and segmentation is developed to decompose the structure and its skeleton. Then, a skeleton segment-based length scale control method is developed to achieve the piecewise length scale control effect. In addition, a special type of length scale constrained topology optimization problem that involves an irregular design domain will be addressed, wherein the complex design domain plus the length scale constraint may make the conventional length scale control methods fail to work. Effectiveness of the proposed method will be proved through a few numerical examples.

Manufacturing cost constrained topology optimization for additive manufacturing

This paper presents a manufacturing cost constrained topology optimization algorithm considering the laser powder bed additive manufacturing process. Topology optimization for additive manufacturing was recently extensively studied, and many related topics have been addressed. However, metal additive manufacturing is an expensive process, and the high manufacturing cost severely hinders the widespread use of this technology. Therefore, the proposed algorithm in this research would provide an opportunity to balance the manufacturing cost while pursuing the superior structural performance through topology optimization. Technically, the additive manufacturing cost model for laser powder bed-based process is established in this paper and real data is collected to support this model. Then, this cost model is transformed into a level set function-based expression, which is integrated into the level set topology optimization problem as a constraint. Therefore, by properly developing the sensitivity result, the metallic additive manufacturing part can be optimized with strictly constrained manufacturing cost. Effectiveness of the proposed algorithm is proved by numerical design examples.

Constrained Topology Optimization of Dynamic Mechanical Systems

AbstractWe present a topology optimization procedure for flexible multibody systems, which is subjected to time‐ and space‐dependent constraints. The flexible bodies are parameterized using the solid isotropic material with penalization approach. For the efficient solution of the optimization problem, the pointwise time‐dependent constraints are discretized and aggregated into a single constraint using the Kreisselmeier‐Steinhauser function. The procedure is tested by optimizing the mass of a flexible piston rod, while the deformation within the design domain is constrained.

Re-Thinking Design Methodology for Castings: 3D Sand-Printing and Topology Optimization

Additive manufacturing of sand molds and cores for metal castings, often called 3D sand-printing (3DSP), is an efficient “freeform” fabrication process that enables rapid production of sand metal castings. The ability to create highly complex molds and cores for advanced metal casting geometries via 3DSP provides unparalleled design freedom, particularly for low-volume production. However, there is a need to thoroughly understand the opportunities and restrictions of 3DSP in a systematic approach similar to well-established design guidelines for traditional sand casting. This study presents a knowledge-based design framework for 3DSP with the goal of developing new part design guidelines under such 3DSP framework. In particular, constrained topology optimization approach for the part redesign is developed for 3DSP. The presented design framework is compared with traditional sand-casting rules and validated through a case study on an existing metal component. Advantages of the developed 3DSP design framework are illustrated and validated through a case study where a 30% improvement in factor of safety and a 50% reduction in weight of a mechanical part is achieved. Other advantages, such as reduced lead time and production cost, are also observed. This research provides the first known investigation into systematic implementation of simultaneous constraints of 3DSP sand-casting rules mechanical strength through the integration of topology optimization and novel design rules to castings via 3D-printed molds. 3DSP also eliminates multiple design constraints in conventional mold-making and core-box fabrication. Findings from this study can be applied for a wide range of alloy systems, part geometries and loading conditions for sand castings in industrial applications.