Topology Optimization Formulation Research Articles

Pure-displacement formulation and bulk modulus propagation for topology optimization with pressure loads

This work proposes a pure-displacement formulation for topology optimization with pressure loads and the use of a continuous boundary propagation technique. This technique allows controlling the non-solid (fluid or void) constitutive behavior by disseminating the desired bulk modulus to holes in the solid domain. The novelty of the proposed approach is to enable the distribution of solid, fluid, and void simultaneously with one design variable field. A density-based topology optimization approach is considered, and a gradient-based interior point algorithm is used. The linear elasticity equilibrium equations are discretized by the finite element method and solved with a sparse direct solver. The sensitivities are calculated by the discrete adjoint method with automatic differentiation. The optimization problems encompass compliance minimization (in 2D and 3D) and a compliant mechanism design to maximize the output displacement. Three examples evaluate the proposed formulation: the internally-pressurized lid, the piston head, and the inverter-compliant mechanism. The results show that different designs are obtained by considering different constitutive behavior for the non-solid phase.

Influence of topology optimization parameters on the mechanical response of an additively manufactured test structure

Topology Optimization (TO) determines a material distribution within a domain under given conditions and design constraints, and generally generates complex geometries as a result. Complementary to TO, Additive Manufacturing (AM) offers the ability to fabricate complex geometries which may be difficult to manufacture using traditional techniques such as milling. AM has been used in multiple industries including the medical devices area. Hence, TO may be used to create patient-matched devices where the mechanical response is catered to a particular patient. However, during a medical device regulatory 510(k) pathway, demonstrating that worst-cases are known and tested is critical to the review process. Using TO and AM to predict worst-case designs for subsequent performance testing may be challenging and does not appear to have been thoroughly explored. Investigating the effects of TO input parameters when AM is employed may be the first step in determining the feasibility of predicting these worst-cases.In this paper, the effect of selected TO parameters on the resulting mechanical response and geometries of an AM pipe flange structure are investigated. Four different input parameters were chosen in the TO formulation: (1) penalty factor, (2) volume fraction, (3) element size, and (4) density threshold. Topology optimized designs were fabricated using PA2200 polyamide and the mechanical responses (reaction force, stress, and strain) were observed through experiments (universal testing machine and 3D Digital Image Correlation) and in silico environments (finite element analysis). In addition, 3D scanning and mass measurement were performed to inspect the geometric fidelity of the AM structures. A sensitivity analysis is performed to examine the effect of each TO parameters. The sensitivity analysis revealed mechanical responses can have non-monotonic and non-linear relationships between each tested parameter.

Level-set topology optimization for Ductile and Brittle fracture resistance using the phase-field method

This work presents a rigorous mathematical formulation for topology optimization of a macro structure undergoing ductile failure. The prediction of ductile solid materials which exhibit dominant plastic deformation is an intriguingly challenging task and plays an extremely important role in various engineering applications. Here, we rely on the phase-field approach to fracture which is a widely adopted framework for modeling and computing the fracture failure phenomena in solids. The first objective is to optimize the topology of the structure in order to minimize its mass, while accounting for structural damage. To do so, the topological phase transition function (between solid and void phases) is introduced, thus resulting in an extension of all the governing equations. Our second objective is to additionally enhance the fracture resistance of the structure. Accordingly, two different formulations are proposed. One requires only the residual force vector of the deformation field as a constraint, while in the second formulation, the residual force vector of the deformation and phase-field fracture simultaneously have been imposed. An incremental minimization principles for a class of gradient-type dissipative materials are used to derive the governing equations. Thereafter, to obtain optimal topology to enhance the structural resistance due to fracture, the level-set-based formulation is formulated. The level-set-based topology optimization is employed to seek an optimal layout with smooth and clear boundaries. Sensitivities are derived using the analytical gradient-based adjoint method to update the level-set surface for both formulations. Here, the evolution of the level-set surface is realized by the reaction–diffusion equation to maximize the strain energy of the structure while a certain volume of design domain is prescribed. Several three-dimensional numerical examples are presented to substantiate our algorithmic developments.

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.

Topology optimization for fluid flow devices modeled through the Multiple Reference Frame approach

Fluid flow devices have been algorithmically designed through optimization methods, and this task is relatively complex for flow machines. Particularly, one recent development is the use of the topology optimization method for this task, which is capable of achieving possibly non-intuitive geometries and even auxiliary structures, such as splitter blades without ever imposing them as a condition for the design. The design of the rotating part of the rotor has already been previously considered. However, in the presence of stationary parts surrounding the rotor, such as diffuser blades, the fluid flow behavior may change, meaning that it may be interesting to consider this different dynamic in the design. One possible approach for this is the Multiple Reference Frame approach, which considers the fluid flow dynamics inside the rotor being computed in the rotating reference frame, whilst the fluid flow dynamics outside the rotor is being computed in the stationary reference frame. In particular, this implies the need for some changes in the topology optimization formulation, also impacting the choice of objective functions. Therefore, here the topology optimization method is formulated for the Multiple Reference Frame approach, and a new combination for the multi-objective function is proposed. The discrete design variable configuration from the Topology Optimization of Binary Structures approach is adopted with some adjustments to account for the Multiple Reference Frame approach. The fluid flow is chosen to be laminar or turbulent. Numerical examples are presented in 2D for the rotor-diffuser in order to consider some aspects that affect the overall topology optimization design.

On topology optimization with gradient-enhanced damage: An alternative formulation based on linear physics

Numerous topology optimization formulations have been proposed in order to enhance structural resistance to material failure. A clear line can often be drawn between those methods which attempt to constrain local failure criteria and those that explicitly model the failure physics during the optimization process. In this work the former method is extended in a manner inspired by the mathematical form of typical gradient-enhanced damage models. Importantly, the proposed formulation relies on linear physics during the optimization procedure, which greatly increases its speed and robustness, both of which are essential in industrial applications where large numerical models are typically used. The size effect introduced by using such a numerical model is further investigated and select observations are provided, such as spurious “fin-like” patterns that emerge depending on the type of structure and loading conditions. Finally, the load capacity is verified for each optimized design through a post-optimization verification procedure which is unaffected by the density-based design parameterization and associated material interpolation schemes.

Static reinforcement and vibration reduction of structures using topology optimization

This paper presents a topology optimization formulation based on the Solid Isotropic Material with Penalization (SIMP) method and solved by the Modified Optimality Criteria (MOC) algorithm. It addresses mechanical design problems such as structural reinforcement adding elastic material or vibration reduction using viscoelastic layers. The aim is thus to attach on a pre-existing given structure a design domain in order to improve the behavior of this elastic structure, according to an objective function. This can be useful when one wants to use, for example, additive manufacturing to reinforce a pre-existing structure or to maximize structural damping. Two objective functions are tested in both linear statics and dynamics: a compliance based objective function and a displacement based one. In the dynamic case, written in the frequency domain, the two proposed objective functions include the viscoelastic material model (a Zener fractional derivative one) used to fill the design domain. The displacement criteria is developed using a general formula able to take into account as many degree of freedom as necessary. Finally, some applications based on beams and CubeSat-like structures are shown in this article. The proposed examples show that in both statics and dynamics, the optimization of a restrained design domain attached to an existing structure can improve its behavior: stiffness improvement or vibration reduction.

A topology optimization methodology for the offshore wind turbine jacket structure in the concept phase

This paper proposes a topology optimization methodology for offshore wind turbine (OWT) jacket structure design in the concept phase. First of all, modal analysis and load calculation of a 5 MW reference OWT are performed. On the basis of finite element analysis, the displacement and von Mises stress of multi-jacket support structures in various ultimate loading conditions are explored. Furthermore, structural topology optimization formulation aims to minimize a weighted normalized objective for the OWT jacket structure. Therefore, several optimum structures are obtained by adjusting weighting coefficients. For the coupling effect between the structure and applied load, load recalculation, displacement, and stress reanalysis are performed on the jacket structure based on the topological configuration under a specific weighting factor. Comparisons are made between the reference structure and optimized structure. Without the addition mass, the fundamental natural frequency of the optimized structure is marginally increased, but the maximum displacement and von Mises stress are drastically lowered, indicating that the optimized structure outperforms its reference counterpart. These findings conclusively demonstrate that the proposed TO technique for the design of OWT jacket structures is feasible and preferable.

Two-scale topology optimization with heterogeneous mesostructures based on a local volume constraint

A new approach to produce optimal porous mesostructures and at the same time optimizing the macro structure subject to a compliance cost functional is presented. It is based on a phase-field formulation of topology optimization and uses a local volume constraint (LVC). The main novelty is that the radius of the LVC may depend both on space and a local stress measure. This allows for creating optimal topologies with heterogeneous mesostructures enforcing any desired spatial grading and accommodating stress concentrations by stress dependent pore size. The resulting optimal control problem is analysed mathematically, numerical results show its versatility in creating optimal macroscopic designs with tailored mesostructures.

Influence of integer design variables in topology optimization of incompressible turbulent flow

• Optimize incompressible turbulent flow at bidimensional and tridimensional domains employing the continuous adjoint approach • Analyze the influence of integer design variables in topology optimization of incompressible turbulent fluids • Develop the adjoint near wall distance calculation to treat the adjoint spalart allmaras turbulence model Turbulent fluid flow is a challenging physical problem to be addressed with topology optimization due to the vortex generation and the near wall distance calculation that difficult the calibration of topology optimization parameters when continuous design variables based-optimizers are used. The intrinsic nature of topology optimization inverse permeability interpolation clashes with the accuracy requirements from turbulent fluid flow dynamics. To solve the mentioned challenge, the present paper investigates the influence of an integer continuous variable-based-optimizer, i.e. a [0, 1] binary behaviour, that avoids grey regions during the optimization process. Its influence on the near wall distance calculation is analyzed as it affects the permeability of the fluid domain. The turbulence phenomenon is treated via the Spalart-Allmaras model and its near wall distance calculation is approached by the Hamilton-Jacobi formulation. Permeability independence on the topology optimization formulation constraints is proposed firstly at the continuous adjoint approach to avoid miscalculations in the fluid cells. The finite volume method is employed to solve the Reynolds Averaged Navier-Stokes governing equations and 2D and 3D tests are developed, where discussions on the solid-fluid boundaries definition, the material penalization dependence, objective function comparisons and accuracy of the CFD analysis and the wall distance calculation are presented.

MILP-based discrete sizing and topology optimization of truss structures: new formulation and benchmarking

Discrete sizing and topology optimization of truss structures subject to stress and displacement constraints has been formulated as a Mixed-Integer Linear Programming (MILP) problem. The computation time to solve a MILP problem to global optimality via a branch-and-cut solver highly depends on the problem size, the choice of design variables, and the quality of optimization constraint formulations. This paper presents a new formulation for discrete sizing and topology optimization of truss structures, which is benchmarked against two well-known existing formulations. Benchmarking is carried out through case studies to evaluate the influence of the number of structural members, candidate cross sections, load cases, and design constraints (e.g., stress and displacement limits) on computational performance. Results show that one of the existing formulations performs significantly worse than all other formulations. In most cases, the new formulation proposed in this work performs best to obtain near-optimal solutions and verify global optimality in the shortest computation time.

Defect-Driven topology optimization for fatigue design of additive manufacturing structures: Application on a real industrial aerospace component

In this paper, a generalized formulation of defect-driven topology optimization (TO) for fatigue design, named TopFat, is proposed, where the first principal stress, that causes the crack propagation, is limited in accordance with the defect size distribution of the additive manufacturing (AM) process. As the largest defect depends on the final volume of the component, which changes according to the topology, an iterative procedure is adopted to avoid volume constraints in the optimization problem. The procedure is applied to an actual aerospace bracket component and its strength and limitations are discussed in comparison with the TO formulations presented in the literature. Results show that considering the defect population significantly affects the final topology, while leading to a feasible optimum. Further, even though the computational time increases, an additional 15% of mass saving is achieved by adopting the proposed iterative procedure with respect to a volume constraint-based formulation of TopFat.

Layout design of thin-walled structures with lattices and stiffeners using multi-material topology optimization

In this paper, the thin-walled structures with lattices and stiffeners manufactured by additive manufacturing are investigated. A design method based on the multi-material topology optimization is proposed for the simultaneous layout optimization of the lattices and stiffeners in thin-walled structures. First, the representative lattice units of the selected lattices are equivalent to the virtual homogeneous materials whose effective elastic matrixes are achieved by the energy-based homogenization method. Meanwhile, the stiffeners are modelled using the solid material. Subsequently, the multi-material topology optimization formulation is established for both the virtual homogeneous materials and solid material to minimize the structural compliance under mass constraint. Thus, the optimal layout of both the lattices and stiffeners could be simultaneously attained by the optimization procedure. Two applications, the aircraft panel structure and the equipment mounting plate, are dealt with to demonstrate the detailed design procedure and reveal the effect of the proposed method. According to numerical comparisons and experimental results, the thin-walled structures with lattices and stiffeners have significant advantages over the traditional stiffened thin-walled structures and lattice sandwich structures in terms of static, dynamic and anti-instability performance.

A Simple Matlab Code for Material Design Optimization Using Reduced Order Models.

The main part of the computational cost required for solving the problem of optimal material design with extreme properties using a topology optimization formulation is devoted to solving the equilibrium system of equations derived through the implementation of the finite element method (FEM). To reduce this computational cost, among other methodologies, various model order reduction (MOR) approaches can be utilized. In this work, a simple Matlab code for solving the topology optimization for the design of materials combined with three different model order reduction approaches is presented. The three MOR approaches presented in the code implementation are the proper orthogonal decomposition (POD), the on-the-fly reduced order model construction and the approximate reanalysis (AR) following the combined approximations approach. The complete code, containing all participating functions (including the changes made to the original ones), is provided.

Topology optimization of structures with infill-supported enclosed voids for additive manufacturing

This paper proposes a novel topology optimization procedure for designing structures with infill-supported enclosed voids for additive manufacturing (AM). In such structures, the open and enclosed regions are separately treated, where the open regions are the standard voids but the enclosed ones are filled by porous materials. The applied porous infill can work as supports in the AM process, but at the same time be part of the final structures and will never be removed. In this way, the difficulty of removing the supports in enclosed voids can be naturally avoided. However, whether the enclosed voids should be removed or kept if the porous material must be filled inside becomes a challenging problem, and thus this work develops a rational procedure to automatically determine it, and optimize the layout of these kept enclosed voids with infill. In the proposed procedure, a nonlinear virtual temperature (N-VTM) filter is utilized to separate the enclosed voids and open regions. Then, by combining the N-VTM filter, PDE filter and projection, a novel interpolation for structures with infill-supported enclosed voids is proposed, which at the same time can be used to provide structures without enclosed voids. The robust topology optimization formulation is applied to enhance the performance of the proposed procedure, and the sensitivity equations are deduced based on the adjoint method. Several numerical and experimental examples in two and three dimensions illustrate the effectiveness of the procedure. The results show that the procedure is capable of automatically determining whether to directly eliminate an enclosed void or to use infill to support it.

Towards intentional aesthetics within topology optimization by applying the principle of unity-in-variety

Topology optimization is increasingly applied to design consumer products, for which aesthetics plays an important role to consumer acceptance. In industrial design, it is known that preferences or taste judgement obey certain rules or principles. These principles are not directly quantifiable, but can qualitatively predict and explain aesthetic responses. In this paper, we empirically evaluate whether or not these design principles are effective for increasing the appealingness of topology optimized shapes. Our starting point is an overarching principle known as Unity-in-Variety. Variety stimulates our interests, whilst unity helps us make sense of a design in its entirety. According to this principle, aesthetic appreciation is maximized when a balance in unity and variety is attained. Since designs from topology optimization often exhibit remarkable complexity and variety, we hypothesize that increasing unity is the key to reach a balance and thus to elevate aesthetic appreciation in topology optimization. In our experimental setup, designs from topology optimization were manually post-processed, with the intention to increase unity, by following the “principles of perceptual grouping”, known as Gestalt principles. Our user study shows that in 11 out of the 12 pairs of topology optimized designs and their modified counterparts, the modified designs are perceived by the majority as visually more appealing, confirming our hypothesis. These findings provide a good basis for improving the aesthetic pleasure of topology optimized designs, either manually or ultimately by integrating them in the topology optimization formulation. It is expected that this eventually will contribute to a wider acceptance of topology optimization for consumer product design.

Topology optimization including a model of the layer-by-layer additive manufacturing process

A topology optimization formulation including a model of the layer-by-layer additive manufacturing (AM) process is considered. Defined as a multi-objective minimization problem, the formulation accounts for the performance and cost of both the final and partially manufactured designs and allows for considering AM-related issues such as overhang and residual stresses in the optimization. The formulation is exemplified by stiffness optimization in which the overhang is limited by adding mechanical or thermal compliance as a measure of the cost of partially manufactured designs. Convergence of the model as the approximate layer-by-layer model is refined is shown theoretically, and an extensive numerical study indicates that this convergence can be fast, thus making it a computationally viable approach useful for including AM-related issues into topology optimization. The examples also show that drips and sharp corners associated with some geometry-based formulations for overhang limitation can be avoided. The codes used in this article are written in Python using only open sources libraries and are available for reference.

An Efficient Sequential Strategy for Non-probabilistic Reliability-based Topology Optimization (NRBTO) of Continuum Structures with Stress Constraints

In the study, the sequential strategy is proposed for non-probabilistic reliability-based topology optimization (NRBTO) of continuum structures with stress constraints. Firstly, the qp-relaxation criterion is involved to circumvent the stress singularity. Then the deterministic topology optimization formulation by Stress-Influence-Function (SIF) approach is described. Uncertainty quantification analysis is conducted to determine the feasible bounds of the elemental stresses under unknown-but-bounded uncertainties. For safety reasons, the non-probabilistic reliability concept is introduced in the optimization formulation. The sequential strategy is proposed to handle the NRBTO problem and thus the original nested process of topology optimization and reliability analysis was decoupled, in which the update of design variables and the shift of the strength feature in the stress influence function are conducted alternately in the optimization process. The Performance Measure Approach (PMA) is applied to calculate the shift value of the strength feature and a shift value function is presented to avoid the numerical oscillations caused by the large variation of strength feature. Three numerical examples are eventually given to illustrate the applicability and validity of the developed sequential strategy for NRBTO with stress constraints.

Stiffness and strength topology optimization for bi‐disc systems based on dual sequential quadratic programming

AbstractAiming at solving the lightweight design problems for bi‐disc systems with consideration of the stiffness and strength simultaneously, a two stage topology optimization framework based on independent continuous and mapping (ICM) method is developed. First, topology optimization formulations are established to guarantee the optimized design satisfying the key nodal displacement and stress constraints. Second, an information function is introduced to realize the transformation of the material properties between elements and components. Composite exponential filter functions are selected to establish the relationship between the properties of elements and corresponding topology design variables. Third, approximate explicit quadratic programming model is obtained by the sensitivity analysis and Taylor expansion. In the process of solving, the duality theory is used to establish the approximate dual model with less design variables. Finally, a series of topology optimization design problems for bi‐disc systems are presented and discussed to illustrate the robustness and feasibility of the method. This study finds some new torsional bi‐disc designs and can also provide some theoretical references for the lightweight topology optimization of a multicomponent system.

Simultaneous topology and fiber path optimization of composite structures with MAC constraints

The pursuit of stiffer lightweight structures has led to the development of new concepts such as the Variable Stiffness Composite Laminates (VSCL), which further enhance the design potential of these highly tailored materials. As the number of design variables rise, more robust topology optimization formulations are necessary to achieve optimal solutions. In this work, a design optimization framework is developed by assigning material and fiber orientation on discrete patches of the VSCL structures. A gradient-based optimization method is used in the framework by applying the Discrete Material Optimization (DMO) formulation. The main goal is to design structures to meet dynamic performance objectives, with particular emphasis on the synthesis of modes with a target design. Accordingly, the objective function depends on the eigenvalues (natural frequencies), while one set of constraints depends on the eigenvectors (mode shapes). The Modal Assurance Criterion (MAC) is simultaneously used to perform mode tracking between the different design iterations and to evaluate the correlation between the synthetized modes and the pre-defined objective shapes. In this study, the calculation of the weight derivatives is also revised for the DMO formulation to show the contribution of the missing terms in the final optimal results. The optimization framework is examined for some potential applications related to dynamic decoupling and dynamic synthesis. It is shown that the design approach tailors the dynamic response of the composite structure and it can effectively be used to design resonators based on a reference model.