Structural Topology Optimization Method Research Articles

Truss layout design under nonprobabilistic reliability‐based topology optimization framework with interval uncertainties

SummaryA nonprobabilistic reliability‐based topology optimization (NRBTO) method for truss structures with interval uncertainties (or unknown‐but‐bounded uncertainties) is proposed in this paper. The cross‐sectional areas of levers are defined as design variables, while the material properties and external loads are regard as interval parameters. A modified perturbation method is applied to calculate structural response bounds, which are the prerequisite to obtain structural reliability. A deviation distance between the current limit state plane and the objective limit state plane, of which the expression is explicit, is defined as the nonprobabilistic reliability index, which serves as a constraint function in the optimization model. Compared with the deterministic topology optimization problem, the proposed NRBTO formulation is still a single‐loop optimization problem, as the reliability index is explicit. The sensitivity results are obtained from an analytical approach as well as a direct difference method. Eventually, the NRBTO problem is solved by a sequential quadratic programming method. Two numerical examples are used to testify the validity and effectiveness of the proposed method. The results show significant effects of uncertainties to the topology configuration of truss structures.

A NURBS-based finite cell method for structural topology optimization under geometric constraints

In this work, we present a NURBS-based finite cell approach for topology optimizations of structures with arbitrary shaped trimmed design domains. The combination of isogeometric analysis and finite cell method allows for a topology optimization of trimmed geometries directly derived from CAD systems which eliminates the time consuming preprocessing work such as mesh generations. We further propose a variationally consistent Nitsche's method for the weak enforcement of essential boundary conditions along trimming curves. Independent from the analysis domain, a high order and continuous NURBS represented density field is introduced which not only provides an intrinsic filter but also allows for the realization of multi-resolution topology optimizations. Additionally, we also propose a simple and straight forward method to enforce geometric constraints of arbitrary shapes during topology optimizations. We demonstrate the efficiency and accuracy of the proposed method with a number of benchmark problems and engineering oriented models.

Topology optimization of thermal conductive support structures for laser additive manufacturing

This paper presents a topology optimization approach to design support structures with efficient thermal conductive capabilities for powder-bed based laser additive manufacturing (AM). A transient heat transfer model is proposed to simulate the temperature distribution of the point-by-point and layer-upon-layer AM process. The process model is seamlessly integrated into a density based structural topology optimization method with an adjoint based sensitivity analysis and a gradient based optimization. For a given prototype shape, the layout of the corresponding support structure is designed such that the thermal energy during the fabrication process can be efficiently transferred from the laser path to a prescribed heat sink. Besides, an overhang constraint is proposed and used together with an AM filter to avoid overhang features in both the optimized support structure and the prototype, ensuring the overall manufacturability. Numerical examples and discussions are given to demonstrate its applicability.

A two-material topology optimization method for structures under steady thermo-mechanical loading

This article explores a topology optimization method for the design of two-material structures that must operate under mechanical and thermal loads, including heat fluxes at the boundaries. A three...

Lightweight Topology Optimization with Buckling and Frequency Constraints Using the Independent Continuous Mapping Method

This research focuses on the lightweight topology optimization method for structures under the premise of meeting the requirements of stability and vibration characteristics. A new topology optimization model with the constraints of natural frequencies and critical buckling loads and the objective of minimizing the structural volume is established and solved based on the independent continuous mapping method. The eigenvalue equations and composite exponential filter functions are applied to convert the optimization formulation into a continuous, solvable mathematical programming model. In the process of topology optimization, suitable initial values of the filter functions are chosen to avoid local modes, and the dynamic frequency gap constraints are added in the optimal model to prevent mode switches. Furthermore, for the optimal structures with grey elements obtained by the continuous optimization model, the bisection–inverse iteration is applied to search the optimal discrete structures. Finally, a detailed scheme is given for the buckling and frequency topology optimization problem. Numerical examples illustrate that the modelling method of minimizing the economic index with given performance requirements is practical and feasible for multi-performance topology optimization problems.

Topology Optimization of Periodic Structures With Substructuring

Topology optimization of macroperiodic structures is traditionally realized by imposing periodic constraints on the global structure, which needs to solve a fully linear system. Therefore, it usually requires a huge computational cost and massive storage requirements with the mesh refinement. This paper presents an efficient topology optimization method for periodic structures with substructuring such that a condensed linear system is to be solved. The macrostructure is identically partitioned into a number of scale-related substructures represented by the zero contour of a level set function (LSF). Only a representative substructure is optimized for the global periodic structures. To accelerate the finite element analysis (FEA) procedure of the periodic structures, static condensation is adopted for repeated common substructures. The macrostructure with reduced number of degree of freedoms (DOFs) is obtained by assembling all the condensed substructures together. Solving a fully linear system is divided into solving a condensed linear system and parallel recovery of substructural displacement fields. The design efficiency is therefore significantly improved. With this proposed method, people can design scale-related periodic structures with a sufficiently large number of unit cells. The structural performance at a specified scale can also be calculated without any approximations. What’s more, perfect connectivity between different optimized unit cells is guaranteed. Topology optimization of periodic, layerwise periodic, and graded layerwise periodic structures are investigated to verify the efficiency and effectiveness of the presented method.

Multicomponent Topology Optimization for Additive Manufacturing With Build Volume and Cavity Free Constraints

Topology optimization for additive manufacturing has been limited to the design of single-piece components that fit within the printer's build volume. This paper presents a gradient-based multicomponent topology optimization method for structures assembled from components built by powder bed additive manufacturing (MTO-A), which enables the design of multipiece assemblies larger than the printer's build volume. Constraints on component geometry for powder bed additive manufacturing are incorporated in a density-based topology optimization framework, with an additional design field governing the component partitioning. For each component, constraints on the maximum allowable build volume (i.e., length, width, and height) and the elimination of enclosed cavities are imposed during the simultaneous optimization of the overall topology and component partitioning. Numerical results of the minimum compliance designs revealed that manufacturing constraints, previously applied to single-piece topology optimization, can unlock richer design exploration space when applied to multicomponent designs.

Study on the characteristics of biology force line of proximal femur based on structural topology optimization

Internal fixator is usually adopted in the treatment of bone fractures. In order to achieve anatomical reduction and effective fixation of fractures, the placement of internal fixators should comply with the biology force line of the bone and adapt to the specific anatomical morphological characteristics of the cortical bone. In order to investigate the distribution characteristics and formation regularity of biology force line and cortical thickness of human bone, three-dimensional model of proximal femur is established by using three-dimensional reconstruction technique in this paper. The normal physiological stress distribution of proximal femur is obtained by finite element analysis under three kinds of behavior conditions: one-legged stance, abduction and adduction. The structural topology optimization method is applied to simulate the cortex of the proximal femur under the combined action of three kinds of behavior conditions, and the anatomic morphological characteristics of the proximal femur are compared. The distribution trend of biology force line of proximal femur and the characteristics of cortex are analyzed. The results show that the biology force lines of bone structure and the morphological characteristics of cortex depend on the load of human activities. The distribution trend of biology force line is related to the direction of trabecular bone and the ridge trend and firmness of cortex when bone is loaded physiologically. The proposed analytical method provides a solution to determine the biology force line of bone and the distribution characteristics of cortex. The conclusions obtained may guide the reasonable placement of internal fixator components of fracture.

Multiscale concurrent topology optimization of structures and microscopic multi-phase materials for thermal conductivity

PurposeThe optimal material microstructures in pure material design are no longer efficient or optimal when accounting macroscopic structure performance with specific boundary conditions. Therefore, it is important to provide a novel multiscale topology optimization framework to tailor the topology of structure and the material to achieve specific applications. In comparison with porous materials, composites consisting of two or more phase materials are more attractive and advantageous from the perspective of engineering application. This paper aims to provide a novel concurrent topological design of structures and microscopic materials for thermal conductivity involving multi-material topology optimization (material distribution) at the lower scale.Design/methodology/approachIn this work, the effective thermal conductivity properties of microscopic three or more phase materials are obtained via homogenization theory, which serves as a bridge of the macrostructure and the periodic material microstructures. The optimization problem, including the topological design of macrostructures and inverse homogenization of microscopic materials, are solved by bi-directional evolutionary structure optimization method.FindingsAs a result, the presented framework shows high stability during the optimization process and requires little iterations for convergence. A number of interesting and valid macrostructures and material microstructures are obtained in terms of optimal thermal conductive path, which verify the effectiveness of the proposed mutliscale topology optimization method. Numerical examples adequately consider effects of initial guesses of the representative unit cell and of the volume constraints of adopted base materials at the microscopic scale on the final design. The resultant structures at both the scales with clear and distinctive boundary between different phases, making the manufacturing straightforward.Originality/valueThis paper presents a novel multiscale concurrent topology optimization method for structures and the underlying multi-phase materials for thermal conductivity. The authors have carried out the concurrent multi-phase topology optimization for both 2D and 3D cases, which makes this work distinguished from existing references. In addition, some interesting and efficient multi-phase material microstructures and macrostructures have been obtained in terms of optimal thermal conductive path.

A stress-based topology optimization method for heterogeneous structures

In this work, we introduce a method to incorporate stress considerations in the topology optimization of heterogeneous structures. More specifically, we focus on using functionally graded materials (FGMs) to produce compliant mechanism designs that are not susceptible to failure. Local material properties are achieved through interpolating between material properties of two or more base materials. Taking advantage of this method, we develop relationships between local Young’s modulus and local yield stress, and apply stress criterion within the optimization problem. A solid isotropic material with penalization (SIMP)–based method is applied where topology and local element material properties are optimized simultaneously. Sensitivities are calculated using an adjoint method and derived in detail. Stress formulations implement the von Mises stress criterion, are relaxed in void regions, and are aggregated into a global form using a p-norm function to represent the maximum stress in the structure. For stress-constrained problems, we maintain local stress control by imposing m p-norm constraints on m regions rather than a global constraint. Our method is first verified by solving the stress minimization of an L-bracket problem, and then multiple stress-constrained compliant mechanism problems are presented. Results suggest that good designs can be produced with the proposed method and that heterogeneous designs can outperform their homogeneous counterparts with respect to both mechanical advantage and reduced stress concentrations.

Non-intrusive polynomial chaos expansion for topology optimization using polygonal meshes

This paper deals with the applications of stochastic spectral methods for structural topology optimization in the presence of uncertainties. A non-intrusive polynomial chaos expansion is integrated into a topology optimization algorithm to calculate low-order statistical moments of the mechanical-mathematical model response. This procedure, known as robust topology optimization, can optimize the mean of the compliance while simultaneously minimizing its standard deviation. In order to address possible variabilities in the loads applied to the mechanical system of interest, magnitude and direction of the external forces are assumed to be uncertain. In this probabilistic framework, forces are described as a random field or a set of random variables. Representation of the random objects and propagation of load uncertainties through the model are efficiently done through Karhunen-Lo\`{e}ve and polynomial chaos expansions. We take advantage of using polygonal elements, which have been shown to be effective in suppressing checkerboard patterns and reducing mesh dependency in the solution of topology optimization problems. Accuracy and applicability of the proposed methodology are demonstrated by means of several topology optimization examples. The obtained results, which are in excellent agreement with reference solutions computed via Monte Carlo method, show that load uncertainties play an important role in optimal design of structural systems, so that they must be taken into account to ensure a reliable optimization process.

An Efficient Topology Optimization Method for Structures with Uniform Stress

This paper focuses on two kinds of bi-objective topology optimization problems with uniform-stress constraints: compliance-volume minimization and local frequency response–volume minimization problems. An adaptive volume constraint (AVC) algorithm based on an improved bisection method is proposed. Using this algorithm, the bi-objective uniform-stress-constrained topology optimization problem is transformed into a single-objective topology optimization problem and a volume-decision problem. The parametric level set method based on the compactly supported radial basis functions is employed to solve the single-objective problem, in which a self-organized acceleration scheme based on shape derivative and topological sensitivity is proposed to adaptively adjust the derivative of the objective function and the step length during the optimization. To solve the volume-decision problem, an improved bisection method is proposed. Numerical examples are tested to illustrate the feasibility and effectiveness of the self-organized acceleration scheme and the AVC algorithm based on the improved bisection method. An extended application to the bi-objective stress-constrained topology optimization of a structure with stress concentration is also presented.

Multiscale concurrent topology optimization for cellular structures with multiple microstructures based on ordered SIMP interpolation

This paper presents a novel multiscale concurrent topology optimization method for cellular structures with various types of microstructures to obtain a superior structural performance at an affordable computation cost. At macroscale, different microstructures are regarded as different materials in a multi-material topology optimization. An ordered solid isotropic material with penalization (SIMP) interpolation model, which is efficient for solving multi-material topology optimization problems without introducing any new variables, is employed to generate an overall structure layout with predetermined multiple materials. At microscale, each macroscale element is viewed as an individual microstructure. All macroscale elements with the same material are represented by a unique microstructure. A parametric level set method (PLSM) is applied to evolve the shape and topology of the representative microstructures, whose effective properties are evaluated by a numerical homogenization method. The connectivity of adjacent microstructures is guaranteed by a kinematical connective constraint. Using the proposed method, the macrostructural topology as well as the locations and configurations of microstructures can be simultaneously optimized. Numerical examples are provided to test the effectiveness of the proposed method.

Multi-component topology and material orientation design of composite structures (MTO-C)

This paper presents a topology optimization method for structures made of multiple composite components (substructures) with tailored material orientations. It is capable of simultaneously optimizing the overall topology, component partitioning, and material orientation for each component, via a vector field variable that specifies fractional membership to each component, together with the conventional density and orientation variables. The convergence towards the non-fractional, zero/one membership during optimization is achieved by a hypercube-to-simplex projection of the membership variables with a penalization scheme. Through the integration with a continuous material orientation design method, the proposed method is capable of generating multi-component composite structures with tailored material orientations for each component, without a prescribed set of alternative discrete angles. The new method extends the existing continuous material orientation design methods by introducing the concept of components (substructures) with the component-level orientation control, which is suitable for economical production with the conventional high-volume composite manufacturing processes.

Multiobjective Structure Topology Optimization of Wind Turbine Brake Pads Considering Thermal-Structural Coupling and Brake Vibration

Brake pads of disc brake play an important role in the stable braking process of a large-megawatt wind turbine. There is always vibration, screaming, and severe nonuniform wear under the effect of both retardation pressure and friction. To solve these issues, this article aims to find a new structure of the brake pads to improve brake performance. A multiobjective structure topology optimization method considering thermal-structural coupling and brake vibration is carried out in this article. Based on topology optimization method of Solid Isotropic Microstructures with Penalization (SIMP), the compromise planning theory is applied to meet the stiffness requirement and vibration performance of brake pads. Structure of brake pads is optimized, and both the stiffness and vibration performance of brake pads are also improved. The disadvantages of single-objective optimization are avoided. Thermal-structural coupling analysis is tested with the actual working conditions. The results show that the new structure meets the stiffness requirement and improves the vibration performance well for the large-megawatt wind turbine. The effectiveness of the proposed method has been proved by the whole optimization process.

A non-probabilistic reliability-based topology optimization (NRBTO) method of continuum structures with convex uncertainties

This paper develops a non-probabilistic reliability-based topology optimization (NRBTO) framework for continuum structures under multi-dimensional convex uncertainties. Combined with the solid isotropic material with penalization (SIMP) model and the set-theoretical convex method, the uncertainty quantification (UQ) analysis is firstly conducted to obtain mathematical approximations and boundary laws of considered displacement responses. By normalization treatment of the limit-state function, a new quantified measure of the non-probabilistic reliability is then defined and further deduced by the principle of the hyper-volume ratio. For circumventing optimization difficulties arising from large-scale design variables, the adjoint vector scheme for sensitivity analysis of the reliability index with respect to design variables are discussed as well. Numerical applications eventually illustrate the applicability and the validity of the present problem statement as well as the proposed numerical techniques.

An asymptotically concentrated method for structural topology optimization based on the SIMLF interpolation

SummaryIn this work, an asymptotically concentrated topology optimization method based on the solid isotropic material with logistic function interpolation is proposed. The asymptotically concentrated method is introduced into the process of optimization cycle after updating the design variables. At the same time, with the use of the solid isotropic material with logistic function interpolation, all the candidate densities are reasonably polarized, relying on the characteristic of the interpolation curve itself. The asymptotically concentrated method can effectively suppress the generation of intermediate density and speed up the process of updating the design variables, hence improving the optimization efficiency. Moreover, the above polarization can weaken the influence of low‐related‐density elements and enhance the influence of high‐related‐density elements. For the single‐material topology optimization problem, gray‐scale elements can be effectively eliminated, and clear boundary and smaller compliance can be obtained by this method. For the multimaterial topology optimization problem, minimum compliance with high efficiency can be achieved by this method. The proposed method mainly includes the following advantages: concentrated density variables, reasonable interpolation, high computational efficiency, and good topological results.

Eigenvalue topology optimization via efficient multilevel solution of the frequency response

SummaryThe article presents an efficient solution method for structural topology optimization aimed at maximizing the fundamental frequency of vibration. Nowadays, this is still a challenging problem mainly because of the high computational cost required by spectral analyses. The proposed method relies on replacing the eigenvalue problem with a frequency response one, which can be tuned and efficiently solved by a multilevel procedure. Connections of the method with multigrid eigenvalue solvers are discussed in details. Several applications demonstrating more than 90% savings of the computational time are presented as well.

Layout Design of Conductive Heat Channel by Emulating Natural Branch Systems

To design effective and easy-to-manufacture conductive heat channels, a heuristic method by emulating the natural branch systems is suggested. The design process of the method is divided into two steps, which are the principal channel design and the lateral channel design. During the process, the width of each channel is controlled by the bifurcation law, and the end point of the channel is located at the point with the maximum temperature while the start points of the principal channel and the lateral channel are respectively determined by the location of the heat sink and the law of the minimum thermal resistance. Four design examples with different boundary conditions are studied by the suggested method, and the design results are compared with that of the traditional structural topology optimization method. Not only lower maximum temperature and relatively uniform distribution of temperature are obtained by the suggested method, but also straight channels are achieved without gray element, which is easy to manufacture. The suggested method inspired by the natural branch systems can provide an effective solution for heat channel design in the heat dissipation structures.

Structural topology optimization of a stamping die made from high-strength steel sheet metal based on load mapping

Even though high-strength steel is a favourable material because of its high strength and good formability, this material poses new challenges on the structure of stamping dies owing to potential damages of the die during production. Stamping dies are conventionally produced according to design manuals and standard manufacturing guidelines, where a high safety factor is specified to ensure that the stamping die is heavy and thick. A structural topology optimization method for a stamping die is presented in this study based on finite element simulations of the sheet metal stamping process. The finite element model of the stamping die is first established. Next, the boundary forces acting on the sheet metal are obtained from simulations and these forces are applied to the punch surface by means of load mapping during topology optimization. These forces are equivalent to the interaction between the blank and die. The objective function is to maximize the static stiffness under multi-conditions, which is defined by using the compromise programming approach. The analytic hierarchy process method is used to determine the weight ratio of the body stiffness in various load conditions and conduct topology optimization of the comprehensive objective function. The results show that the weight of the optimal punch is reduced while its performance is improved. More importantly, the reconstructed punch can be manufactured practically.