Nonlinear topology optimization of layered shell structures

Topology stiffness (compliance) design of linear and geometrically nonlinear shell structures is solved using the SIMP approach together with a filtering scheme. A general anisotropic multi-layer shell model is employed to allow the formation of through-the-thickness holes or stiffening zones. The finite element analysis is performed using nine-node Mindlin-type shell elements based on the degenerated shell approach, which are capable of modeling both single and multi-layered structures exhibiting anisotropic or isotropic behavior. The optimization problem is solved using analytical compliance and constraint sensitivities together with the Method of Moving Asymptotes (MMA). Geometrically nonlinear problems are solved using iterative Newton–Raphson methods and an adjoint variable approach is used for the sensitivity analysis. Several benchmark tests are presented in order to illustrate the difference in optimal topologies between linear and geometrically nonlinear shell structures.     

Simultaneous design of components layout and supporting structures using coupled shape and topology optimization technique

The purpose of this paper was to study the layout design of the components and their supporting structures in a finite packing space. A coupled shape and topology optimization (CSTO) technique is proposed. On one hand, by defining the location and orientation of each component as geometric design variables, shape optimization is carried out to find the optimal layout of these components and a finite-circle method (FCM) is used to avoid the overlap between the components. On the other hand, the material configuration of the supporting structures that interconnect components is optimized simultaneously based on topology optimization method. As the FE mesh discretizing the packing space, i.e., design domain, has to be updated itertively to accommodate the layout variation of involved components, topology design variables, i.e., density variables assigned to density points that are distributed regularly in the entire design domain will be introduced in this paper instead of using traditional pseudo-density variables associated with finite elements as in standard topology optimization procedures. These points will thus dominate the pseudo-densities of the surrounding elements. Besides, in the CSTO, the technique of embedded mesh is used to save the computing time of the remeshing procedure, and design sensitivities are calculated w.r.t both geometric variables and density variables. In this paper, several design problems maximizing structural stiffness are considered subject to the material volume constraint. Reasonable designs of components layout and supporting structures are obtained numerically.     

Maximization of eigenvalues using topology optimization

Topology optimization is used to optimize the eigenvalues of plates. The results are intended especially for MicroElectroMechanical Systems (MEMS) but can be seen as more general. The problem is not formulated as a case of reinforcement of an existing structure, so there is a problem related to localized modes in low density areas. The topology optimization problem is formulated using the SIMP method. Special attention is paid to a numerical method for removing localized eigenmodes in low density areas. The method is applied to numerical examples of maximizing the first eigenfrequency. One example is a practical MEMS application; a probe used in an Atomic Force Microscope (AFM). For the AFM probe the optimization is complicated by a constraint on the stiffness and constraints on higher order eigenvalues. Received June 10, 1999     

Design of phononic materials/structures for surface wave devices using topology optimization

We develop a topology optimization approach to design two- and three-dimensional phononic (elastic) materials, focusing primarily on surface wave filters and waveguides. These utilize propagation modes that transmit elastic waves where the energy is contained near a free surface of a material. The design of surface wave devices is particularly attractive given recent advances in nano- and micromanufacturing processes, such as thin-film deposition, etching, and lithography, which make it possible to precisely place thin film materials on a substrate with submicron feature resolution. We apply our topology optimization approach to a series of three problems where the layout of two materials (silicon and aluminum) is sought to achieve a prescribed objective: (1) a grating to filter bulk waves of a prescribed frequency in two and three dimensions, (2) a surface wave device that uses a patterned thin film to filter waves of a single or range of frequencies, and (3) a fully three-dimensional structure to guide a wave generated by a harmonic input on a free surface to a specified output port on the surface. From the first to the third example, the resulting topologies increase in sophistication. The results demonstrate the power and promise of our computational framework to design sophisticated surface wave devices.     

Minimization of sound radiation in fully coupled structural-acoustic systems using FEM-BEM based topology optimization

This paper presents a method to locally constrain multiple material volume domains for structural optimization with the Level Set Method (LSM). Two different Lagrangian formulations and multiplier update methods are used, for both the global and local problem. The local volume domains can be constrained by both equality and inequality constraints. The optimization objective is compliance minimization for well-posed statically loaded structures. For validation, several example problems are established and solved using the proposed method. Results show that the volume ratios for user established sub-domains can be controlled successfully. The local constraint values are met accurately in the case of equality constraints and remain in their feasible domain in the case of inequality constraints. Optimization results are not significantly hindered by the introduction of local volume constraints for comparable problems.     

Evolutionary topology optimization of continuum structures with a global displacement control

The conventional compliance minimization of load-carrying structures does not directly deal with displacements that are of practical importance. In this paper, a global displacement control is realized through topology optimization with a global constraint that sets a displacement limit on the whole structure or certain sub-domains. A volume minimization problem is solved by an extended evolutionary topology optimization approach. The local displacement sensitivities are derived following a power-law penalization material model. The global control of displacement is realized through multiple local displacement constraints on dynamically located critical nodes. Algorithms are proposed to secure the stability and convergence of the optimization process. Through numerical examples and by comparing with conventional stiffness designs, it is demonstrated that the proposed approach is capable of effectively finding optimal solutions which satisfy the global displacement control. Such solutions are of particular importance for structural designs whose deformed shapes must comply with functioning requirements such as aerodynamic performances.     

Note on topology optimization of continuum structures including self-weight

This paper proposes to investigate topology optimization with density-dependent body forces and especially self-weight loading. Surprisingly the solution of such problems cannot be based on a direct extension of the solution procedure used for minimum-compliance topology optimization with fixed external loads. At first the particular difficulties arising in the considered topology problems are pointed out: non-monotonous behaviour of the compliance, possible unconstrained character of the optimum and the parasitic effect for low densities when using the power model (SIMP). To get rid of the last problem requires the modification of the power law model for low densities. The other problems require that the solution procedure and the selection of appropriate structural approximations be revisited. Numerical applications compare the efficiency of different approximation schemes of the MMA family. It is shown that important improvements are achieved when the solution is carried out using the gradient-based method of moving asymptotes (GBMMA) approximations. Criteria for selecting the approximations are suggested. In addition, the applications also provide the opportunity to illustrate the strong influence of the ratio between the applied loads and the structural weight on the optimal structural topology.     

Viscoelastic material design with negative stiffness components using topology optimization

An application of topology optimization to design viscoelastic composite materials with elastic moduli that soften with frequency is presented. The material is a two-phase composite whose first constituent is isotropic and viscoelastic while the other is an orthotropic material with negative stiffness but stable. A concept for this material based on a lumped parameter model is used. The performance of the topology optimization approach in this context is illustrated using three examples.     

Reliability optimization of topology communication network design using an improved ant colony optimization

Network design problem is a well-known NP-hard problem which involves the selection of a subset of possible links or a network topology in order to minimize the network cost subjected to the reliability constraint. To overcome the problem, this paper proposes a new efficiency algorithm based on the conventional ant colony optimization (ACO) to solve the communication network design when considering both economics and reliability. The proposed method is called improved ant colony optimizations (IACO) which introduces two addition techniques in order to improve the search process, i.e. neighborhood search and re-initialization process. To show its efficiency, IACO is applied to test with three different topology network systems and its results are compared with those obtained results from the conventional approaches, i.e. genetic algorithm (GA), tabu search algorithm (TSA) and ACO. Simulation results, obtained these test problems with various constraints, shown that the proposed approach is superior to the conventional algorithms both solution quality and computational time.     

Adhesive surface design using topology optimization

Tailoring adhesive properties between surfaces is of great importance for micro-scale systems, ranging from managing stiction in MEMS devices to designing wall-scaling gecko-like robots. A methodology is introduced for designing adhesive interfaces between structures using topology optimization. Structures subjected to external loads that lead to delamination are studied for situations where displacements and deformations are small. Only the effects of adhesive forces acting normal to the surfaces are considered. An interface finite element is presented that couples a penalty contact formulation and a Lennard–Jones model of van der Waals adhesive forces. Two- and three dimensional design optimization problems are presented in which adhesive force distributions are designed such that load-displacement curves of delaminating structures match target responses. The design variables describe the adhesive energy per area of the interface between the surfaces, as well as the geometry of the delaminating structure. A built-in length scale in the formulation of the adhesion forces eliminates the need for filtering to achieve comparable optimal adhesive designs over a range of mesh densities. The resulting design problem is solved by gradient based optimization algorithms evaluating the design sensitivities by the adjoint method. Results show that the delamination response can be effectively manipulated by the method presented. Varying simultaneously both adhesive and geometric parameters yields a wider range of reachable target load-displacement curves than in the case varying adhesive energy alone.     

Growth method for size,topology, and geometry optimization of truss structures

The problem of optimally designing the topology of plane trusses has, in most cases, been dealt with as a size problem in which members are eliminated when their size tends to zero. This article presents a novel growth method for the optimal design in a sequential manner of size, geometry, and topology of plane trusses without the need of a ground structure. The method has been applied to single load case problems with stress and size constraints. It works sequentially by adding new joints and members optimally, requiring five basic steps: (1) domain specification, (2) topology and size optimization, (3) geometry optimization, (4) optimality verification, and (5) topology growth. To demonstrate the proposed growth method, three examples were carried out: Michell cantilever, Messerschmidt–Bölkow–Blohm beam, and Michell cantilever with fixed circular boundary. The results obtained with the proposed growth method agree perfectly with the analytical solutions. A Windows XP program, which demonstrates the method, can be downloaded from http://www.upct.es/~deyc/software/tto/.     

Experimental validation and prototyping of optimum designs obtained from topology optimization

This paper provides, through both numerical analyses and physical tests, a validation of the optimality of structural designs obtained from a topology optimization process. Issues related to the manufacturability of the topology-optimized design are first addressed in order to develop structural specimens suitable for experimental validation. Multidomain and multistep topology optimization techniques are introduced that, by embedding the designer’s intuition and experience into the design process, allow for the simplification of the design layout and thus for a better manufacturability of the design. A boundary identification method is also proposed that is applied to produce a smooth boundary for the design. An STL (STereo Lithography) file is then generated and used as input to a rapid prototyping machine, and physical specimens are fabricated for the experiments. Finally, the experimental measurements are compared with the theoretical and numerical predictions. Results agree extremely well for the example problems considered, and thus the optimum designs pass both virtual and physical tests. It is also shown that the optimum design obtained from topology optimization can be independent of the material used and the dimensions assumed for the structural design problem. This important feature extends the applicability of a single optimum design to a range of different designs of various sizes, and it simplifies the prototyping and experimental validation since small, inexpensive prototypes can be utilized. This could result in significant cost savings when carrying out proof-of-concept in the product development process.     

Compliant mechanism design using multi-objective topology optimization scheme of continuum structures

Topology optimization problems for compliant mechanisms using a density interpolation scheme, the rational approximation of material properties (RAMP) method, and a globally convergent version of the method of moving asymptotes (GCMMA) are primarily discussed. First, a new multi-objective formulation is proposed for topology optimization of compliant mechanisms, in which the maximization of mutual energy (flexibility) and the minimization of mean compliance (stiffness) are considered simultaneously. The formulation of one-node connected hinges, as well as checkerboards and mesh-dependency, is typically encountered in the design of compliant mechanisms. A new hybrid-filtering scheme is proposed to solve numerical instabilities, which can not only eliminate checkerboards and mesh-dependency efficiently, but also prevent one-node connected hinges from occurring in the resulting mechanisms to some extent. Several numerical applications are performed to demonstrate the validity of the methods presented in this paper.     

A review of optimization of cast parts using topology optimization

This is the second part of a two-paper review of optimization of cast parts. In the first paper, we focused on the application of the original topology optimization codes. The problems with the interpretation has been shown and discussed. In this paper, we introduce TopShape to overcome this lack. It is able to take manufacturing constraints for cast parts into account. The features of TopShape and its results will be discussed and compared with the result of the commercial code OptiStruct. Furthermore, a new design alternative for cast parts will be derived.     

A review of optimization of cast parts using topology optimization

In recent years, there has been considerable progress in the optimization of cast parts with respect to strength, stiffness, and frequency. Here, topology optimization has been the most important tool in finding the optimal features of a cast part, such as optimal cross-section or number and arrangement of ribs. An optimization process with integrated topology optimization has been used very successfully at Adam Opel AG in recent years, and many components have been optimized. This two-paper review gives an overview of the application and experience in this area. This is the first part of a two-paper review of optimization of cast parts.Here, we want to focus on the application of the original topology optimization codes, which do not take manufacturing constraints for cast parts into account. Additionally, the role of shape optimization as a fine-tuning tool will be briefly analyzed and discussed.     

A review of homogenization and topology optimization III-topology optimization using optimality criteria

This is the first part of a three-paper review of homogenization and topology optimization, viewed from an engineering standpoint and with the ultimate aim of clarifying the ideas so that interested researchers can easily implement the concepts described. In the first paper we focus on the theory of the homogenization method where we are concerned with the main concepts and derivation of the equations for computation of effective constitutive parameters of complex materials with a periodic micro structure. Such materials are described by the base cell, which is the smallest repetitive unit of material, and the evaluation of the effective constitutive parameters may be carried out by analysing the base cell alone. For simple microstructures this may be achieved analytically, whereas for more complicated systems numerical methods such as the finite element method must be employed. In the second paper, we consider numerical and analytical solutions of the homogenization equations. Topology optimization of structures is a rapidly growing research area, and as opposed to shape optimization allows the introduction of holes in structures, with consequent savings in weight and improved structural characteristics. The homogenization approach, with an emphasis on the optimality criteria method, will be the topic of the third paper in this review.     

A new boundary search scheme for topology optimization of continuum structures with design-dependent loads

The identification of the load surface is a key problem in solving topology optimization of continuum structures with design-dependent loads. In this paper, an element-based search scheme is introduced to identify the load surface. The load surfaces are formed by the connection of the real boundary of elements and the pressures are transferred directly to corresponding element nodes. The search scheme is very convenient to apply and is found to be efficient and effective in identifying the load surfaces. Only slight modifications to the load codes in the routine procedure are required and there is no need to calculate the sensitivities of the load with respect to the material density changes. Numerical examples are presented to demonstrate the efficiency of the boundary search scheme.     

On the maximization of the fundamental eigenvalue in topology optimization

The paper considers a classic formulation of the topology optimization problem of discrete or discretized structures. The objective function to be maximized is the smallest natural frequency of the structure. We develop non-heuristic mathematical models paying special attention to the situation when some design variables take zero values. These models take into account multiple load conditions, equilibrium of forces, constraints on compliance and volume, and the effect of possible non-structural mass. We discuss serious obstacles for a successful numerical treatment of this formulation such as non-Lipschitzean behavior and even discontinuity of the objective function. As a cure, we present an equivalent reformulation as a bilinear semidefinite programming problem without the pitfalls of the original problem. An algorithm is presented for finding an approximation of a globally optimal solution up to a user-defined accuracy. The key ingredient of this algorithm is the treatment of a sequence of linear semidefinite programs. Numerical examples are provided for truss structures. Examples of both academic and larger size illustrate the theoretical results achieved and demonstrate the practical use of this approach. We conclude with an extension on multiple non-structural mass conditions.     

Multiobjective topology optimization of structures using genetic algorithms with chromosome repairing

In this work, a genetic algorithm (GA) for multiobjective topology optimization of linear elastic structures is developed. Its purpose is to evolve an evenly distributed group of solutions to determine the optimum Pareto set for a given problem. The GA determines a set of solutions to be sorted by its domination properties and a filter is defined to retain the Pareto solutions. As an equality constraint on volume has to be enforced, all chromosomes used in the genetic GA must generate individuals with the same volume value; in the coding adopted, this means that they must preserve the same number of “ones” and, implicitly, the same number of “zeros” along the evolutionary process. It is thus necessary: (1) to define chromosomes satisfying this propriety and (2) to create corresponding crossover and mutation operators which preserve volume. Optimal solutions of each of the single-objective problems are introduced in the initial population to reduce computational effort and a repairing mechanism is developed to increase the number of admissible structures in the populations. Also, as the work of the external loads can be calculated independently for each individual, parallel processing was used in its evaluation. Numerical applications involving two and three objective functions in 2D and two objective functions in 3D are employed as tests for the computational model developed. Moreover, results obtained with and without chromosome repairing are compared.     

Reinforcement layout and sizing optimization of composite submarine sail structures

A topology optimization approach that makes use of nonlinear design variable-to-sizing relationship is presented. A finite element (FE) model is used to describe the loaded structure, but unlike the microstructure approach, the decision is whether an element in the continuum should have maximum or minimum cross-sectional dimension while its material density and moduli are held constant. This approach is applied to reinforcement layout optimization of a very large and geometrically complex Composite Advanced Sail (CAS) structure under an asymmetric wave slap loading condition. A high-complexity model in the form of multilayered shell and a low-complexity model in the form of stiffened shell are developed for the layout optimization of the CAS and solved for minimum strain energy. The effects of constraints such as buckling instability on optimal placement of internal stiffeners are also explored. Based on the results of the layout optimization, a new FE model of the CAS is developed and optimized for minimum weight. Depending upon the degree of variability in skin thickness, the results show a weight saving of up to 19% over the original model.