Interpolation/penalization applied for strength design of 3D thermoelastic structures

With design independent loads and only a constrained volume (no local bounds), the same optimal design leads simultaneously to minimum compliance and maximum strength. However, for thermoelastic structures this is not the case and a maximum volume may not be an active constraint for minimum compliance. This is proved for thermoelastic structures by sensitivity analysis of compliance that facilitates localized determination of sensitivities, and the compliance is not identical to the total elastic energy (twice strain energy). An explicit formula for the difference is derived and numerically illustrated with examples. In compliance minimization for thermoelastic structures it may be advantageous to decrease the total volume, but for strength maximization it is argued to keep the total permissible volume. Linear interpolation (no penalization) may to a certain extent be argued for 2D thickness optimized designs, but for 3D design problems interpolation must be included and not only from the penalization point of view to obtain 0–1 designs. Three interpolation types are presented in a uniform manner, including the well known one parameter penalizations, named SIMP and RAMP. An alternative two parameter interpolation in explicit form is preferred, and the influence of interpolation on compliance sensitivity analysis is included. For direct strength maximization the sensitivity analysis of local von Mises stresses is demanding. An applied recursive procedure to obtain uniform energy density is presented in details, and it is shown by examples that the obtained designs are close to fulfilling also strength maximization. Explicit formulas for equivalent thermoelastic loads in 2D and 3D finite element analysis are derived and applied, including the sensitivity analysis.     

Design objectives with non-zero prescribed support displacements

When non-zero prescribed support displacements are involved in addition to design independent loads for a continuum/structure, then the objectives of minimum compliance (total elastic energy) and of maximum strength lead to different designs. This is verified by the presented sensitivities. Designs from neither of the two objectives are characterized by uniformly distributed energy density. However, simple iterations with the goal of obtaining uniform energy density show that the strength is favored by this approach. These observations leads to a rejection of the objectives of compliance minimization as well as that of direct strength maximization; we choose the objective of obtaining uniform energy density and show by examples that the obtained solutions are close to fulfilling also strength maximization, with the price of increased compliance. Optimal design examples are presented and discussed in detail for different combinations of non-zero prescribed support displacements and design independent loads.     

Multi-objective concurrent topology optimization of thermoelastic structures composed of homogeneous porous material

The present paper studies multi-objective design of lightweight thermoelastic structure composed of homogeneous porous material. The concurrent optimization model is applied to design the topologies of light weight structures and of the material microstructure. The multi-objective optimization formulation attempts to find minimum structural compliance under only mechanical loads and minimum thermal expansion of the surfaces we are interested in under only thermo loads. The proposed optimization model is applied to a sandwich elliptically curved shell structure, an axisymmetric structure and a 3D structure. The advantage of the concurrent optimization model to single scale topology optimization model in improving the multi-objective performances of the thermoelastic structures is investigated. The influences of available material volume fraction and weighting coefficients are also discussed. Numerical examples demonstrate that the porous material is conducive to enhance the multi-objective performance of the thermoelastic structures in some cases, especially when lightweight structure is emphasized. An “optimal” material volume fraction is observed in some numerical examples.     

Multi-material topology optimization for practical lightweight design

Topology optimization is one of the most effective tools for conducting lightweight design and has been implemented across multiple industries to enhance product development. The typical topology optimization problem statement is to minimize system compliance while constraining the design space to an assumed volume fraction. The traditional single-material compliance problem has been extended to include multiple materials, which allows increased design freedom for potentially better solutions. However, compliance minimization has the limitations for practical lightweight design because compliance lacks useful physical meanings and has never been a design criterion in industry. Additionally, the traditional compliance minimization problem statement requires volume fraction constraints to be selected a priori; however, designers do not know the optimized balance among materials. In this paper, a more practical method of multi-material topology optimization is presented to overcome the limitations. This method seeks the optimized balance among materials by minimizing the total weight while satisfying performance constraints. This paper also compares the weight minimization approach to compliance minimization. Several numerical examples prove the success of weight minimization and demonstrate its benefit over compliance minimization.     

A new method based on adaptive volume constraint and stress penalty for stress-constrained topology optimization

This paper focuses on the stress-constrained topology optimization of minimizing the structural volume and compliance. A new method based on adaptive volume constraint and stress penalty is proposed. According to this method, the stress-constrained volume and compliance minimization topology optimization problem is transformed into two simple and related problems: a stress-penalty-based compliance minimization problem and a volume-decision problem. In the former problem, stress penalty is conducted and used to control the local stress level of the structure. To solve this problem, the parametric level set method with the compactly supported radial basis functions is adopted. Meanwhile, an adaptive adjusting scheme of the stress penalty factor is used to improve the control of the local stress level. To solve the volume-decision problem, a combination scheme of the interval search and local search is proposed. Numerical examples are used to test the proposed method. Results show the lightweight design, which meets the stress constraint and whose compliance is simultaneously optimized, can be obtained by the proposed method.     

A compliance based design problem of structures under multiple load cases

There are two popular methods concerning the optimal design of structures. The first is the minimization of the volume of the structure under stress constraints. The second is the minimization of the compliance for a given volume. For multiple load cases an arising issue is which energy quantity should be the objective function. Regarding the sizing optimization of trusses, Rozvany proved that the solution of the established compliance based problems leads to results which are awkward and not equivalent to the solutions of minimization of the volume under stress constraints, unlike under single loading (the layouts would be the same if in the compliance problem the volume is set equal to the result of the first problem). In this paper, we introduce the “envelope strain energy” problem where we minimize the volume integral of the worst case strain energy of each point of the structure. We also prove that in the case of sizing optimization of statically non-indeterminate (the term non-indeterminate includes both statically determinate trusses and mechanisms) trusses, this compliance method gives the same optimal design as the stress based design method.     

Topology optimization using a reaction–diffusion equation

This paper presents a structural topology optimization method based on a reaction–diffusion equation. In our approach, the design sensitivity for the topology optimization is directly employed as the reaction term of the reaction–diffusion equation. The distribution of material properties in the design domain is interpolated as the density field which is the solution of the reaction–diffusion equation, so free generation of new holes is allowed without the use of the topological gradient method. Our proposed method is intuitive and its implementation is simple compared with optimization methods using the level set method or phase field model. The evolution of the density field is based on the implicit finite element method. As numerical examples, compliance minimization problems of cantilever beams and force maximization problems of magnetic actuators are presented to demonstrate the method’s effectiveness and utility.     

Topological optimization of continuum structures with design-dependent surface loading – Part I: new computational approach for 2D problems

This paper describes a new computational approach for optimum topology design of 2D continuum structures subjected to design-dependent loading. Both the locations and directions of the loads may change as the structural topology changes. A robust algorithm based on a modified isoline technique is presented that generates the appropriate loading surface which remains on the boundary of potential structural domains during the topology evolution. Issues in connection with tracing the variable loading surface are discussed and treated in the paper. Our study indicates that the influence of the variation of element material density is confined within a small neighbourhood of the element. With this fact in mind, the cost of the calculation of the sensitivities of loads may be reduced remarkably. Minimum compliance is considered as the design problem. There are several models available for such designs. In the present paper, a simple formulation with weighted unit cost constraints based on the expression of potential energy is employed. Compared to the traditional models (i.e., the SIMP model), it provides an alternative way to implement the topology design of continuum structures. Some 2D examples are tested to show the differences between the designs obtained for fixed, design-independent loading, and for variable, design-dependent loading. The general and special features of the optimization with design-dependent loads are shown in the paper, and the validity of the algorithm is verified. An algorithm dealing with 3D design problems is described in Part II, which is developed from the 2D algorithm in the present Part I of the paper.     

Shape optimization of curves and surfaces considering fairness metrics and elastic stiffness

A method is presented for generating round curves and surfaces allowing discontinuities in tangent vectors and curvatures. The distance of the centre of curvature from the specified point is used for formulating the objective function which is a continuous function of the design variables through convex and concave shapes. It is shown that a shell with and without ribs can be generated within the same problem formulation if the minimization problem is converted into a maximization problem and the parameter region where integration is to be carried out is restricted in view of the curvature. Optimal shapes are also found under constraints on the compliance against static loads. A multiobjective optimization problem is solved by the constraint method to generate a trade-off design between roundness and mechanical performance.An erratum to this article can be found at     

Analytical optimal designs for long and short statically determinate beam structures

Analytical solutions for optimal beam design may serve as benchmarks for numerical studies and for basic understanding. The influence of load case, of boundary conditions and of cross sectional type is severe, so many cases are studied. Short beams based on Timoshenko theory are included in the present energy approach. With a given amount of material/volume the objective is to minimize compliance, and the necessary optimality criterion is to obtain the same gradient of elastic energy along the beam axis x, i.e., for all volumes A(x) d x, where the design function is the area A(x). The beams considered in the paper are geometrically unconstrained, i.e., no minimum and/or maximum constraints are specified for the variable cross sectional area function A(x) other than the volume constraint. The obtained explicitly described designs may be used for comparison with obtained distribution of volume densities for two and three dimensional numerical models, or in the sense of topology optimization for the distribution of the number of active design pixels/voxels. Also the presented decrease in compliance, relative to compliance for uniform beam design, show how much it is possible to obtain for these optimal compliance designs of statically determinate cases.     

Parameter-free optimum design method of stiffeners on thin-walled structures

In this paper, we present a shape optimization method for designing stiffeners on thin-walled or shell structures. Solutions are proposed to deal with a stiffness maximization problem and a volume minimization problem, which are subject to a volume constraint and a compliance constraint, respectively. The boundary shapes of the stiffeners are determined under a condition where the stiffeners are movable in the in-plane direction to the surface. Both problems are formulated as distributed-parameter shape optimization problems, and the shape gradient functions are derived using a material derivative method and an adjoint variable method. The optimal free-boundary shapes of the stiffeners are obtained by applying the derived shape gradient function to the $H^{1}$ gradient method for shells, which is a parameter-free shape optimization method proposed by one of the authors. Several stiffener design examples are presented to validate the proposed method and demonstrate its practical utility.     

Shape optimization of curves and surfaces considering fairness metrics and elastic stiffness

A method is presented for generating round curves and surfaces allowing discontinuities in tangent vectors and curvatures. The distance of the center of curvature from the specified point is used for formulating the objective function that is a continuous function of the design variables through convex and concave shapes. It is shown that a shell with and without ribs can be generated within the same problem formulation if the minimization problem is converted into a maximization problem and the parameter region where integration is to be carried out is restricted in view of the sign of the curvature. Optimal shapes are also found under constraints on the compliance against static loads. A multiobjective optimization problem is solved by the constraint method to generate a trade-off design between roundness and mechanical performance. The online version of the original article can be found at:     

ENTROPY MINIMAX MULTIVARIATE STATISTICAL MODELING–I: THEORY

The theory of entropy mimmax, an information theoretic approach to predictive modeling, is reviewed. Comparisons are given to other methodologies, including: Neyman-Pearson hypothesis testing, James-Stein “empirical Bayes” estimation, maximum likelihood estimation, least-squares fitting, linear regression and logistic regression. Examples are provided showing how maximum entropy expectation probabilities are computed and how minimum entropy partitions are determined. The importance of the a priori weight normalization, in establishing the coarse-grain of the minimum entropy partition, is discussed. The trial crossvalidation procedure for determining the normalization is described. Generalizations utilizing Zadeh's fuzzy entropies are provided for variables involving indistinguishability, partial or total. Specific cases are discussed of maximum likelihood estimation, illustrating ils “data range irregularity” which is avoided by methods such as entropy minimization that account for residuals over the full distribution range. Discriminant analysis and polynomial fitting are discussed as examples of areas of application of the principles of entropy minimax. In curve fitting, the order of the polynomial is determined by entropy minimization, and the coefficients are determined by entropy maximization. Entropy maximization operates as a goodness-of-fit criterion, while entropy minimization operates as a mathematical formulation of Ockham's razor, controlling the level of smoothness in fitting models to data.     

On the equivalence of minimum compliance and stress-constrained minimum weight design of trusses under multiple loading conditions

This article is devoted to topology optimization of trusses under multiple loading conditions. Compliance minimization with material volume constraint and stress-constrained minimum weight problem are considered. In the case of a single loading condition, it has been shown that the two problems have the same optimal topology. The possibility of extending this result for problems involving multiple loading conditions is examined in the present work. First, the compliance minimization problem is formulated as a multicriterion optimization problem, where the conflicting criteria are the compliances of the different loading conditions. Then, the optimal topologies of the stress-constrained minimum weight problem and the multicriterion compliance minimization problem for a simple test example are compared. The results verify that when multiple loading conditions are involved, the stress-constrained minimum weight topology cannot be obtained in general by solving the compliance minimization problem.     

Evolutionary structural optimization for stress minimization problems by discrete thickness design

Stress minimization is a major aspect of structural optimization in a wide range of engineering designs. This paper presents a new evolutionary criterion for the problems of variable thickness design whilst minimizing the maximum stress in a structure. On the basis of finite element analysis, a stress sensitivity number is derived to estimate the stress change in an element due to varying the thickness of other elements. Following the evolutionary optimization procedure, an optimal design with a minimum maximum stress is achieved by gradually removing material from those elements, which have the lowest stress sensitivity number or adding material onto those elements, which have the highest stress sensitivity number. The numerical examples presented in this paper demonstrate the capacity of the proposed method for solving stress minimization problems. The results based on the stress criterion are compared with traditional ones based on a stiffness criterion, and an optimization scheme based on the combination of both the stress minimization and the stiffness maximization criteria is presented.     

Simultaneous design of structural layout and discrete fiber orientation using bi-value coding parameterization and volume constraint

The so-called bi-value coding parameterization (BCP) method is developed for the simultaneous optimization of layout design and discrete fiber orientations of laminated structures related to the compliance minimization and natural frequency maximization. Both kinds of problems are transformed into a discrete material selection problem that is then solved as a continuous topology optimization problem with multiphase materials. A new form of the volume constraint is introduced in accordance with the BCP to control the material usage and material removal in the corresponding problem formulation. The BCP scheme assigning the integer value of +1 or -1 to each design variable for the unique “coding” is efficiently used to interpolate discrete fiber orientations and to identify the presence and removal of materials. Meanwhile, a general set-up strategy is proposed by assigning “uniform” weight values in BCP to ensure the feasibility of the initial starting point. Numerical tests illustrate that the BCP is efficient in dealing with both kinds of design problems including the volume constraint.     

Two novel iterative algorithms for interference alignment with symbol extensions in the MIMO interference channel

Interference alignment(IA)with symbol extensions in the quasi-static flat-fading K-user multipleinput multiple-output(MIMO)interference channel(IC)is considered in this paper.In general,long symbol extensions are required to achieve the optimal fractional degrees of freedom(DOF).However,long symbol extensions over orthogonal dimensions produce structured(diagonal or block diagonal)channel matrices from transmitters to receivers.Most of existing approaches are limited in cases where the channels have some special structures,because they align the interference without preserving the dimensionality of the desired signal explicitly.To overcome this common drawback of most existing IA algorithms,two novel iterative algorithms for IA with symbol extensions are proposed.The first algorithm designs transceivers for IA based on the mean square error(MSE)criterion which minimizes the total MSE of the system while preserving the dimensionality of the desired signal.The novel IA algorithm is a constrained optimization problem which can be solved by Lagrangian method.Its convergence is proven as well.Utilizing the reciprocity of alignment,the second algorithm is proposed based on the maximization of the multidimensional case of the generalized Rayleigh Quotient.It maximizes each receiver’s signal to interference plus noise ratio(SINR)while preserving the dimensionality of the desired signal.In simulation results,we show the superiority of the proposed algorithms in terms of four aspects,i.e.,average sum rate,the fraction of the interfering signal power in the desired signal subspace,bit error rate(BER)and the relative power of the weakest desired data stream.     

Topology optimization for worst load conditions based on the eigenvalue analysis of an aggregated linear system

This paper proposes a topology optimization for a linear elasticity design problem subjected to an uncertain load. The design problem is formulated to minimize a robust compliance that is defined as the maximum compliance induced by the worst load case of an uncertain load set. Since the robust compliance can be formulated as the scalar product of the uncertain input load and output displacement vectors, the idea of “aggregation” used in the field of control is introduced to assess the value of the robust compliance. The aggregation solution technique provides the direct relationship between the uncertain input load and output displacement, as a small linear system composed of these vectors and the reduced size of a symmetric matrix, in the context of a discretized linear elasticity problem, using the finite element method. Introducing the constraint that the Euclidean norm of the uncertain load set is fixed, the robust compliance minimization problem is formulated as the minimization of the maximum eigenvalue of the aggregated symmetric matrix according to the Rayleigh–Ritz theorem for symmetric matrices. Moreover, the worst load case is easily established as the eigenvector corresponding to the maximum eigenvalue of the matrix. The proposed structural optimization method is implemented using topology optimization and the method of moving asymptotes (MMA). The numerical examples provided illustrate mechanically reasonable structures and establish the worst load cases corresponding to these optimal structures.     

On some fundamental properties of structural topology optimization problems

We study some fundamental mathematical properties of discretized structural topology optimization problems. Either compliance is minimized with an upper bound on the volume of the structure, or volume is minimized with an upper bound on the compliance. The design variables are either continuous or 0–1. We show, by examples which can be solved by hand calculations, that the optimal solutions to the problems in general are not unique and that the discrete problems possibly have inactive volume or compliance constraints. These observations have immediate consequences on the theoretical convergence properties of penalization approaches based on material interpolation models. Furthermore, we illustrate that the optimal solutions to the considered problems in general are not symmetric even if the design domain, the external loads, and the boundary conditions are symmetric around an axis. The presented examples can be used as teaching material in graduate and undergraduate courses on structural topology optimization.     

Lifting low-lying loads in the sagittal plane.

A simple geometrical model was employed to investigate various elementary techniques during static, sagittal plane lifts. Relationships between the various joint reaction forces were deduced. In particular, it was found that it is possible to simultaneously increase (or decrease) both knee and low back forces. In terms of the total force on the low back, hip, knee, and ankle joints, vertical back lifting is generally not recommended. This is especially true when lifting low-lying objects from the ground. However, minimization of loads is not the only factor to consider when analysing the optimal technique of a certain lifting task. Several other cost functions have been previously proposed. Comparisons derived from minimizing various cost functions suggest that the minimization of the maximum necessary muscle intensity may be the most appropriate in deducing optimal load lifting configurations.