Topology optimization of hyperelastic bodies including non-zero prescribed displacements

Stiffness topology optimization is usually based on a state problem of linear elasticity, and there seems to be little discussion on what is the limit for such a small rotation-displacement assumption. We show that even for gross rotations that are in all practical aspects small (<3 deg), topology optimization based on a large deformation theory might generate different design concepts compared to what is obtained when small displacement linear elasticity is used. Furthermore, in large rotations, the choice of stiffness objective (potential energy or compliance), can be crucial for the optimal design concept. The paper considers topology optimization of hyperelastic bodies subjected simultaneously to external forces and prescribed non-zero displacements. In that respect it generalizes a recent contribution of ours to large deformations, but we note that the objectives of potential energy and compliance are no longer equivalent in the non-linear case. We use seven different hyperelastic strain energy functions and find that the numerical performance of the Kirchhoff–St.Venant model is in general significantly worse than the performance of the other six models, which are all modifications of this classical law that are equivalent in the limit of infinitesimal strains, but do not contain the well-known collapse in compression. Numerical results are presented for two different problem settings.     

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.     

A general formulation of structural topology optimization for maximizing structural stiffness

This paper presents a general formulation of structural topology optimization for maximizing structure stiffness with mixed boundary conditions, i.e. with both external forces and prescribed non-zero displacement. In such formulation, the objective function is equal to work done by the given external forces minus work done by the reaction forces on prescribed non-zero displacement. When only one type of boundary condition is specified, it degenerates to the formulation of minimum structural compliance design (with external force) and maximum structural strain energy design (with prescribed non-zero displacement). However, regardless of boundary condition types, the sensitivity of such objective function with respect to artificial element density is always proportional to the negative of average strain energy density. We show that this formulation provides optimum design for both discrete and continuum structures.     

A note on logspace optimization

We show that computing iterated multiplication of word matrices over {0,1}^*, using the operations maximum and concatenation, is complete for the class optL of logspace optimization functions. The same problem for word matrices over {1}^* is complete for the class FNL of nondeterministic logspace functions. Improving previously obtained results, we furthermore place the class optL in AC¹, and characterize FNL by restricted logspace optimization functions.     

A formulation of the force method in the range of large displacements

The force method is formulated with regard to analysis problems of discrete elastic structures in the range of large displacements (but small strains). It is shown that the usual concepts — degree of redundancy, redundant force, release, etc. — which characterize the force method in the range of infinitesimal displacements are still valid in the present context (second-order geometric effects). The degree of redundancy is affected by the rank of the geometric stiffness matrix, while the (external) flexibility matrix of the primary structure proves to be the sum of an elastic flexibility matrix and a geometric one. A few comments on computational aspects and on the need of future developments conclude the paper.     

Large displacements of shells including material nonlinearities

The paper presents a finite element formulation of shells with large deflections including elastoplastic material behaviour. The elements utilized are the so-called degenerated shell elements, with special emphasis on the simple 4 node quadrilateral. Both the total Lagrangian and the updated Lagrangian formulation are considered. For the treatment of plastic behaviour the concept of a layered element model is proposed and investigated for both the tangential modulus method and the initial load method. In the final section sample problems are presented and compared with reference solutions. It is shown that the 4 node element is very well-suited to the class of problems under consideration. It is characterized by an easy applicability, high accuracy and low rates of computer time.In Appendix A an analytical solution of a cantilever beam, subjected to an end load, including geometrical and material nonlinearities is presented.     

A note on stress-constrained truss topology optimization

The purpose of this brief note is to demonstrate that general-purpose optimization methods and codes should not be discarded when dealing with stress-constrained truss topology optimization. By using a disaggregated formulation of the considered problem, such methods may find also “singular optima”, without using perturbation techniques like the ε-relaxed approach. Received February 19, 2002     

A note on communicating between information systems based on including degrees

In order to study the communication between information systems, Gong and Xiao (2010, International journal of general systems, 39, 189–206) proposed the concept of general relation mappings based on including degrees. Some properties and the extension for fuzzy information systems of the general relation mappings have been investigated there. In this paper, we point out by counterexamples that several assertions (Lemmas 3.1 and 3.2, Theorems 4.1 and 4.3) in the aforementioned work are not true in general.     

A remark on inseparability of min-max systems

Min-max systems, or min-max-plus systems are algebraic models of discrete-event dynamic systems. They are nonlinear extensions of the well known linear max-plus system models. The structural property "inseparability" of min-max systems was proposed by us in a recent paper. It turns out that inseparability is equivalent to the property of "irreducibility" proposed recently by van der Woude and Subiono, although the two properties appear in very different forms. We will prove this fact here.     

一类非线性极小极大问题的改进粒子群算法

针对一类非线性极小极大问题目标函数非光滑的特点给求解带来的困难,利用改进的粒子群算法并结合极大熵函数法给出了此类问题的一种新的有效算法。首先利用极大熵函数将无约束和有约束极小极大问题转化为一个光滑函数的无约束最优化问题,将此光滑函数作为粒子群算法的适应值函数;然后用数学中的外推方法给出一个新的粒子位置更新公式,并应用这个改进的粒子群算法来优化此问题。数值结果表明,该算法收敛快﹑数值稳定性好,是求解非线性极小极大问题的一种有效算法。     

A note on a globalization of Wilson-type optimization methods

A globally and locally superlinearly convergent combination of a feasible direction method with a Wilson-type method proposed in this journal for inequality constrained problems exclusively is modified in such a way that it is applicable to general nonlinear programming problems.     

A note on robustness tolerances for combinatorial optimization problems

We consider so-called generic combinatorial optimization problem, where the set of feasible solutions is some family of nonempty subsets of a finite ground set with specified positive initial weights of elements, and the objective function represents the total weight of elements of the feasible solution. We assume that the set of feasible solutions is fixed, but the weights of elements may be perturbed or are given with errors. All possible realizations of weights form the set of scenarios.A feasible solution, which for a given set of scenarios guarantees the minimum value of the worst-case relative regret among all the feasible solutions, is called a robust solution. The maximum percentage perturbation of a single weight, which does not destroy the robustness of a given solution, is called the robustness tolerance of this weight with respect to the solution considered.In this paper we give formulae for computing the robustness tolerances with respect to an optimal solution obtained for some initial weights and we show that this can be done in polynomial time whenever the optimization problem is polynomially solvable itself.     

A note on the structure of space frame stiffness equations for iterative solution storage schemes

The off-diagonal blocks in space frame stiffness equations are observed to have a structure which allows them to be stored in 21 memory locations instead of 36. This allows iterative and semi-iterative (such as conjugate gradient methods) storage scheme algorithms to store the equations more compactly.     

A linear programming based heuristic framework for min-max regret combinatorial optimization problems with interval costs

This work deals with a class of problems under interval data uncertainty, namely interval robust-hard problems, composed of interval data min-max regret generalizations of classical NP-hard combinatorial problems modeled as 0-1 integer linear programming problems. These problems are more challenging than other interval data min-max regret problems, as solely computing the cost of any feasible solution requires solving an instance of an NP-hard problem. The state-of-the-art exact algorithms in the literature are based on the generation of a possibly exponential number of cuts. As each cut separation involves the resolution of an NP-hard classical optimization problem, the size of the instances that can be solved efficiently is relatively small. To smooth this issue, we present a modeling technique for interval robust-hard problems in the context of a heuristic framework. The heuristic obtains feasible solutions by exploring dual information of a linearly relaxed model associated with the classical optimization problem counterpart. Computational experiments for interval data min-max regret versions of the restricted shortest path problem and the set covering problem show that our heuristic is able to find optimal or near-optimal solutions and also improves the primal bounds obtained by a state-of-the-art exact algorithm and a 2-approximation procedure for interval data min-max regret problems.     

Analytic formulation of stiffness matrix for axisymmetric solids

Stiffness matrices for axisymmetric solids with arbitrary loading were derived through the application of variational principles in analytic form. The singularities in the stiffness matrices were removed through displacement constraints along the axis of symmetry for each circumferential mode. The shear stresses and maximum deflections of a set of Saint-Venant flexural problems were obtained both analytically and numerically. The results indicate that the finite element analysis with the analytic stiffness matrix provides a very good solution. The same problems were solved with a commercial code, ANSYS. and showed that the analytic stiffness matrix contributed to a faster convergence rate as the number of elements increases in an analysis.     

A note on continuity in scalarization for multicriteria optimization problems

We establish sufficient conditions for solutions of scalarization replacements of multicriteria optimization problems to exhibit continuous dependence on scaling factors and other parameters.     

A reinforcement learning approach based on the fuzzy min-max neural network

The fuzzy min-max neural network constitutes a neural architecture that is based on hyperbox fuzzy sets and can be incrementally trained by appropriately adjusting the number of hyperboxes and their corresponding volumes. Two versions have been proposed: for supervised and unsupervised learning. In this paper a modified approach is presented that is appropriate for reinforcement learning problems with discrete action space and is applied to the difficult task of autonomous vehicle navigation when no a priori knowledge of the enivronment is available. Experimental results indicate that the proposed reinforcement learning network exhibits superior learning behavior compared to conventional reinforcement schemes.     

A classification of methods for distributed system optimization based on formulation structure

Augmented Lagrangian coordination (ALC) is a provably convergent coordination method for multidisciplinary design optimization (MDO) that is able to treat both linking variables and linking functions (i.e. system-wide objectives and constraints). Contrary to quasi-separable problems with only linking variables, the presence of linking functions may hinder the parallel solution of subproblems and the use of the efficient alternating directions method of multipliers. We show that this unfortunate situation is not the case for MDO problems with block-separable linking constraints. We derive a centralized formulation of ALC for block-separable constraints, which does allow parallel solution of subproblems. Similarly, we derive a distributed coordination variant for which subproblems cannot be solved in parallel, but that still enables the use of the alternating direction method of multipliers. The approach can also be used for other existing MDO coordination strategies such that they can include block-separable linking constraints. This work is funded by MicroNed, grant number 10005898.