Phenomena such as biological growth and damage evolution can be thought of as time evolving processes, the directions of which are governed by descendent of certain goal functions. Mathematically this means using a dynamical systems approach to optimization. We extend such an approach by introducing a field quantity, representing nutrients or other non-mechanical stimuli, that modulate growth and damage evolution. The derivation of a generic model is systematic, starting from a Lyaponov-type descent condition and utilizing a Coleman-Noll strategy. A numerical algorithm for finding stationary points of the resulting dynamical system is suggested and applied to two model problems where the influence of different levels of nutrient sensitivity are observed. The paper demonstrates the use of a new modeling technique and shows its application in deriving a generic problem of growth and damage evolution. (C) 2014 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim

We present a contribution to a relatively unexplored application of topology optimization: structural topology optimization with fatigue constraints. A probability based high-cycle fatigue analysis is combined with principal stress calculations in order to find the topology with minimal mass that can withstand prescribed loading conditions for a specific life time. This allows us to generate optimal conceptual designs of structural components where fatigue life is the dimensioning factor.

We describe the fatigue analysis and present ideas that makes it possible to separate the fatigue analysis from the topology optimization. The number of constraints is kept low as they are applied to stress clusters, which are created such that they give adequate representations of the local stresses. Optimized designs constrained by fatigue and static stresses are shown and a comparison is also made between stress constraints based on the von Mises criterion and the highest tensile principal stresses.

The homogenized material optimization (HMO) problem is a novel structural optimization problem that we have developed for optimization of fiber reinforced composite structures. In the HMO problem we apply a smeared-out approach to model the material properties of fiber reinforced composite materials. The objective of the HMO problem is to maximize the stiffness of a composite structure by means of finding the optimal distribution of composite material, belonging to a fixed set of fiber orientations, across the design domain. In order to obtain manufacturable solutions, we have introduced a linear density filter as a restriction method to control the thickness variation across the design domain. To examine the effect of the density filter on the thickness variation and the objective function value of composite structures, obtained in the HMO problem, we have performed numerical tests for different load cases, mesh densities and range of the filter radius.

It is observed that for the present problem the thickness variation was mesh-independent. Both the thickness variation and objective function value depend on the load case used in the HMO problem. For all load cases the thickness variations exhibits an approximately piece-wise linear behaviour for increased filter radius. Furthermore, it was observed that an increase of filter radius would result in an moderate increase in objective function value for the solutions obtained from the HMO problem. From these results we conclude that by using a density filter, the HMO problem can be used to obtain manufacturable designs for composite structures.

We present a global (one constraint) version of the clustered approach previously developed for stress constraints, and also applied to fatigue constraints, in topology optimization. The global approach gives designs without large stress concentrations or geometric shapes that would cause stress singularities. For example, we solve the well known L-beam problem and obtain a radius at the internal corner.

The main reason for using a global stress constraint in topology optimization is to reduce the computational cost that a high number of constraints impose. In this paper we compare the computational cost and the results obtained using a global stress constraint versus using a number of clustered stress constraints.

We also present a method for deactivating those design variables that are not expected to change in the current iteration. The deactivation of design variables provides a considerable decrease of the computational cost and it is made in such a way that approximately the same final design is obtained as if all design variables are active.

In the present work we propose a two phase composite structure optimization method based on a novel material homogenization approach. It consists of a stiffness and a lay-up optimization problem, respectively, with the aim of obtaining manufacturable composite structures with maximized stiffness properties. The method is applied to a cantilever plate, and numerical tests were performed for three load cases and for a number of parameters settings. The results show that the proposed method can obtain manufacturable composite structures with maximized stiffness properties. In the first phase of the method, the stiffness optimization problem provides an optimal distribution of the composite material, such that the stiffness properties of the structureare maximized. The second phase, the lay-up optimization problem, provides a manufacturable lay-up sequence of discrete plies which attempts to retain the stiffness properties of the structure from the first phase.

This paper develops and evaluates a method for handling stress constraints in topology optimization. The stress constraints are used together with an objective function that minimizes mass or maximizes stiffness, and in addition, the traditional stiffness based formulation is discussed for comparison. We use a clustering technique, where stresses for several stress evaluation points are clustered into groups using a modified P-norm to decrease the number of stress constraints and thus the computational cost. We give a detailed description of the formulations and the sensitivity analysis. This is done in a general manner, so that different element types and 2D as well as 3D structures can be treated. However, we restrict the numerical examples to 2D structures with bilinear quadrilateral elements. The three formulations and different approaches to stress constraints are compared using two well known test examples in topology optimization: the L-shaped beam and the MBB-beam. In contrast to some other papers on stress constrained topology optimization, we find that our formulation gives topologies that are significantly different from traditionally optimized designs, in that it actually manage to avoid stress concentrations. It can therefore be used to generate conceptual designs for industrial applications.

The dynamical systems approach to sizing and SIMP topology optimization, introduced in a previous paper, is extended to the case of time-varying loads. A general dynamical system, satisfying a Lyaponov-type descent condition, is derived and specialized to a goal function combining stiffness and mass. For a cyclic time-dependent load it is indicated how, in the limit of short cycles compared to the overall time scale, this can be handled by multiple load cases. Numerical examples, both for a convex and a non-convex case, illustrates the theory.

The connection between apparent density-type bone remodeling theories and density formulations of topology optimization is well known from a large number of publications and its theoretical basis has recently been discussed by making use of a dynamical systems approach to optimization. The present paper takes this connection one step further by showing how the Coleman–Noll procedure of rational thermodynamics can be used to derive general dynamical systems, where a special case includes the lazy zone concept of bone remodeling theory. It is also shown how a numerical solution method for the dynamical system can be developed by using the sequential convex approximation idea. The method is employed to obtain a series of solutions that show the influence of modeling parameters representing elements of plasticity and viscosity in the growth process.

This paper uses a dynamical systems approach for studying the material distribution (density or SIMP) formulation of topology optimization of structures. Such an approach means that an ordinary differential equation, such that the objective function is decreasing along a solution trajectory of this equation, is constructed. For stiffness optimization two differential equations with this property are considered. By simple explicit Euler approximations of these equations, together with projection techniques to satisfy box constraints, we obtain different iteration formulas. One of these formulas turns out to be the classical optimality criteria algorithm, which, thus, is receiving a new interpretation and framework. Based on this finding we suggest extensions of the optimality criteria algorithm. A second important feature of the dynamical systems approach, besides the purely algorithmic one, is that it points at a connection between optimization problems and natural evolution problems such as bone remodeling and damage evolution. This connection has been hinted at previously but, in the opinion of the authors, not been clearly stated since the dynamical systems concept was missing. To give an explicit example of an evolution problem that is in this way connected to an optimization problem, we study a model of bone remodeling. Numerical examples, related to both the algorithmic issue and the issue of natural evolution represented as bone remodeling, are presented.