We propose a topology optimization method that includes high-cycle fatigue as a constraint. The fatigue model is based on a continuous-time approach where the evolution of damage in each point of the design domain is governed by a system of ordinary differential equations, which employs the concept of a moving endurance surface being a function of the stress and back stress. Development of fatigue damage only occurs when the stress state lies outside the endurance surface. The fatigue damage is integrated for a general loading history that may include non-proportional loading. Thus, the model avoids the use of a cycle-counting algorithm. For the global high-cycle fatigue constraint, an aggregation function is implemented, which approximates the maximum damage. We employ gradient-based optimization, and the fatigue sensitivities are determined using adjoint sensitivity analysis. With the continuous-time fatigue model, the damage is load history dependent and thus the adjoint variables are obtained by solving a terminal value problem. The capabilities of the presented approach are tested on several numerical examples with both proportional and non-proportional loads. The optimization problems are to minimize mass subject to a high-cycle fatigue constraint and to maximize the structural stiffness subject to a high-cycle fatigue constraint and a limited mass.

Metal AM (additive manufacturing) components are generally inhomogeneous and have different microstructure in the bulk compared with (contour) regions near the surface. This, as well as rough as-built surfaces, affects mechanical properties. In this paper, we develop a topology optimization method that considers such inhomogeneities. The method is a direct extension of standard density-based methods using linear filtering for regularization, and a second filtering of the design variables is used to identify a surface layer, the thickness of which is given by the filter radius. Domain extension is used in order to properly identify such layers at the boundary of the design domain. The method is generally applicable but is demonstrated for stiffness optimization. Both two- and three-dimensional problems are treated. A general property of the method is that the topological complexity is reduced, i.e. the optimized designs get fewer and thicker structural members as the width of the surface layer is increased.

Several filter approaches that introduce additive manufacturing-related overhang constraints in topology optimization exist. However, a drawback of these is that exact satisfaction of overhang constraints produces sharp inward corners resulting in stress singularities. The present paper therefore modifies such filter approaches by a penalty formulation, where the choice of penalty factor regulates how closely the overhang constraint is satisfied. By appropriately choosing certain weight factors in the penalty function, the cost of support structures is also reflected in the formulation in a simple and computationally inexpensive way. The method is demonstrated by parameter studies using the classical MBB beam, using both structured and unstructured meshes.

We develop a topology optimization method including high-cycle fatigue as a constraint. The fatigue model is based on a continuous-time approach, which uses the concept of a moving endurance surface as a function of the stress history and back stress evolution. The development of damage only occurs when the stress state lies outside the endurance surface. Furthermore, an aggregation function, which approximates the maximum fatigue damage, is implemented. As the optimization workflow is sensitivity-based, the fatigue sensitivities are determined using an adjoint sensitivity analysis. The capabilities of the presented approach are tested on numerical models where the problem is to maximize the stiffness subject to high-cycle fatigue constraints.

Fatigue cracked primary aircraft structural parts that cannot be replaced need to be repaired by other means. A structurally efficient repair method is to use adhesively bonded patches as reinforcements. This paper considers optimal design of such patches by minimizing the crack extension energy release rate. A new topology optimization method using this objective is developed as an extension of the standard SIMP compliance optimization method. The method is applied to a cracked test specimen that resembles what could be found in a real fuselage and the results show that an optimized adhesively bonded repair patch effectively reduces the crack energy release rate.

A design problem of finding an optimally stiff membrane structure by selecting one–dimensional fiber reinforcements is formulated and solved. The membrane model is derived in a novel manner from a particular three-dimensional linear elastic orthotropic model by appropriate assumptions. The design problem is given in the form of two minimization statements. After finite element discretization, the separate treatment of each of the two statements follows from classical results and methods of structural optimization: the stiffest orientation of reinforcing fibers coincides with principal stresses and the separate selection of density of fibers is a convex problem that can be solved by optimality criteria iterations. Numerical solutions are shown for two particular configurations. The first for a statically determined structure and the second for a statically undetermined one. The latter shows related but non-unique solutions.

Thermal barrier coatings (TBCs) are used in gas turbines to reduce creep, thermo-mechanical fatigue, and oxidation, or to allow for reduced air cooling. TBCs may fail due to fatigue. Structural optimization methods were applied to optimize the TBC thickness in such a way as to increase the life of the TBC. The TBC thickness was varied for three cases: 1) minimizing TBC volume, 2) minimizing TBC maximum effective stress, and 3) minimizing compliance (minimizing the strain energy). The results from the optimization were used to estimate the relative change in TBC life via a strain energy based failure criterion and a Coffin–Manson-like expression. Minimization of volume had limited use due to limitations in the current implementation. Minimization of effective stress did not give any significant increase in life. The minimization of compliance increased the estimated TBC life at highly stressed regions.

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.

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 minimum mass that can withstand prescribed variable-amplitude 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 make 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 paper is written with focus on structural parts in the avionic industry, but the method applies to any load carrying structure, made of linear elastic isotropic material, subjected to repeated loading conditions.

The increasing use of bre-reinforced composite materials in the manufacturing of high performance structures is primarily driven by their superior strength-toweight ratio when compared to traditional metallic alloys. This provides the ability to design and manufacture lighter structures with improved mechanical properties. However, the specic manufacturing process of composite structures, along with the orthotropic material properties exhibited by bre-reinforced composite materials, result in a complex structural design process where a number of dierent design parameters and manufacturing issues, which aect the mechanical properties of the composite structure, have to be considered. An ecient way to do this is to implement structural optimization techniques in the structural design process thus improving the ability of the design process to nd design solutions which satisfy the structural requirements imposed on the composite structure.This thesis describes a two phase composite structure optimization method based on a novel material homogenization approach. The proposed method consists of a stiness optimization problem and a lay-up optimization problem, respectively, with the aim to obtain a manufacturable composite structure with maximized stiness properties. The homogenization material approach is applied in both optimization problems, such that the material properties of the composite structure are homogenized. In the proposed method the stiness optimization problem provides a composite structure with maximized stiness properties by nding the optimal distribution of composite material across the design domain. The aim of the lay-up optimization problem is to obtain a manufacturable lay-up sequence of bre-reinforced composite plies for the composite structure which, as far as possible, retains the stiness properties given by the stiness optimization problem. The ability of the composite structure optimization method to obtain manufacturable composite structures is tested and conrmed by a number of numerical tests.

The increasing use of bre-reinforced composite materials in the manufacturing of high performance structures is primarily driven by their superior strength-toweight ratio when compared to traditional metallic alloys. This provides the ability to design and manufacture lighter structures with improved mechanical properties. However, the specic manufacturing process of composite structures, along with the orthotropic material properties exhibited by bre-reinforced composite materials, result in a complex structural design process where a number of dierent design parameters and manufacturing issues, which aect the mechanical properties of the composite structure, have to be considered. An ecient way to do this is to implement structural optimization techniques in the structural design process thus improving the ability of the design process to nd design solutions which satisfy the structural requirements imposed on the composite structure.This thesis describes a two phase composite structure optimization method based on a novel material homogenization approach. The proposed method consists of a stiness optimization problem and a lay-up optimization problem, respectively, with the aim to obtain a manufacturable composite structure with maximized stiness properties. The homogenization material approach is applied in both optimization problems, such that the material properties of the composite structure are homogenized. In the proposed method the stiness optimization problem provides a composite structure with maximized stiness properties by nding the optimal distribution of composite material across the design domain. The aim of the lay-up optimization problem is to obtain a manufacturable lay-up sequence of bre-reinforced composite plies for the composite structure which, as far as possible, retains the stiness properties given by the stiness optimization problem. The ability of the composite structure optimization method to obtain manufacturable composite structures is tested and conrmed by a number of numerical tests.

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.

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.

We present a global (one constraint) version of the clustere d approach developed for stress constraints in [1] and applied to fatigue constraints in [2] . A global stress constraint has previously been discussed by several authors, but in the present a pproach we manage to avoid large stress concentrations and geometric shapes that would caus e stre s singularities. For example, we solve the standard L-beam problem and obtain a radius t the internal corner; a result which, to the authors knowledge, previously has not been obt ained with just one global stress constraint. The clustered approach means that stress constraints are ap pli d to stress clusters, where each cluster contains the stresses from several stress evaluati on points. This gives an acceptable control of the local stresses even though a small number of co nstraints are used; the global approach is obtained as a special case if only one cluster is u sed. A global stress constraint allows for creating, at low computational cost, light weigh t conceptual designs that are free from large stress concentrations. Using our formulation, t he local stresses will be higher than the stress limit [1]; however, topology optimization is a to ol for conceptual design and the aim is to obtain a good structural shape, which will be sized in la ter design stages. The clustered approach is based on a P-norm of penalized loca l stresses which are divided by the number of members in the cluster. The stress penalizatio n increases the stress for intermediate design variable values and together with the SIMP-p enalization it forces the solution to a black-and-white design. The method is formulated so tha t different element types as well as three-dimensional problems can be treated, even though t wo-dimensional problems using four-node elements are considered in the examples. The global and the clustered approaches will be shown and com pared for stress and fatigue constrained problems, where the objective is to minimize th e mass. The stress constraints are based on either von Mises stresses or principal stresses , wh re the latter allows for highcycle fatigue constraints according to the methodology in [ 2]. The settings of the solver and the P-norm based stress measure, required in order to obtain good solutions with the global approach, will be discussed. The theoretical discussions w ill be confirmed by well known examples, such as the L-beam and the MBB-beam.

We present a new method for topology optimization with design-dependent loads. In traditional topology optimization a fixed design domain on which the loads and constraints are specified is used. S ...

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.

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.

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.