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 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 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.
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.
The dynamical systems approach to sizing and SIMP topology optimization, introduced in a previous paper, is extended to the case of timevarying 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.
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.
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