In this article, we propose an optimisation framework that can contribute to the prevention of premature failure or damage to building structures and can thereby strengthen their longevity. We concentrate on structures that are contaminated by chemical substances and that have strong destructive effects on the material. The aim of this contribution is a mathematical algorithm that allows the optimisation of a structure exposed to chemical influences and thus the assurance of the static load capacity. To achieve this, we present a coupled mechanical-diffusion-degradation approach embedded in a finite element (FE) framework. Furthermore, we integrate an optimisation algorithm to reduce material degradation. In this paper, we establish shape optimisation of a structure with a gradient based optimisation algorithm. © 2021, The Author(s).

This contribution presents a theoretical and computational framework for two-scale shape optimisation of nonlinear elastic structures. Particularly, minimum compliance optimisation problems with composite (matrix-inclusion) microstructures subjected to static loads and volume-type design constraints are focused. A homogenisation-based FE2 scheme is extended by an enhanced formulation of variational (shape) sensitivity analysis based on Noll’s intrinsic, frame-free formulation of continuum mechanics. The obtained overall two-scale sensitivity information couples shape variations across micro- and macroscopic scales. A numerical example demonstrates the capabilities of the proposed variational sensitivity analysis and the (shape) optimisation framework. The investigations involve a mesh morphing scheme for the design parametrisation at both macro- and microscopic scales. © 2021, The Author(s).

An isotropic gradient-enhanced damage model is applied to shape optimisation in order to establish a computational optimal design framework in view of optimal damage distributions. The model is derived from a free Helmholtz energy density enriched by the damage gradient contribution. The Karush–Kuhn–Tucker conditions are solved on a global finite element level by means of a Fischer–Burmeister function. This approach eliminates the necessity of introducing a local variable, leaving only the global set of equations to be iteratively solved. The necessary steps for the numerical implementation in the sense of the finite element method are established. The underlying theory as well as the algorithmic treatment of shape optimisation are derived in the context of a variational framework. Based on a particular finite deformation constitutive model, representative numerical examples are discussed with a focus on and application to damage optimised designs. © 2020, The Author(s).

The paper deals with the gradient based shape optimization of the biaxial X0-specimen, which has been introduced and examined in various papers, under producibility restrictions and the related experimental verification. The original, engineering based design of the X0-specimen has been applied successfully to different loading conditions persisting the question if relevant stress states could be reached by optimizing the geometry. Specimens with the initial as well as with the two load case dependent optimized geometries have been fabricated of the aluminum alloy sheets (AlSi1MgMn; EN AW 6082-T6) and tested. The strain fields in critical regions of the specimens have been recorded by digital image correlation technique. In addition, scanning electron microscope analysis of the fracture surfaces clearly indicate the stress state dependent damage processes. Consequently, the presented gradient based optimization technique facilitated significant improvements to study the damage and fracture processes in a more purposeful way. © 2020, The Author(s).

The aim of this paper is to improve the shape of specimens for biaxial experiments with respect to optimal stress states, characterized by the stress triaxiality. Gradient-based optimization strategies are used to achieve this goal. Thus, it is crucial to know how the stress state changes if the geometric shape of the specimen is varied. The design sensitivity analysis (DSA) of the stress triaxiality is performed using a variational approach based on an enhanced kinematic concept that offers a rigorous separation of structural and physical quantities. In the present case of elastoplastic material behavior, the deformation history has to be taken into account for the structural analysis as well as for the determination of response sensitivities. The presented method is flexible in terms of the choice of design variables. In a first step, the approach is used to identify material parameters. Thus, material parameters are chosen as design variables. Subsequently, the design parameters are chosen as geometric quantities so as to optimize the specimen shape with the aim to obtain a preferably homogeneous stress triaxiality distribution in the relevant cross section of the specimen. © 2020, The Author(s).

Design optimisation of structures and materials becomes more important in most engineering disciplines, especially in forming. The treatment of inelastic, path-dependent materials is a recent topic in this connection. Unlike purely elastic materials, it is necessary to store and analyse the deformation history in order to appropriately describe path-dependent material behaviour. For structural optimisation with design variables such as the outer shape of a structure, the boundary conditions and the material properties, it is necessary to compute sensitivities of all quantities of influence to use gradient based optimisation algorithms. Considering path-dependent materials, this includes the sensitivities of internal variables that represent the deformation history. We present an algorithm to compute afore-mentioned sensitivities, based on variational principles, in the context of finite deformation elastoplasticity. The novel approach establishes the possibility of design exploration using singular value decomposition. © 2017 Author(s).

The paper is concerned with the sensitivity analysis of structural responses in context of linear and non-linear stability phenomena like buckling and snapping. The structural analysis covering these stability phenomena is summarised. Design sensitivity information for a solid shell finite element is derived. The mixed formulation is based on the Hu-Washizu variational functional. Geometrical non-linearities are taken into account with linear elastic material behaviour. Sensitivities are derived analytically for responses of linear and non-linear buckling analysis with discrete finite element matrices. Numerical examples demonstrate the shape optimisation maximising the smallest eigenvalue of the linear buckling analysis and the directly computed critical load scales at bifurcation and limit points of non-linear buckling analysis, respectively. Analytically derived gradients are verified using the finite difference approach. © 2016 Springer-Verlag Berlin Heidelberg

The authors' variant of variational design sensitivity analysis in structural optimisation is highlighted in detail. A rigorous separation of physical quantities into geometry and displacement mappings based on an intrinsic presentation of continuum mechanics build up the first step. The variations with respect to design and displacements are easily available in a second step. The subsequent discrete matrix expressions are used to formulate the finite element equations in a third step. The fourth step elaborates the derived Matlab implementation while the fifth step shows the computational behaviour for an academic example. Both, the general case of nonlinear structural behaviour and the linearised approximation are outlined. The advocated scheme is compared with the well-known analytical differentiation approach of the discrete finite element equations.

This paper describes a modified extended finite element method (XFEM) approach, which is designed to ease the challenge of an analytical design sensitivity analysis in the framework of structural optimisation. This novel formulation, furthermore labelled YFEM, combines the well-known XFEM enhancement functions with a local sub-meshing strategy using standard finite elements. It deviates slightly from the XFEM path only at one significant point but thus allows to use already derived residual vectors as well as stiffness and pseudo load matrices to assemble the desired information on cut elements without tedious and error-prone re-work of already performed derivations and implementations. The strategy is applied to sensitivity analysis of interface problems combining areas with different linear elastic material properties. © 2015 John Wiley & Sons, Ltd.

The paper is concerned with variational sensitivity analysis of a nonlinear solid shell element, which is based on the Hu-Washizu variational principle. The sensitivity information is derived on the continuous level and discretized to yield the analytical expressions on the computational level. Especially, the pseudo load matrix and the sensitivity matrix, which dominate design sensitivity analysis of shape optimization problems, are derived. Because of the mixed formulation, condensation of the pseudo load matrix on the element level is performed to compute the sensitivity matrix. An illustrative example from the field of geometry-based shape optimization demonstrates the possible application of the presented formulation. © 2013 John Wiley & Sons, Ltd.

Evolutionary solid bodies undergoing changes of mass, of properties, and of shapes are considered in models of growth and adaptation and similarily in structural optimisation. A fundamental separation of different growth phenomena and a subsequent parametrisation using independent design variables for the amount of substance as well as for molar mass and molar volume facilitates an efficient formulation of the design space. Thus, the effects of design variations, i.e. change of amount of substance, on the variations of the structural response, i.e. The deformation in physical space, can be clearly described. Overall, a novel treatment of growth processes based on an evolution of the amount of substance is outlined. The parallelism of variations in physical and design space are highlighted and compared with the multiplicative decomposition of the deformation gradient into a growth and an elastic part incorporating an incompatible intermediate configuration. This drawback is overcome by a compatible manifold based on material points modelling the amount of substance outside of any geometrical space.

This paper outlines an enhanced analysis of the design sensitivities beyond the standard computation of the gradient values. It is based on the analytical derivation and efficient computation of the Fréchet derivatives of objectives and constraints with respect to the full space of all possible design variables. This overhead of sensitivity information is examined by a singular value decomposition (SVD) in order to detect major and minor influence and response modes of the considered structure. Thus, this methodology leads to valuable qualitative and quantitative insight which is so far unused in standard approaches to structural optimization. This knowledge enables the optimiser to understand and improve the models systematically which are usually set up entirely by engineering experience and intuition. Furthermore, a reduction of the complete design space to the most valuable subspace of design modifications demonstrates the information content of the decomposed sensitivities. The generic concept is applied to topology optimization which is a challenging model problem due to the large number of independent design variables. The details specific to topology optimization are outlined and the pros and cons are discussed. An illustrative example shows that reasonable optimal designs can be obtained with a small percentage of properly defined design variables. Nevertheless, further research is necessary to improve the overall computational efficiency. © Springer-Verlag 2012.

The pseudo load matrix and the sensitivity matrix dominate design sensitivity analysis of shape optimisation problems. They describe how a structure reacts on an imposed design modification. We analyse these matrices for the model problem of nodal based shape optimisation by a singular value decomposition and show that they contain additional valuable information which is not yet used either in theory or computation of shape optimisation. The inner structure of the sensitivities is capable to formulate reduced quadratic sub-problems within the sequential quadratic programming approach. We also tackle the problem of indefinite Hessian matrices in nodal based shape optimisation. Furthermore, we avoid jagged boundaries and obtain meshindependent optimised structures applying density filtering technique to shape optimisation. Overall, we emphasise an enhanced analysis of sensitivities and point to unused substantial capabilities. © Springer-Verlag 2011.