Elsevier

Composite Structures

Volume 287, 1 May 2022, 115289
Composite Structures

On the structural stiffness maximisation of anisotropic continua under inhomogeneous Neumann–Dirichlet boundary conditions

https://doi.org/10.1016/j.compstruct.2022.115289Get rights and content

Abstract

This work discusses three aspects of topology optimisation (TO) problems dealing with structural stiffness maximisation of anisotropic continua under mixed inhomogeneous Neumann–Dirichlet boundary conditions (BCs). Firstly, the total potential energy (TPE) is introduced as intuitive measure of the structural stiffness, instead of the work of applied forces and displacements (WAFD). Secondly, it is proven that the WAFD under mixed BCs is not a self-adjoint functional, while the one related to the TPE is always a self-adjoint functional, regardless of the BCs nature. Thirdly, the influence of the anisotropy, of the applied BCs and of the design requirement on the volume fraction on the optimised topology is investigated: depending on these features, the optimal solutions of the two problem formulations, i.e., minimisation of the functional involving the TPE or minimisation of the WAFD subject to a constraint on the volume fraction, can coincide. The problem is formulated in the context of a special density-based TO approach wherein a Non-Uniform Rational Basis Spline (NURBS) hyper-surface is used to represent the topological descriptor, i.e., the pseudo-density field. The properties of NURBS entities are exploited to derive the gradient of the physical responses involved in the problem formulation and to easily satisfy the minimum length scale requirement (related to manufacturing needs). The differences between TPE-based and WAFD-based formulations and the effectiveness of the proposed method are shown on 2D and 3D problems.

Introduction

Nowadays topology optimisation (TO) methods are increasingly used in several industrial sectors during the preliminary design phase of a product/system. TO is, indeed, knowing a new “era” mainly because of the development of modern additive manufacturing (AM) technologies, which can be exploited to manufacture products of complex shapes. Moreover, during the last 30 years, TO has gained an increasing attention to such an extent that, today, it constitutes a widespread research topic in different fields of study. The goal of TO is to determine the optimal distribution of the material, within a prescribed domain, to minimise a given merit function, while satisfying a set of design requirements. Among the different TO methods available in the literature, the solid isotropic material with penalisation (SIMP) approach [1], [2], and the level set method (LSM) [3], [4], [5], [6], are, undoubtedly, the most popular methodologies.

In the context of the SIMP approach [1], [2], the goal is to find the optimal distribution of a fictitious, heterogeneous material by introducing a pseudo-density field, ρ(x)[0,1], which affects, via a penalisation law, the stiffness tensor of the elements constituting the finite element (FE) model.

In the framework of the LSM, the FE model is used solely to evaluate the physical responses involved into the problem formulation. The topological descriptor is a level-set function, whose sign is conventionally associated to solid or void zones, while the zero value represents the boundary of the optimised structure [6]. A detailed discussion of the LSM for TO is available in [3], [4], [5], [6].

Further TO methods available in the literature include: the evolutionary structural optimisation (ESO) method [7] and its extension, i.e., the bi-directional evolutionary structural optimisation (BESO) method [8]. The ESO method is based on the combination of a metaheuristic algorithm and the FE method, whilst the BESO method is a generalisation of the former, which includes operators able to produce mesh-independent results (without checker-board pattern) and a sensitivity number averaging method, which allows avoiding convergence issues [9], [10].

Among the most recent TO methods allowing for both a design variables count reduction during optimisation and an explicit representation of the boundary of the optimised topology, it is worth to mention the moving morphable component (MMC) approach [11] and its dual counterpart, i.e., the moving morphable void (MMV) method [12]. An efficient and versatile method belonging to this class is the SIMP method reformulated in the framework of non-uniform rational basis spline (NURBS) hyper-surfaces [13], [14], [15], [16], [17], [18], [19], [20], [21]. Unlike classical density-based TO approaches [2], the NURBS-based SIMP method separates the pseudo-density field, describing the topology of the continuum, from the mesh of the FE model. More precisely, for general 3D problems, a 4D NURBS hyper-surface is used as a topological descriptor, whilst for 2D problems a standard 3D NURBS surface is employed. In this way, the topological descriptor, i.e., the pseudo-density field, relies on a purely geometric entity.

In the framework of the NURBS-based SIMP method, the computer aided design (CAD) reconstruction phase of the optimised topology becomes a trivial task [22], [23] because the topology boundary is available (at each iteration of the optimisation process) in a CAD-compatible format (thanks to the use of NURBS entities). Moreover, some fundamental properties of the NURBS basis functions, like the local support property, can be conveniently exploited to determine the gradient of the physical responses with respect to the topological variables, i.e., the pseudo-density evaluated at control points (CPs) and the related weights. Indeed, as discussed in [16], [17], NURBS entities allow for handling in the most efficient way the design requirements of geometrical nature (like minimum and maximum length scale requirements) during the optimisation process.

Regardless of the adopted TO method, it is a well-known fact that the most studied problem in the literature is the one dealing with the maximisation of the structural stiffness subject to a design requirement on the total volume/mass [2]. It is noteworthy that, in the vast majority of cases [1], [2], [4], [5], [7], [8], [10], [11], [12], [13], [14], a combination of non-zero boundary conditions (BCs) of the Neumann type (i.e., on generalised forces) and null BCs of the Dirichlet type (i.e., on generalised displacements) are imposed on the nodes belonging to the boundary. In such cases, the work of applied forces and displacements (WAFD) can be adopted as a measure of the structural stiffness.

Nevertheless, the WAFD cannot be considered as a measure of the structural stiffness when non-zero Dirichlet’s BCs are considered. To the best of the author’s knowledge, the effect of mixed non-zero Neumann-Dirichlet (ND) BCs on the structural maximisation problem formulation has been investigated only by few authors [24], [25], [26], [27]. In [24], the authors studied the influence of non-zero Dirichlet’s type BCs on both minimum compliance and maximum strength problems. However, the merit function used as a measure of the compliance in [24] was related to the total elastic energy (defined as twice the strain energy) of the continuum which equals the WAFD (under static equilibrium condition). The authors noticed that the two problem formulations lead to different design and the optimised topologies obtained in both cases are not characterised by a uniform distribution of the strain energy density.

Niu et al. [25] proposed a new formulation of structural maximisation problems under non-zero mixed BCs by introducing a new functional obtained as a linear combination of the WAFD and of the work done by reaction forces at nodes where non-zero Dirichlet’s BCs are applied. They provided a proof to determine the gradient of this functional, by showing that it is equal to the gradient of the compliance in the standard case of non-zero Neumann’s BCs and zero Dirichlet’s BCs. Moreover, they conducted a sensitivity analysis of the optimised topology to the applied non-zero displacement.

Later, and independently, Klarbring and Strömberg [26] and Barbarosie and Lopes [27] proved that the formulation of the structural maximisation problem proposed in [25] can be related, from a calculus of variations standpoint, to the minimisation of a merit function proportional to the total potential energy (TPE) of the continuum. In particular, Klarbring and Strömberg [26] formalised this concept by using a notation related to the discrete algebraic formulation of the FE method, by confirming the results found in [25]. On the other hand, Barbarosie and Lopes [27] proposed a formulation (referred to as generalised compliance) making use of the formalism of variational mechanics. By using a low-resolution density-based TO method, they highlighted that the results obtained by minimising the generalised compliance are different from those obtained by minimising the WAFD.

This work aims to shed a light on the influence of mixed non-zero BCs on the structural stiffness maximisation problem of anisotropic continua. Particularly, three theoretical/numerical aspects are discussed in this study.

Firstly, a proof simpler than the one proposed in [25] is provided to determine the gradient of the merit function related to the TPE.

Secondly, it is shown that, under mixed non-zero ND BCs, the optimisation problem making use of the TPE-related functional as structural stiffness descriptor is self-adjoint, while the one making use of the WAFD as merit function is not self-adjoint.

Finally, three sensitivity analyses are conducted for both problem formulations (i.e., minimisation of the WAFD and minimisation of the functional related to the TPE). The first one aims to assess the influence of the elastic symmetry type of the continuum on the optimised topology. The second one aims to investigate the influence of the non-zero Dirichlet’s type BCs on the solution. The last one aims to determine the influence of the design requirement on the volume fraction on the optimised solution.

A large campaign of numerical tests is conducted on both 2D and 3D benchmark problems: results highlight that, depending on the elastic symmetry type of the continuum and on the value of the constraint on the volume fraction, the optimal topologies resulting from the two problem formulations can be the same.

The reminder of the paper is as follows. The fundamentals of NURBS hyper-surfaces are briefly recalled in Section 2. The main features of the NURBS-based SIMP method and the two problem formulations involving the WAFD and the TPE as merit functions, respectively, are presented in Section 3. The numerical results are presented and discussed in Section 4, whilst Section 5 ends the paper with meaningful conclusions and prospects.

Upper-case bold letters and symbols are used to indicate tensors and matrices, while lower-case bold letters and symbols indicate column vectors. S denotes the cardinality of the generic set S.

Section snippets

NURBS hyper-surfaces

A NURBS hyper-surface is a polynomial-based function, defined as h:RNRD, where N is the dimension of the parametric space, whilst D is the dimension of the co-domain. The formula of a NURBS hyper-surface reads: h(ζ1,,ζN)i1=0n1iN=0nNRi1iN(ζ1,,ζN)yi1,,iN,where Ri1iN(ζ1,,ζN) are the piece-wise rational basis functions, which are related to the standard Bernstein’s polynomials Nik,pk(ζk), (k=1,,N) by means of the relationship: Ri1iNωi1iNk=1NNik,pk(ζk)j1=0n1jN=0nNωj1jNk=1NNjk,pk(

The NURBS-based SIMP method

A detailed description of the mathematical background of the NURBS-based SIMP method is available in [13], [14]. The main features of the approach are briefly described here only for 3D TO problems.

Numerical results

The effectiveness of the proposed method is illustrated on 2D and 3D benchmark problems taken from the literature [25], [27]. For each case, the pseudo-density field and the optimum topology are shown. The results presented here are obtained through the code SANTO (SIMP and NURBS for topology optimisation) developed at the I2M laboratory in Bordeaux [13], [14]. SANTO is coded in the Python® environment and can be interfaced with any FE code. In this study, the FE code ANSYS® is used to generate

Conclusions

In this work, three theoretical aspects of TO problems dealing with structural stiffness maximisation of anisotropic continua under mixed non-zero ND BCs have been discussed.

Firstly, it has been shown that the most natural measure of the structural stiffness is the so-called generalised compliance, i.e., a functional related to the TPE of the continuum, rather than the WAFD.

Secondly, it has been proved that the WAFD is not a self-adjoint functional under mixed non-zero ND BCs, whilst the

CRediT authorship contribution statement

Marco Montemurro: Conceptualisation, Methodology, Software, Validation, Writing – original draft, Writing – review & editing, Supervision, Funding acquisition.

Declaration of Competing Interest

The author declares that he has no known competing financial interests that could have appeared to influence the work reported in this paper.

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