The stiffest designs of elastic plates: Vector optimization for two loading conditions

doi:10.1016/j.cma.2011.01.003

Computer Methods in Applied Mechanics and Engineering

Volume 200, Issues 17–20, 1 April 2011, Pages 1708-1728

https://doi.org/10.1016/j.cma.2011.01.003 Get rights and content

Abstract

The paper deals with optimal design of linearly elastic plates of the Kelvin moduli being distributed according to a given pattern. The case of two loading conditions is discussed. The optimal plate is characterized by the minimum value of the weighted sum of the compliances corresponding to the two kinds of loads. The problem is reduced to the equilibrium problem of a hyperelastic mixture of properties expressed in terms of two stress fields. The stress-based formulation (P) is rearranged to the displacement-based form (P^∗). The latter formulation turns out to be well-posed due to convexity of the relevant potential expressed in terms of strains. Due to monotonicity of the stress–strain relations the problem (P^∗) is tractable by the finite element method, using special Newton’s solvers. Exemplary numerical results are presented delivering layouts of variation of elastic characteristics for selected values of the weighting factors corresponding to two kinds of loadings.

Introduction

The problem of maximization of the overall stiffness of a structure can be formulated in several manners. One can assume that two or three materials are at our disposal, the amount of both being given. The aim is to find the optimal placement of these materials, see e.g. Cherkaev [11] and Allaire [1]. The first paper on two-material plates is due to Gibiansky and Cherkaev [15], where the problem was formulated in a proper setting. Majority of papers in this vein are targeted at the shape design, see Bendsøe and Sigmund [6] and Allaire [1]. The papers mentioned are based on the relaxation by homogenization method, with using a specific energy estimate, called translation method, cf. Chapter VI in Ref. [21].

The alternative method is FMD (free material design approach) where all the anisotropic properties are treated as design variables of non-uniform distribution, cf. Ringertz [29], Bendsøe et al. [4], [5], see chapter 3 in the book by Bendsøe [3], cf. Banichuk [2]; an alternative approach can be found in Taylor [38] and Du and Taylor [13], Gaile et al. [14]. The Hooke tensor encompasses all these properties. In FMD no point-wise conditions are imposed on this tensor; instead an integral isoperimetric condition is assumed involving e.g. the Frobenius norm as the integrand. This approach is still developed in papers by Hörnlein et al. [16] and Kočvara and Stingl [17], see the web page of the PLATO-N Project coordinated by Bendsøe.

The essential drawback of the FMD approach as outlined above is the unclear physical meaning of the isoperimetric condition. The relaxation by homogenization approach was free from this drawback, since the isoperimetric condition is there imposed on the volume of the materials being at our disposal. The mass density was also linked with Hooke’s tensor. The FMD does not use the density of mass, since the Hooke tensor itself is viewed as design variable, with no link to the underlying microstructure. The other drawback of FMD is the very result which leads to one non-zero elastic modulus. This degeneracy is a direct consequence of the formulation in which the moduli must follow the one stress field.

The present paper puts forward a new version of the FMD in which the Kelvin moduli are viewed as distributed according to a given pattern, as announced in Lewiński [20] and in Czarnecki and Lewiński [12]. The other elastic characteristics ω_K, K = 1, 2, 3, the so called eigen-states, are design variables. We refer here to the theory of spectral decomposition of Hooke’s tensor developed in: Rychlewski [31], [32], [33], [34], Blinowski and co-workers [7], [8], [9], Theocaris and Sokolis [39], Sutcliffe [37], Moakher [24], Norris [26], Moakher and Norris [23]. No integral type isoperimetric condition is imposed. Two loading conditions are considered. The merit function is the weighted sum of the total compliances corresponding to two loading conditions. Introduction of two loading states will make the distribution of Hooke tensor unique, of lesser degeneracy than in the one loading case. Just this is the main reason to consider two types of loads. As in early paper by Gibiansky and Cherkaev [15] the departure point is stress-based. This makes the formulation a minimum problem with respect to: statically admissible stress fields and design variables.

The problem discussed brings about local minimization problems which turn out to be solvable by relatively elementary methods. By solving the local problems explicitly, the main problem is rearranged to the form of an equilibrium problem of a hypothetic hyperelastic body composed of two constituents, within a mixture theory. The constitutive equations couple two strain fields with two stress fields in a nonlinear manner. These equations are inverted to the primal form, which makes it possible to recover the strain-based potential. This effective potential turns out to be convex thus corresponding to monotonicity of the stress–strain equations and to solvability of the optimization problem considered. The FEM is applied using quadrilateral elements with bilinear shape functions. We show that a special Newton type algorithm is a right tool to solve a rich family of optimization problems for various loading cases and contrasts between the Kelvin moduli.

The following conventions are adopted: the space of symmetric tensors of degree 2 is denoted by $E_{s}^{2}$ . The summation convention applies to repeated indices at all levels. Small Latin indices run over 1, 2.

Section snippets

The plane problem of linear elasticity. Equilibrium for two loading cases

Consider a thin elastic plate loaded in plane. Assume that the plane stress assumptions are fulfilled. The plate is subject to a boundary loading $\overset{α}{T}$ acting on a part Γ₁ of the boundary of a given plane domain Ω. Two kinds of the loading will be considered, indexed by α = 1 or 2. The plate is fixed along Γ₂ being a part of ∂Ω. Assume that Ω is parameterized by Cartesian coordinates (x₁, x₂) with the orthonormal basis (e₁, e₂). Let the virtual displacement field v = (v₁, v₂), referred to Ω, be

Formulation of an optimal design problem for two loading conditions

In the present section the case of two kinds of loads acting independently will be discussed. The aim is to find optimal distribution of the eigenstates ω₁(x), ω₂(x), ω₃(x) to make the plate as stiff as possible with respect to both the load cases. The distribution of the Kelvin moduli λ₁(x), λ₂(x), λ₃(x) within Ω is viewed as prescribed. These moduli do not undergo optimization. Since ω₁(x) is determined by ω₂(x) and ω₃(x), only the latter two fields will be design variables. The fields ω₂(x)

Construction of potential W_η

Let $σ, τ \in E_{s}^{2}$ and choose (a, b) ∈ Q. Let us define the auxiliary function $g (σ, τ, a, b) = η k (σ, a, b) + (1 - η) k (τ, a, b),$ which can be put in the form $g (σ, τ, a, b) = \frac{η}{λ_{1}} ‖ σ ‖^{2} + \frac{1 - η}{λ_{1}} ‖ τ ‖^{2} + h (σ, τ, a, b)$ with $h (σ, τ, a, b) = H (\sqrt{η} σ, \sqrt{1 - η} τ, a, b),$ where $H (σ, τ, a, b) = ν_{2} [(a \cdot σ)^{2} + (a \cdot τ)^{2}] + ν_{3} [(b \cdot σ)^{2} + (b \cdot τ)^{2}] .$ The problem (3.5) can be rewritten as follows: $W_{η} (σ^{'}, σ^{″}) = \frac{η}{λ_{1}} ‖ σ^{'} ‖^{2} + \frac{1 - η}{λ_{1}} ‖ σ^{″} ‖^{2} + W_{η}^{'} (σ^{'}, σ^{″}),$ where $W_{η}^{'} (σ, τ) = \min \{H (\sqrt{η} σ, \sqrt{1 - η} τ, a, b)| (a, b) \in Q\} .$ We see that $W_{η}^{'} (σ^{'}, σ^{″}) = U_{η} (\sqrt{η} σ^{'}, \sqrt{1 - η} σ^{″})$ with $U_{η} (σ, τ) = \min \{H (σ, τ, a, b)| (a, b) \in Q\} .$ The latter function will be found explicitly. To this end it is

Construction of the optimal Hooke tensor C

Let us come back to the result (4.40). The minimizing value of y = y^∗ is such that $y^{*} = - \hat{φ} + π,$ where $\tan \hat{φ} = - \frac{ζ \sin y_{0}}{1 + ζ \cos y_{0}}$ and y₀ = 2α. Hence $y^{*} = π + \arctan (\frac{ζ \sin y_{0}}{1 + ζ \cos y_{0}}) .$ Minimum in (4.37) is attained for $x^{*} = \frac{1}{2} y^{*}$ with y₀ = 2α, being determined by (4.44). Thus the angle x in Fig. 3 is given as above. Our task is now to express ω₁ and ω₃ in terms of $\hat{σ}$ and $\tilde{τ}$ being non-colinear. Let us redraw Fig. 3 as in Fig. 4.

We introduce the unit vector τ^⊥, orthogonal to $\tilde{τ}$ . $τ^{⊥} = z_{1} \tilde{τ} + z_{2} \hat{σ} .$ We shall find z₁, z₂ from the

Question of well-posedness of problem (P)

Let us focus our attention on problem (P) cf. (3.7). The relevant Lagrangian has the form $L_{η} = \int_{Ω} W_{η} (\overset{1}{τ} (x), \overset{2}{τ} (x)) dx + \int_{Γ_{1}} \overset{1}{T} \cdot \overset{1}{v} ds - \int_{Ω} {\overset{1}{τ}}_{ij} ε_{ij} (\overset{1}{v}) dx + \int_{Γ_{1}} \overset{2}{T} \cdot \overset{2}{v} ds - \int_{Ω} {\overset{2}{τ}}_{ij} ε_{ij} (\overset{2}{v}) dx$ with $ε_{ij} (v) = \frac{1}{2} (\frac{\partial v_{i}}{\partial x_{j}} + \frac{\partial v_{j}}{\partial x_{i}}) .$ Here $\overset{1}{v}, \overset{2}{v}$ are kinematically admissible; they vanish on Γ₂. These fields are also Lagrangian multipliers for the statical admissibility conditions of $\overset{1}{τ}$ and $\overset{2}{τ}$ . The variation δL_η = 0 for all $δ {\overset{1}{τ}}_{ij}$ , $δ {\overset{2}{τ}}_{ij}$ . Let ${\overset{2}{τ}}_{ij} = 0$ . Then $δ L_{η} = \int_{Ω} [\frac{\partial W_{η}}{\partial {\overset{1}{τ}}_{ij}} - ε_{ij} (\overset{1}{v})] δ {\overset{1}{τ}}_{ij} dx .$ The above variational equation implies $ε_{ij} (\overset{α}{v}) = \frac{\partial W_{η} (\overset{1}{τ}, \overset{2}{τ})}{\partial {\overset{α}{τ}}_{ij}}$

Displacement-based formulation of the optimal problem considered

The optimal design problem (3.2) has been rearranged to the form (3.7) or (P). Problem (P) is expressed in terms of the stress fields. Since the available numerical methods of nonlinear mechanics are based on displacement fields as the primal unknowns, it is expedient to work with the problem dual to (P). The aim of this section is to put forward this formulation and deliver its equations explicitly. To this end we invert the nonlinear constitutive equations (6.20) arriving at $\begin{matrix} σ = \frac{1}{1 - ν^{2}} [ε + ν L (ε, ϰ)], \\ τ = \end{matrix}$

Numerical treatment of the problem (P^∗)

The problem (P^∗) formulated in Section 7 has not its counterpart in the literature. All what is known has been said above, the main feature being monotonicity of the constitutive equations (7.17). The degree of complexity of this problem resembles that of the equilibrium of strongly curved thin hyperelastic shells, where two stress resultant fields and two strain fields are involved. The constitutive equations are simultaneously coupled and nonlinear. The hitherto existing papers on numerical

Case studies

The aim of this section is to construct the optimal layouts of elastic moduli within rectangular plates of various kinematic boundary conditions subject to two kinds of in-plane surface loads: $\overset{1}{T} = ({\overset{1}{T}}_{1}, {\overset{1}{T}}_{2}), \overset{2}{T} = ({\overset{2}{T}}_{1}, {\overset{2}{T}}_{2})$ contributing to the compliance functional (3.22) with the weight factors $\overset{1}{η} = η$ and $\overset{2}{η} = 1 - η$ , respectively; η ∈ [0, 1]. The optimal moduli C_ijkl are chosen among all possible moduli referring to a given distribution of the Kelvin moduli λ₁, λ₂, λ₃ within the plate domain. In the examples

Final remarks

The main problem of the present paper, e.g. problem (3.2) of minimization of the weighted sum of compliances (corresponding to two kinds of loads applied non-simultaneously) over all fields of projectors P₁, P₂, P₃ which determine Hooke’s tensor (2.4) in each point of Ω has been reduced to problem (P), see (3.7) which can be viewed as an equilibrium problem of an effective two- component mixture of specific non-linear and anisotropic properties. This result makes it possible to solve the

Acknowledgements

The paper was prepared within the Research Grant no N506 071338, financed by the Polish Ministry of Science and Higher Education, entitled: Topology Optimization of Engineering Structures. Simultaneous shaping and local material properties determination.

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The stiffest designs of elastic plates: Vector optimization for two loading conditions

Abstract

Introduction

Section snippets

The plane problem of linear elasticity. Equilibrium for two loading cases

Formulation of an optimal design problem for two loading conditions

Construction of potential Wη

Construction of the optimal Hooke tensor C

Question of well-posedness of problem (P)

Displacement-based formulation of the optimal problem considered

Numerical treatment of the problem (P∗)

Case studies

Final remarks

Acknowledgements

Aeorosp. Sci. Technol.

Shape Optimization by the Homogenization Method

Optimization of anisotropic properties for continuum bodies and structural elements using spectral methods of tensor analysis

Mech. Struct. Mach.

Optimization of Structural Topology, Shape and Material

An analytical model to predict optimal material properties in the context of optimal structural design

J. Appl. Mech. Trans. ASME

Optimal design of material properties and material distribution for multiple loading conditions

Int. J. Numer. Methods Engrg.

Material interpolation schemes in topology optimization

Arch. Appl. Mech.

On the decomposition of the isotropic tensorial function in orthogonal bases

Bull. Pol. Acad. Sci. Tech. Sci.

Two-dimensional Hooke’s tensors-isotropic decomposition. Effective symmetry criteria

Arch. Mech.

On the elastic orthotropy

Arch. Mech.

Taschenbuch der Mathematik

Variational Methods for Structural Optimization

Application of an energy-based model for the optimal design of structural materials and topology

Struct. Multidiscip. Optim.

Free material optimization for stress constraints

Struct. Multidiscip. Optim.

Foam structures with a negative Poisson’s ratio

Science

Advances in negative Poisson’s ratio materials

Adv. Mater.

Construction of potential W_η

Numerical treatment of the problem (P^∗)