Approximation of Periodic PDE Solutions with Anisotropic Translation Invariant Spaces

TL;DR

This paper presents a unified framework for approximating solutions to periodic PDEs in linear elasticity using anisotropic translation invariant spaces, extending FFT-based methods and connecting them to finite element approaches.

Contribution

It introduces a general approach that unifies various FFT-based discretizations and extends them to anisotropic lattices, linking to finite element methods.

Findings

Demonstrates the numerical advantages of the proposed framework.

Shows the connection between discrete solution spaces and finite element methods.

Extends FFT-based discretizations to anisotropic lattices.

Abstract

We approximate the quasi-static equation of linear elasticity in translation invariant spaces on the torus. This unifies different FFT-based discretisation methods into a common framework and extends them to anisotropic lattices. We analyse the connection between the discrete solution spaces and demonstrate the numerical benefits. Finite element methods arise as a special case of periodised Box spline translates.

Figures

Equations

$$\mathbf{h}\equiv \mathbf{k}\mathrm{mod}\mathbf{M}\iff \exists \mathbf{z}\in {\mathbb{Z}}^d:\mathbf{k}=\mathbf{h}+\mathbf{Mz}\text{.}$$

$$\Lambda (\mathbf{M}):={\mathbf{M}}^{-1}{\mathbb{Z}}^d=\{\mathbf{y}\in {\mathbb{R}}^d:\mathbf{My}\in {\mathbb{Z}}^d\},$$

$$V_{\mathbf{M}}^{f}\coloneqq\operatorname{span}\bigl{\{}\operatorname{\mathcal{T}}(\mathbf{y})f\,;\,\mathbf{y}\in\operatorname{\mathcal{P}}(\mathbf{M})\bigr{\}}$$

$$g=\sum _{\mathbf{y}\in P(\mathbf{M})}{a}_{\mathbf{y}}T(\mathbf{y})f,$$

$$\mathcal{F}(\mathbf{M})\coloneqq\frac{1}{\sqrt{m}}\Bigl{(}{\,\mathrm{e}}^{-2\pi{\mathrm{i}}\mathbf{h}^{\mathrm{T}}\mathbf{y}}\Bigr{)}_{\mathbf{h}\in\operatorname{\mathcal{G}}(\mathbf{M}^{\mathrm{T}}),\,\mathbf{y}\in\operatorname{\mathcal{P}}(\mathbf{M})},$$

$$\hat{\mathbf{a}}=({\hat{a}}_{\mathbf{h}}{)}_{\mathbf{h}\in G({\mathbf{M}}^{\mathrm{T}})}=\mathcal{F}(\mathbf{M})\mathbf{a},$$

$$c_{\mathbf{h}+{\mathbf{M}}^{\mathrm{T}}\mathbf{z}}(g)={\hat{a}}_{\mathbf{h}}{c}_{\mathbf{h}+{\mathbf{M}}^{\mathrm{T}}\mathbf{z}}(f)\text{for all }\mathbf{h}\in G({\mathbf{M}}^{\mathrm{T}}),\mathbf{z}\in {\mathbb{Z}}^d\text{,}$$

$$\bigl{[}\mathbf{a}\bigr{]}_{\mathbf{k}}^{\mathbf{M}}\coloneqq\sum_{\mathbf{z}\in\mathbb{Z}^{d}}a_{\mathbf{k}+\mathbf{M}^{\mathrm{T}}\mathbf{z}},\qquad\mathbf{k}\in\mathbb{Z}^{d}.$$

$$c_{\mathbf{k}}(D_{\mathbf{M}})=\{\begin{array}{ll}1,& \mathbf{k}\in G({\mathbf{M}}^{\mathrm{T}})\\ 0,& \text{else}\end{array},\mathbf{k}\in {\mathbb{Z}}^d$$

$$c_{\mathbf{k}}(f_{\mathbf{M},\alpha })=\frac{1}{\sqrt{m}}{B}_{\Xi }({\mathbf{M}}^{-\mathrm{T}}\mathbf{k}),\mathbf{k}\in {\mathbb{Z}}^d.$$

$${\Gamma }^0:\epsilon$$

$${\hat{\Gamma }}_{\mathbf{k}}^0:c_{\mathbf{k}}(\epsilon )$$

$${\nabla }_{\mathbf{k}}{c}_{\mathbf{k}}(u)$$

$$\bigl{\langle}\mathcal{C}^{0}\Gamma^{0}\mathcal{C}:\varepsilon,\gamma\bigr{\rangle}=-\bigl{\langle}\mathcal{C}^{0}\Gamma^{0}\mathcal{C}:\varepsilon^{0},\gamma\bigr{\rangle}$$

$$\bigl{\langle}\varepsilon+\Gamma^{0}\bigl{(}\mathcal{C}-\mathcal{C}^{0}\bigr{)}:\bigl{(}\varepsilon+\varepsilon^{0}\bigr{)},\gamma\bigr{\rangle}=0$$

$${\mathcal{C}}^{\mathrm{eff}}:{\epsilon }^0:={\int }_{{\mathbb{T}}^d}\mathcal{C}:\epsilon \mathrm{d}x$$

$${\Gamma }^{\mathrm{p}}:{\gamma }_{\mathbf{M}}$$

$${\hat{\Gamma }}_{\mathbf{h}}^{\mathrm{p}}:{\hat{G}}_{\mathbf{h}}$$

$$\bigl{\langle}\varepsilon_{\mathbf{M}}+\Gamma^{0}\bigl{(}\mathcal{C}-\mathcal{C}^{0}\bigr{)}:\bigl{(}\varepsilon_{\mathbf{M}}+\varepsilon^{0}\bigr{)},\gamma_{\mathbf{M}}\bigr{\rangle}=0$$

$$\sum_{\mathbf{y}\in\operatorname{\mathcal{P}}(\mathbf{M})}\Bigl{(}E_{\mathbf{y}}+\Gamma^{\mathrm{p}}_{\mathbf{y}}\bigl{(}\mathcal{C}(\mathbf{y})-\mathcal{C}^{0}\bigr{)}:\bigl{(}E_{\mathbf{y}}+\varepsilon^{0}\bigr{)}\Bigr{)}\bigl{(}\operatorname{\mathcal{T}}(\mathbf{y})\operatorname{I}_{\mathbf{M}}\bigr{)}(\mathbf{x})=\mathbf{0}$$

$$\bigl{\langle}\mathcal{C}^{0}\Gamma^{0}\mathcal{C}:\bigl{(}\varepsilon_{\mathbf{M}}+\varepsilon^{0}\bigr{)},\gamma_{\mathbf{M}}\bigr{\rangle}=0$$

$$\sum_{\mathbf{y}\in\operatorname{\mathcal{P}}(\mathbf{M})}\mathcal{C}^{0}\Gamma^{\mathrm{p}}_{\mathbf{y}}\mathcal{C}(\mathbf{y}):\bigl{(}E_{\mathbf{y}}+\varepsilon^{0}\bigr{)}\bigl{(}\operatorname{\mathcal{T}}(\mathbf{y})\operatorname{I}_{\mathbf{M}}\bigr{)}(\mathbf{x})=\mathbf{0}$$

$$\sum_{\mathbf{y}\in\operatorname{\mathcal{P}}(\mathbf{M})}\Bigl{(}E_{\mathbf{y}}+\Gamma^{\mathrm{p}}_{\mathbf{y}}\bigl{(}\mathcal{C}(\mathbf{y})-\mathcal{C}^{0}\bigr{)}:\bigl{(}E_{\mathbf{y}}+\varepsilon^{0}\bigr{)}\Bigr{)}\bigl{(}\operatorname{\mathcal{T}}(\mathbf{y})\operatorname{I}_{\mathbf{M}}\bigr{)}(\mathbf{x})=\mathbf{0}$$

$$\sum_{\mathbf{y}\in\operatorname{\mathcal{P}}(\mathbf{M})}\mathcal{C}^{0}\Gamma^{\mathrm{p}}_{\mathbf{y}}\mathcal{C}(\mathbf{y}):\bigl{(}E_{\mathbf{y}}+\varepsilon^{0}\bigr{)}\bigl{(}\operatorname{\mathcal{T}}(\mathbf{y})\operatorname{I}_{\mathbf{M}}\bigr{)}(\mathbf{x})=\mathbf{0}$$

Full text

1 Introduction

The simulation of composite materials is the foundation of many modern tools and applications. Typical simulated materials are fibre reinforced plastics or metal foams where one is interested in the characterization of their macroscopic elastic properties. These properties are described by the quasi-static equation of linear elasticity in homogenization with periodic boundary conditions. For this equation Moulinec and Suquet [8, 9] developed a FFT-based algorithm which approximates the solution using truncated Fourier series [10].

This manuscript uses anisotropic spaces of periodic translates [7, 1] for the approximation. It unifies and generalizes the approaches of Vondřejc, et.al [10] and Brisard and Dormieux [5]. The detailed analysis can be found in [3].

The continuous equation allows for two equivalent formulations. They have the same discrete solution in a space of translates if and only if the space yields the truncated Fourier series approach which corresponds to using spaces of translates generated by a Dirichlet kernel on anisotropic patterns [2].

The remainder is structured as follows: We introduce patterns and anisotropic spaces of translates and the partial differential equation in Section 2. The approximation of the PDE and connections between solution spaces stemming from the two formulations of the equation are explained in Section 3. The effects of this framework is then demonstrated numerically in Section 4 followed by a conclusion in Section 5.

2 Preliminaries

We first introduce the spaces of interest, namely translation invariant spaces of multivariate periodic square integrable functions $f\colon\mathbb{T}^{d}\to\mathbb{C}$, where $\mathbb{T}^{d}$ denotes the $d$-dimensional torus $\mathbb{T}^{d}\cong[-\pi,\pi)^{d}$. The translation invariance is defined with respect to patterns which generalize the usual rectangular sampling grids. On these patterns, a discrete Fourier transform is defined which can be employed to characterize the spaces of interest using Fourier coefficients of the involved functions. Next, two equivalent formulations of the equation of quasi-static elasticity with periodic boundary conditions are introduced.

2.1 Patterns and translation invariant spaces and the Fourier transform

For any regular matrix $\mathbf{M}\in\mathbb{Z}^{d\times d}$, we define the congruence relation for $\mathbf{h},\mathbf{k}\in\mathbb{Z}^{d}$ with respect to $\mathbf{M}$ by

[TABLE]

We define the lattice

[TABLE]

and the pattern $\operatorname{\mathcal{P}}(\mathbf{M})$ as any set of congruence representants of the lattice with respect to $\bmod\ 1$, e.g. $\Lambda(\mathbf{M})\cap[0,1)^{d}$ or \Lambda(\mathbf{M})\cap\bigl{[}-\tfrac{1}{2},\tfrac{1}{2}\bigr{)}^{d}. For the rest of the paper we will refer to the set of congruence class representants in the symmetric unit cube \bigl{[}-\tfrac{1}{2},\tfrac{1}{2}\bigr{)}^{d}, i.e. \operatorname{\mathcal{P}}(\mathbf{M})=\Lambda(\mathbf{M})\cap\bigl{[}-\tfrac{1}{2},\tfrac{1}{2}\bigr{)}. The generating set $\operatorname{\mathcal{G}}(\mathbf{M})$ is defined by $\operatorname{\mathcal{G}}(\mathbf{M})\coloneqq\mathbf{M}\operatorname{\mathcal{P}}(\mathbf{M})$ for any pattern $\operatorname{\mathcal{P}}(\mathbf{M})$. For both, the number of elements is given by $\lvert\operatorname{\mathcal{P}}(\mathbf{M})\rvert=\lvert\operatorname{\mathcal{G}}(\mathbf{M})\rvert=\lvert\det{\mathbf{M}}\rvert\eqqcolon m$, cf. [4, Lemma II.7].

A space of square integrable functions $V\subset L^{2}(\mathbb{T}^{d})$ is called $\mathbf{M}$-invariant, if for all $\mathbf{y}\in\operatorname{\mathcal{P}}(\mathbf{M})$ and all functions $f\in V$ the translates $\operatorname{\mathcal{T}}({\mathbf{y}})f\coloneqq f(\cdot-2\pi\mathbf{y})\in V$, i.e. especially

[TABLE]

is $\mathbf{M}$-invariant. A function $g\in V_{\mathbf{M}}^{f}$ is of the form

[TABLE]

where $\mathbf{a}=(a_{\mathbf{y}})_{y\in\operatorname{\mathcal{P}}(\mathbf{M})}\in\mathbb{C}^{m}$.

A function $\operatorname{I}_{\mathbf{M}}\in V^{f}_{\mathbf{M}}$ is called fundamental interpolant if $\operatorname{I}_{\mathbf{M}}(2\pi\mathbf{y})=1$ if $\mathbf{y}\equiv\mathbf{0}\mod\mathbf{M}$ and [math] else.

The discrete Fourier matrix on the pattern $\operatorname{\mathcal{P}}(\mathbf{M})$ is defined [6] by

[TABLE]

where $\mathbf{h}\in\operatorname{\mathcal{G}}(\mathbf{M}^{\mathrm{T}})$ the rows and $\mathbf{y}\in\operatorname{\mathcal{P}}(\mathbf{M})$ indicate the columns of the Fourier matrix $\mathcal{F}(\mathbf{M})$, and the discrete Fourier transform is defined by

[TABLE]

where $\mathbf{a}=(a_{\mathbf{y}})_{\mathbf{y}\in\operatorname{\mathcal{P}}(\mathbf{M})}\in\mathbb{C}^{m}$ and $\mathbf{\hat{a}}$ are arranged in the same ordering as the columns and rows in (2), respectively. A fast Fourier transform can be derived, see [1, Theorem 2].

For $f\in L^{2}(\mathbb{T}^{d})$ an easy calculation on the Fourier coefficients $c_{\mathbf{k}}(f)=\langle f,{\,\mathrm{e}}^{{\mathrm{i}}\mathbf{k}^{\mathrm{T}}\circ}\rangle$, $\mathbf{k}\in\mathbb{Z}^{d}$, using the unique decomposition of $\mathbf{k}\in\mathbb{Z}^{d}$ into $\mathbf{k}=\mathbf{h}+\mathbf{M}^{\mathrm{T}}\mathbf{z}$, $\mathbf{h}\in\operatorname{\mathcal{G}}(\mathbf{M}^{\mathrm{T}}),\ \mathbf{z}\in\mathbb{Z}^{d}$, yields, that $g\in V_{\mathbf{M}}^{f}$ holds if and only if [7, Theorem 3.3]

[TABLE]

holds, where $\mathbf{\hat{a}}=\mathcal{F}(\mathbf{M})\mathbf{a}$ for $\mathbf{a}$ from (1). Finally we define for a generalized sequence $\mathbf{a}=\{a_{\mathbf{k}}\}_{\mathbf{k}\in\mathbb{Z}^{d}}$ and a regular integer matrix $\mathbf{M}\in\mathbb{Z}^{d\times d}$ the bracket sum

[TABLE]

The discrete Fourier coefficients $c_{\mathbf{h}}^{\mathbf{M}}$ are related to the Fourier coefficients for a function $f\in A(\mathbb{T}^{d})$, where $A(\mathbb{T}^{d})$ denotes the Wiener Algebra, i.e. the space of functions with an absolutely convergent Fourier series.

Special cases for the choice of the space of translated $V_{\mathbf{M}}^{f}$ are given by the Dirichlet kernel $f=D_{\mathbf{M}}$ with

[TABLE]

and the de la Vallée Poussin means $f=f_{\mathbf{M},\alpha}$ given by

[TABLE]

The function $B_{\Xi}$ denotes a Box spline with $\Xi=(\operatorname{diag}(\alpha)\operatorname{Id}_{d})\in\mathbb{R}^{d\times 2d}$ and $\alpha\in[0,1]^{d}$.

2.2 Periodic homogenization in linear elasticity

FFT-based methods to solve the equations of periodic homogenization in linear elasticity were first introduced by Moulinec and Suquet [8, 9] and later interpreted as a Galerkin projection onto trigonometric sums by Vondřejc et.al. [10]. The partial differential equation (PDE) is formulated as a variational equation in terms of the symmetric strain $\varepsilon\in L^{2}(\mathbb{T}^{d})^{d\times d}$ with zero mean given a symmetric macroscopic strain $\varepsilon^{0}\in\mathbb{R}^{d\times d}$ and a stiffness distribution $\mathcal{C}\in L^{\infty}(\mathbb{T}^{d})^{d\times d\times d\times d}$. This stiffness is assumed to be uniformly elliptic, i.e. there exists a constant $c\in\mathbb{R}$ such that $\langle\mathcal{C}:\gamma,\gamma\rangle\geq c\lVert\gamma\rVert^{2}$ for all $\gamma\in L^{2}(\mathbb{T}^{d})^{d\times d}$ symmetric. Further, assume that $\mathcal{C}$ has the symmetries $\mathcal{C}_{ijkl}=\mathcal{C}_{jikl}=\mathcal{C}_{ijlk}=\mathcal{C}_{klij}$ for $i,j,k,l=1,\dots,d$.

The equation makes use of the Green operator $\Gamma^{0}$ defined by its action in frequency domain

[TABLE]

The positive definite reference stiffness $\mathcal{C}^{0}\in\mathbb{R}^{d\times d\times d\times d}$ with the same symmetries as $\mathcal{C}$ is a free parameter and $\Gamma^{0}:\varepsilon$ denotes the product of the fourth-order tensor $\Gamma^{0}$ with the second-order tensor $\varepsilon$.

With these definitions at hand there are two equivalent formulations of the PDE which yield the strain $\varepsilon$, cf. [10]: The variational equation (VE) states

[TABLE]

for all symmetric $\gamma\in L^{2}(\mathbb{T}^{d})^{d\times d}$ and the Lippmann-Schwinger equation (LS) requires

[TABLE]

to hold true for all symmetric $\gamma\in L^{2}(\mathbb{T}^{d})^{d\times d}$.

The effective stiffness matrix $\mathcal{C}^{\mathrm{eff}}$ is given by

[TABLE]

where $\varepsilon$ solves the PDE (12) with macroscopic strain $\varepsilon^{0}$.

3 Approximation of periodic PDE solutions

In the following the discretisation of the variational equation (11) and the Lippmann-Schwinger equation (12) on anisotropic translation invariant spaces is introduced. The choice of the function $f$ generating the space $V_{\mathbf{M}}^{f}$ influences properties of the discrete solution spaces. The connections between these spaces are detailed in Figure 1. The rest of this section follows this diagram and explains the relations therein. Further details and proofs can be found in [3].

In the following, the discretised version of the strain $\varepsilon$ is denoted by \varepsilon_{\mathbf{M}}\in\bigl{(}V_{\mathbf{M}}^{f}\bigr{)}^{d\times d} with $\varepsilon_{\mathbf{M}}=\sum_{\mathbf{y}\in\operatorname{\mathcal{P}}(\mathbf{M})}E_{\mathbf{y}}\operatorname{\mathcal{T}}(\mathbf{y})\operatorname{I}_{\mathbf{M}}$. Test functions $\gamma_{\mathbf{M}}$ are also expanded in terms of translates of $f$ by $\gamma_{\mathbf{M}}=\sum_{\mathbf{y}\in\operatorname{\mathcal{P}}(\mathbf{M})}G_{\mathbf{y}}\operatorname{\mathcal{T}}(\mathbf{y})\operatorname{I}_{\mathbf{M}}$. We further denote the discrete Fourier transforms of $\mathbf{E}=(E_{\mathbf{y}})_{\mathbf{y}\in\operatorname{\mathcal{P}}(\mathbf{M})}$ and $\mathbf{G}=(G_{\mathbf{y}})_{\mathbf{y}\in\operatorname{\mathcal{P}}(\mathbf{M})}$ by $\hat{\mathbf{E}}=(\hat{E}_{\mathbf{h}})_{\mathbf{h}\in\operatorname{\mathcal{G}}(\mathbf{M}^{\mathrm{T}})}=\operatorname{\mathcal{F}}(\mathbf{M})\mathbf{E}$ and $\hat{\mathbf{G}}=\operatorname{\mathcal{F}}(\mathbf{M})\mathbf{G}$, respectively.

The approximation of the solution of (11) and (12) requires the definition of the periodised Green operator.

Definition 1**.**

Let $f\in L^{2}(\mathbb{T}^{d})$, then the periodised Green operator $\Gamma^{\mathrm{p}}$ on $V_{\mathbf{M}}^{f}$ is defined in terms of its action on a symmetric field \gamma_{\mathbf{M}}\in\bigl{(}V_{\mathbf{M}}^{f}\bigr{)}^{d\times d} by

[TABLE]

With these definitions the discretisation of the PDEs on spaces of translates can be performed as follows:

Theorem 2**.**

Let the translates of $f$ be orthonormal and let $\mathcal{C}\in A(\mathbb{T}^{d})^{d\times d\times d\times d}$ with the usual symmetries. Then $\varepsilon_{\mathbf{M}}$ fulfils the weak form

[TABLE]

for all symmetric \gamma\in\bigl{(}V_{\mathbf{M}}^{f}\bigr{)}^{d\times d} if and only if the equation (VE in $V_{\mathbf{M}}^{f}$)

[TABLE]

holds true for all $\mathbf{x}\in\operatorname{\mathcal{P}}(\mathbf{M})$.

Theorem 3**.**

Let the translates of $f$ be orthonormal and let $\mathcal{C}\in A(\mathbb{T}^{d})^{d\times d\times d\times d}$ with the usual symmetries. Then $\varepsilon_{\mathbf{M}}$ fulfils the weak form

[TABLE]

for all symmetric \gamma\in\bigl{(}V_{\mathbf{M}}^{f}\bigr{)}^{d\times d} if and only if the equation (LS in $V_{\mathbf{M}}^{f}$)

[TABLE]

holds true for all $\mathbf{x}\in\operatorname{\mathcal{P}}(\mathbf{M})$.

Choosing the function $f$ as the Dirichlet kernel $f=D_{\mathbf{M}}$ reduces (18) and (20) to discretised equations which correspond to truncating the Fourier series arising in the continuous equations (11) and (12). The effect of the pattern matrix $\mathbf{M}$ in this special case is analysed in [2]. For the Dirichlet kernel on a tensor product grid, which corresponds to a diagonal pattern matrix, Vondřejc et.al. [10] prove the equivalence of the continuous equations and show that it carries over to the discrete solutions as well. The equations (VE) and (LS) discretised on such a tensor product grid using the Dirichlet kernel, i.e. $f=D_{\operatorname{diag}(\mathbf{m})}$ for $\mathbf{m}\in\mathbb{N}^{d}$ are denoted by (VE with $\Gamma^{0}$) and (LS with $\Gamma^{0}$) in the diagram, respectively.

The Dirichlet kernel proves to play a special role when analysing the equivalence of the discrete solutions.

Theorem 4**.**

Let the translates of $f$ be orthonormal and let $\mathcal{C}\in A(\mathbb{T}^{d})^{d\times d\times d\times d}$ with the usual symmetries. Then $\varepsilon_{\mathbf{M}}$ fulfils the equations

[TABLE]

and

[TABLE]

for all $\mathbf{x}\in\operatorname{\mathcal{P}}(\mathbf{M})$ if and only if $V_{\mathbf{M}}^{f}=V_{\mathbf{M}}^{D_{\mathbf{M}}}$.

The equivalence of the solutions in case of the space generated by the Dirichlet kernel is due to the representation of constant functions in frequency domain. In case of the Dirichlet kernel, i.e. in case of the discrete Fourier transform, a constant function $g$ is only characterized by the Fourier coefficient $c_{\mathbf{0}}(g)$. This is no longer true for other spaces of translates which leads to different solutions of (18) and (19).

Functions $f$ with compact support in frequency domain allow for an easy evaluation of the Bracket sum in (16). This is no longer the case when choosing $f$ compactly supported in space domain. However, such functions allow to include (simplified) finite elements into the framework by choosing $f$ to be a Box spline on the unit cell \mathbf{M}^{-1}\bigl{[}-\frac{1}{2},\frac{1}{2}\bigr{)}^{d}. The case of constant Box splines was already analysed by Brisard and Dormieux [5] in case of the Lippmann-Schwinger equation.

4 Numerical examples

The effect of choosing different spaces of translates $V_{\mathbf{M}}^{f}$ is demonstrated in the following. The generalized Hashin structure that consists of two confocal ellipsoids ($\Omega_{c}$) and ($\Omega_{e}$) embedded in a matrix material ($\Omega_{m}$) and is depicted in Figure 2 (left). For this structure an analytic expression for the strain $\varepsilon$ and for the effective stiffness $\mathcal{C}^{\mathrm{eff}}:\varepsilon^{0}$ is known and described in [2, Section 4.2]. The examples were computed using (18) using an iterative scheme based on a Neumann series approach, originally proposed by Moulinec and Suquet [9].

The following computation was performed with the pattern matrix \mathbf{M}=\bigl{(}\begin{smallmatrix}128&272\\ 0&128\end{smallmatrix}\bigr{)}, showing the minimal $\ell^{2}$-error in [2].

The relative logarithmic error $e_{\log}=\log(1+\lvert\varepsilon+\tilde{\varepsilon}\rvert)$ with $\tilde{\varepsilon}$ being the analytic solution is depicted for the Dirichlet kernel (middle) and de la Vallée Poussin means with slopes $\alpha=(0.4,0)^{\mathrm{T}}$ (right). The error using de la Vallée Poussin means shows significantly less Gibbs phenomenon and thus a smoother solution. The benefits of choosing $f=f_{\mathbf{M},\alpha}$ also get visible when looking at the $\ell^{2}$-error and the error in the effective stiffness. These errors are defined via $e_{\ell^{2}}=\lVert\varepsilon-\tilde{\varepsilon}\rVert\lVert\tilde{\varepsilon}\rVert^{-1}$ and $e_{\mathrm{eff}}=\lVert\tilde{\mathcal{C}}^{\mathrm{eff}}:\varepsilon^{0}-\mathcal{C}^{\mathrm{eff}}:\varepsilon^{0}\rVert\lVert\tilde{\mathcal{C}}^{\mathrm{eff}}:\varepsilon^{0}\rVert^{-1}$, respectively. The analytic effective stiffness matrix is denoted by $\tilde{\mathcal{C}}^{\mathrm{eff}}$.

The slopes $\alpha$ of the de la Vallée Poussin means were optimized to yield the smallest $e_{\mathrm{eff}}$ which is reduced by more than $33\%$. The $\ell^{2}$-error increases slightly from $0.022$ to $0.025$.

5 Conclusion

The introduced framework extends the truncated Fourier series approach to anisotropic spaces of translates and thus incorporates amongst others simplified finite elements. The Green operator $\Gamma^{0}$ emerges as a special case of Dirichlet kernel translates. Especially, the discrete solutions of the Lippmann-Schwinger equation and the variational equation coincide if and only if the space of translates $V_{\mathbf{M}}^{f}$ is the one generated by the Dirichlet kernel. Choosing de la Vallée Poussin means for $f$ leads to a reductions of the Gibbs phenomenon and allows for a better prediction of the effective stiffness of a composite material.

Continuing the work of [3] a direct extension towards periodic wavelets and finite elements with full integration are two points of future work. Furthermore, the convergence of the discretisation towards the continuous solution and convergence speed have to be investigated.