Mixed boundary conditions for FFT-based homogenization at finite strains

Abstract

In this article we introduce a Lippmann–Schwinger formulation for the unit cell problem of periodic homogenization of elasticity at finite strains incorporating arbitrary mixed boundary conditions. Such problems occur frequently, for instance when validating computational results with tensile tests, where the deformation gradient in loading direction is fixed, as is the stress in the corresponding orthogonal plane. Previous Lippmann–Schwinger formulations involving mixed boundary can only describe tensile tests where the vector of applied force is proportional to a coordinate direction. Utilizing suitable orthogonal projectors we develop a Lippmann–Schwinger framework for arbitrary mixed boundary conditions. The resulting fixed point and Newton–Krylov algorithms preserve the positive characteristics of existing FFT-algorithms. We demonstrate the power of the proposed methods with a series of numerical examples, including continuous fiber reinforced laminates and a complex nonwoven structure of a long fiber reinforced thermoplastic, resulting in a speed-up of some computations by a factor of 1000.

The Lippmann–Schwinger equation with mixed boundary conditions and gradient descent

Similarly to [25], we exhibit the fixed point scheme for the generalized Lippmann–Schwinger equation (12) as a gradient descent method, explaining its basic mechanisms and particularly proofing convergence of Algorithm 1.

Using the $\mathbb {C}^0$—weighted $L^2$—inner product

$$\begin{aligned} (S,T)\mapsto \int _Y S:\mathbb {C}^0:T\, \mathrm{d}x \end{aligned}$$

we want to compute the gradient of

$$\begin{aligned} f(F)=\int _Y {W(x,F(x))- \bar{P}:F(x)}\, \mathrm{d}x, \end{aligned}$$

i.e. setting

$$\begin{aligned} {\text {GRAD }} f = H_0 + \nabla w, \quad H_0\in \mathbb {R}^{3\times 3}, \quad \mathbb {P}:H_0=0 \end{aligned}$$

we need to solve

$$\begin{aligned} \int _Y {\text {GRAD }} f(F) :\mathbb {C}^0:G \, dx=Df(F)[G] \end{aligned}$$

(26)

for all G of the form $G=\langle G\rangle _Y+\nabla v$ with $\mathbb {P}:\langle G\rangle _Y=0$. Inserting the definition, (26) reads

$$\begin{aligned}&\int _Y (H_0 + \nabla w) :\mathbb {C}^0:(\langle G\rangle _Y+\nabla v) \, \mathrm{d}x \\&\quad =\int _Y (P(F)- \bar{P}):(\langle G\rangle _Y+\nabla v)\, \mathrm{d}x, \end{aligned}$$

which decomposes into the two equations

$$\begin{aligned}&\int _Y H_0 :\mathbb {C}^0:\langle G\rangle _Y \, \mathrm{d}x=\int _Y (\langle P(F)\rangle _Y - \bar{P}):\langle G\rangle _Y\, \mathrm{d}x\\&\int _Y \nabla w :\mathbb {C}^0:\nabla v \, \mathrm{d}x=\int _Y P(F):\nabla v\, \mathrm{d}x \end{aligned}$$

The first equation becomes

$$\begin{aligned} \mathbb {Q}:\mathbb {C}^0:\mathbb {Q}: H_0 = \mathbb {Q}:\big (\langle P(F)\rangle _Y - \bar{P}\big ), \end{aligned}$$

i.e. with $\mathbb {P}:\bar{P}=0$ and $\mathbb {M}=(\mathbb {Q}:\mathbb {C}^0:\mathbb {Q})^\dagger$ we have

$$\begin{aligned} H^0 = \mathbb {M}:\mathbb {Q}:\big (\langle P(F)\rangle _Y - \bar{P}\big ). \end{aligned}$$

The second equation implies

$$\begin{aligned} \nabla w=\varGamma ^0:P(F), \end{aligned}$$

see [25]. Consequently, the gradient reads

$$\begin{aligned} {\text {GRAD }} f(F)= \mathbb {M}:\mathbb {Q}:(\langle P(F)\rangle _Y - \bar{P}) + \varGamma ^0:P(F). \end{aligned}$$

A forward Euler discretization (with time step $\Delta t$) of the negative gradient flow for f becomes

$$\begin{aligned} F^{k+1}=F^k - \Delta t \left[ \mathbb {M}:\mathbb {Q}:(\langle P(F)\rangle _Y - \bar{P}) + \varGamma ^0:P(F)\right] . \end{aligned}$$

Inserting

$$\begin{aligned} F^k = \bar{F} + \mathbb {Q}:\langle F^k \rangle _Y + \varGamma ^0:\mathbb {C}^0:F^k \end{aligned}$$

we get

$$\begin{aligned} F^{k+1}= & {} (1-\Delta t)F^k - \Delta t \left[ -\bar{F} + \varGamma ^0:(P(F)-\mathbb {C}^0:F^k) \right. \\&\left. +\mathbb {M}:\mathbb {Q}:(\langle P(F)\rangle _Y - \bar{P}) - \mathbb {Q}:\langle F^k \rangle _Y \right] \end{aligned}$$

Using

$$\begin{aligned}&\mathbb {M}:\mathbb {Q}:(\langle P(F)\rangle _Y - \bar{P}) - \mathbb {Q}:\langle F^k \rangle _Y \\&\quad = \mathbb {M}:\mathbb {Q}:(\langle P(F) - C^0 F^k \rangle _Y - \bar{P}) \\&\qquad +\,\mathbb {M}:\mathbb {Q}:C^0:\langle F^k \rangle _Y- \mathbb {Q}:\langle F^k \rangle _Y \\&\quad = \mathbb {M}:\mathbb {Q}:(\langle P(F) \rangle _Y - C^0: \langle F^k \rangle _Y - \bar{P})\\&\qquad +\, \mathbb {M}:\mathbb {Q}:C^0:\bar{F} \end{aligned}$$

we arrive at the expression

$$\begin{aligned} F^{k+1}= & {} (1-\Delta t)F^k - \Delta t \left[ -\bar{F} - \mathbb {M}:(\bar{P}-\mathbb {Q}:C^0:\bar{F}) \right. \\&\left. +(\varGamma ^0+\mathbb {M}:\mathbb {Q}:\langle \cdot \rangle _Y) :(P(F)-\mathbb {C}^0: F^k)\right] , \end{aligned}$$

which, for time step $\Delta t=1$, is nothing but the Lippmann–Schwinger equation (12).