Efficient Solutions for Nonlocal Diffusion Problems Via Boundary-Adapted Spectral Methods

Abstract

We introduce an efficient boundary-adapted spectral method for peridynamic transient diffusion problems with arbitrary boundary conditions. The spectral approach transforms the convolution integral in the peridynamic formulation into a multiplication in the Fourier space, resulting in computations that scale as O(N log N). The limitation of regular spectral methods to periodic problems is eliminated using the volume penalization method. We show that arbitrary boundary conditions or volume constraints can be enforced in this way to achieve high levels of accuracy. To test the performance of our approach we compare the computational results with analytical solutions of the nonlocal problem. The performance is tested with convergence studies in terms of nodal discretization and the size of the penalization parameter in problems with Dirichlet and Neumann boundary conditions.

Appendices

Appendix 1. Boundary-adapted spectral method implementation for PD diffusion in MATLAB

Here the MATLAB implantation for boundary-adapted spectral method with volume penalization (BASM-VP) is provided. First, note that Eq. (23) can be directly used when the periodic domain of computation is [0, S), meaning the origin locates on the left end of the domain. If the domain of choice is [b, b + S), then the following modified form of Eq. (23) should be used:

$$ \frac{d{u}_i^N}{dt}=\nu {\mathbf{\mathcal{F}}}_{\boldsymbol{D}}^{-\mathbf{1}}\left({\overset{\sim }{\mu^s}}_k\tilde{u}_{k}\Delta x\right)-\nu \beta {u}_i^N+{f}_i^N $$

(67)

where ${\overset{\sim }{\mu^s}}_k$ is the DFT of the shifted kernel function:

$$ {\mu}^s(x)=\mu \left(x-b\right) $$

(68)

The reason is that the DFT definitions that govern the FFT solvers are based on [0, S) domain. If b = 0, then the kernel function does not shift and Eq. (67) becomes identical to Eq. (23).

A MATLAB implementation of the peridynamic BASM with VP for the transient diffusion example in Section 5.2 is as follows:

• Inputs:

• Physical parameters: ν, δ, f(x, t), μ(x), L, t_max

• Initial and boundary conditions: u(x, 0), $u\left(-\frac{L}{2},0\right)={u}_{b1}$, $u\left(\frac{L}{2},0\right)={u}_{b2}$

• BASM with VP parameters: N, ε

• Initialization:

• Calculate grid size: $\Delta x=\frac{L+2\delta }{N}$ (length of the extended domain divided by N)

• Calculate time step: Δt from Eq. (47) with ν, ε, and μ(x)

• Discretize the extended domain: ${x}_i=-\frac{L}{2}-\delta +\left(i-1\right)\Delta x$ and i = 1, 2, …, N

• Shift the kernel function based on left-end of the extended domain and discretize: ${\mu}_i^s=\mu \left({x}_i+\frac{L}{2}+\delta \right)$

• Discretize the initial condition and the Source term: ${y}_i^0=u\left({x}_i,0\right)$; ${f}_i^0=f\left({x}_i,0\right)$;

• Fast Fourier transform ${\mu}_i^s$ and ${y}_i^0$: ${\overset{\sim }{\mu^s}}_k=\mathbf{FFT}\left({\mu}_i^s\right)$ and ${\overset{\sim }{y^0}}_k=\mathbf{FFT}\left({y}_i^0\right)$

• Define constrained regions: ${\Gamma}_1={x}_i\in \left[-\frac{L}{2}-\delta, -\frac{L}{2}\right)$ and ${\Gamma}_2={x}_i\in \left(\frac{L}{2},\frac{L}{2}+\delta \right)$

• Define the main domain: $\Omega ={x}_i\in \left[-\frac{L}{2},\frac{L}{2}\right]$

• Discretize the mask function: χ_i = χ(x_i) from Eq. (25).

• Calculate volume constraints on Γ₁ and Γ₂ from Eq. (58) and (59): u_Γ1( Γ₁, 0), u_Γ2( Γ₂, 0)

Define ${y}_{\varGamma i}^0=\left\{\begin{array}{cc} Eq.\kern0.33em (58)& {x}_i\in \left[-\frac{L}{2}-\delta, -\frac{L}{2}\right)\\ {}0& {x}_i\in \left[-\frac{L}{2},\frac{L}{2}\right]\\ {} Eq.(59)& {x}_i\in \left(\frac{L}{2},\frac{L}{2}+\delta \right)\end{array}\right.$

• Initialize step counter: n = 0

• Initialize time: tⁿ = 0

• Solve the transient diffusion: while tⁿ < t_max

• Update time: t^{n + 1} = tⁿ + Δt

• Update solution: ${y}_i^{n+1}={y}_i^n+\Delta t\left[\nu \mathbf{FF}{\mathbf{T}}^{-\mathbf{1}}\left({\overset{\sim }{\mu^s}}_k{\overset{\sim }{y^n}}_k\Delta x\right)-\nu \beta {y}_i^n+{f}_i^n-\frac{\chi_i}{\varepsilon}\left({y}_i^n-{y}_{\Gamma, i}^n\right)\right]$

• Update the source term: ${f}_i^{n+1}={f}_i\left({x}_i,{t}^{n+1}\right)$

• Update volume constraints:${y}_{\Gamma i}^{n+1}=\left\{\begin{array}{cc}\mathrm{Eq}.(58)&\ {x}_i\in \left[-\frac{L}{2}-\delta, -\frac{L}{2}\right)\\ {}0\kern3em &\ {x}_i\in \left[-\frac{L}{2},\frac{L}{2}\right]\kern3em \\ {}\mathrm{Eq}.(59)& {x}_i\in \left(\frac{L}{2},\frac{L}{2}+\delta \right)\kern1.25em \end{array}\right.$

• Fast Fourier transform ${y}_i^{n+1}$: ${\overset{\sim }{y^{n+1}}}_k=\mathbf{FFT}\left({y}_i^{n+1}\right)$

• Update step counter: n = n + 1

The algorithm above is for the example with Dirichlet BCs. The corresponding MATLAB code is provided in Online Resource 1. In the case of Neumann BCs Eqs. (58) and (59) are replaced with Eqs. (65) and (66).

Appendix 2. Discretization error versus penalization error in BASM-VP

To obtain a better understanding of error distribution on the domain for the example in Section 5.2, and the evolution of maximum error during the diffusion process (see Figs. 6 and 7), we conducted two more simulations: one simulation with a much smaller ε compared with the test Section 5.2, but the same N, and one simulation with a much larger N compared with that test, but the same ε. The first simulation reveals the error behavior with respect to the discretization, while the second one is focused on the penalization error.

Results for the first simulation with ε = 5 × 10⁻⁶, N = 2⁹ are given in Fig. 11.

Fig. 11

Time snapshots of the relative error dominated by discretization in 1D nonlocal diffusion problem in Section 5.2 at t = 5 (a), and t = 15 (b). The time-evolution of the maximum error in (c)

The second simulation is performed with ε = 5 × 10⁻⁴ and N = 2¹⁵. Results are given in Fig. 12.

Fig. 12

Time snapshots of the relative error dominated by penalization in 1D nonlocal diffusion problem shown in Section 5.2 at t = 5 (a), and t = 15 (b). The time-evolution of maximum error in (c)

The discretization error rapidly grows and then decays, while the penalization error grows near the boundaries and approaches a constant value in time. Comparing Figs. 11 and 12 with Figs. 6 and 7 in Section 5.2 (see also video 1) helps us to clearly identify the “mixture” of the penalization and discretization errors in the example corresponding to Figs. 6 and 7.