A fast fully discrete mixed finite element scheme for fractional viscoelastic models of wave propagation (2410.01467v5)

Published 2 Oct 2024 in math.NA and cs.NA

Abstract: Due to the nonlocal feature of fractional differential operators, the numerical solution to fractional partial differential equations usually requires expensive memory and computation costs. This paper develops a fast scheme for fractional viscoelastic models of wave propagation. We first apply the Laplace transform to convert the time-fractional constitutive equation into an integro-differential form that involves the Mittag-Leffler function as a convolution kernel. Then we construct an efficient sum-of-exponentials (SOE) approximation for the Mittag-Leffler function. We use mixed finite elements for the spatial discretization and the Newmark scheme for the temporal discretization of the second time-derivative of the displacement variable in the kinematical equation and finally obtain the fast algorithm. Compared with the traditional L1 scheme for time fractional derivative, our fast scheme reduces the memory complexity from $\mathcal O(N_sN) $ to $\mathcal O(N_sN_{exp})$ and the computation complexity from $\mathcal O(N_sN^2)$ to $\mathcal O(N_sN_{exp}N)$, where $N$ denotes the total number of temporal grid points, $N_{exp}$ the number of exponentials in SOE, and $N_s$ the complexity of memory and computation related to the spatial discretization. Numerical experiments confirm the theoretical results.

Summary

The paper presents a fast numerical scheme that reformulates the viscoelastic constitutive law using an SOE approximation to reduce computational costs.
It integrates the SOE approximation with Newmark-beta and mixed finite element methods to achieve significant gains in memory and computational efficiency.
Numerical experiments validate the scheme’s accuracy and superior performance over traditional L1-based methods in long-time wave propagation simulations.

An Efficient Numerical Scheme for Fractional Viscoelastic Wave Propagation Models

Introduction

The paper of viscoelastic materials has gained prominence due to their unique properties, which exhibit a combination of both elastic and viscous behaviors. This paper proposes a fast numerical scheme for fractional viscoelastic models of wave propagation by leveraging an efficient Sum-of-Exponentials (SOE) approximation for the Mittag-Leffler function. The approach significantly improves computational and memory efficiency compared to traditional methods.

Background

Viscoelastic models are characterized by their ability to describe materials that exhibit both elastic (spring-like) and viscous (dashpot-like) behaviors. Classical models like Kelvin-Voigt, Maxwell, and Zener employ integer-order derivatives to capture these behaviors. However, fractional differential operators, which generalize integer-order operators, provide a more precise characterization of viscoelastic materials, particularly those with complex rheological properties.

The fractional viscoelastic model studied in this paper is described by a second-order temporal derivative for wave propagation, coupled with a constitutive equation involving a fractional derivative. The numerical challenge arises from the nonlocal nature of fractional derivatives, which demand high computational and memory resources when using traditional numerical methods such as the L1 scheme.

Contributions and Methodology

The authors' principal contributions include:

Constitutive Law Reformulation: Conversion of the constitutive equation into an integro-differential form with a Mittag-Leffler function as the convolution kernel.
SOE Approximation of Mittag-Leffler Function: Usage of Gaussian quadrature to approximate the Mittag-Leffler function efficiently, significantly reducing computational complexity.
Error Estimation: Provision of a theoretical error estimate for the SOE approximation, which was previously lacking in the literature.
Fast Numerical Scheme Development: Integration of the SOE approximation with the Newmark-beta scheme and mixed finite element methods to develop a fast scheme for solving the fractional viscoelastic wave propagation model.

By transforming the constitutive equation into a form amenable to SOE approximation, the authors reduce both the memory complexity from $\mathcal{O}(N_sN)$ to $\mathcal{O}(N_sN_{exp})$ and the computational complexity from $\mathcal{O}(N_sN^2)$ to $\mathcal{O}(N_sN_{exp}N)$ . Here, $N$ denotes the number of temporal grid points, $N_{exp}$ represents the number of terms in the SOE approximation, and $N_s$ captures the complexity related to spatial discretization.

Numerical Results

Numerical experiments are presented to verify the theoretical results and demonstrate the efficiency and accuracy of the proposed scheme. Two key tests include varying the parameters $l$ and $q$ , while keeping the fractional order $\alpha$ and tolerance error $\epsilon$ fixed, and varying $\epsilon$ while holding $\alpha$ , $l$ , and $q$ constant. The results confirm that the SOE approximation achieves the desired accuracy ( $\mathcal{O}(\epsilon)$ ) with significantly fewer terms compared to traditional methods.

Moreover, when applied to a model problem representing wave propagation in a viscoelastic medium, the fast scheme provided computational and memory advantages without compromising accuracy. The results indicate that the fast scheme consistently outperforms the L1-Newmark scheme, particularly for long-time simulations, making it highly suitable for practical applications involving complex viscoelastic materials.

Implications and Future Work

The proposed scheme's implications are notable for both theoretical and practical applications in computational mechanics and material science. The significant reduction in computational and memory resources opens up possibilities for high-fidelity simulations of complex viscoelastic systems over extended periods, which are otherwise infeasible with traditional methods.

Future research may explore extending the fast scheme to three-dimensional problems and other types of fractional differential equations arising in different scientific domains. Additionally, adaptive strategies for optimizing the choice of parameters in the SOE approximation could further enhance computational efficiency.

Conclusion

This paper provides a significant advancement in the numerical simulation of fractional viscoelastic wave propagation models by proposing a fast and efficient scheme based on an SOE approximation of the Mittag-Leffler function. The scheme's theoretical foundations are rigorously validated through numerical experiments, demonstrating its superior performance over traditional methods. This work promises to facilitate more efficient and accurate simulations of complex viscoelastic materials, contributing to advancements in various fields of science and engineering.

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