[Paper Review] A simple and efficient BEM implementation of quasistatic linear visco-elasticity

Christos G. Panagiotopoulos, V. Mantič+1 more|arXiv (Cornell University)|Feb 26, 2014

Elasticity and Material Modeling36 references3 citations

TL;DR

This paper presents a simple, stable, and efficient boundary element method (BEM) formulation for quasistatic linear visco-elasticity using implicit time discretization (Rothe method) and an algebraic variable transformation. The approach enables solution using only the Kelvin elastostatic fundamental solution, avoids inverse Laplace transforms, and supports multiple rheologies including Kelvin-Voigt, Maxwell, Boltzmann, Jeffreys, and Burgers models with minimal code modifications.

ABSTRACT

A simple, yet efficient procedure to solve quasistatic problems of special linear visco-elastic solids at small strains with equal rheological response in all tensorial components, utilizing boundary element method (BEM), is introduced. This procedure is based on the implicit discretisation in time (the so-called Rothe method) combined with a simple "algebraic" transformation of variables, leading to a numerically stable procedure (proved explicitly by discrete energy estimates), which can be easily implemented in a BEM code to solve initial-boundary value visco-elastic problems by using the Kelvin elastostatic fundamental solution only. It is worth mentioning that no inverse Laplace transform is required here. The formulation is straightforward for both 2D and 3D problems involving unilateral frictionless contact. Although the focus is to the simplest Kelvin-Voigt rheology, a generalization to Maxwell, Boltzmann, Jeffreys, and Burgers rheologies is proposed, discussed, and implemented in the BEM code too. A few 2D and 3D initial-boundary value problems, one of them with unilateral frictionless contact, are solved numerically.

Motivation & Objective

• To develop a computationally efficient and numerically stable BEM formulation for quasistatic linear visco-elastic problems.
• To eliminate the need for time-dependent fundamental solutions or inverse Laplace transforms in BEM-based visco-elastic analysis.
• To enable straightforward implementation in existing elastostatic BEM codes through a simple algebraic transformation of variables.
• To extend the formulation to unilateral frictionless contact problems involving visco-elastic bodies.
• To generalize the method to multiple standard visco-elastic rheologies, including Maxwell, Jeffreys, and Burgers models.

Proposed method

• Apply the Rothe method (implicit Euler time discretization) to the governing visco-elastic PDEs, transforming the time-dependent problem into a sequence of quasi-static problems.
• Introduce an algebraic variable transformation at each time step to convert the visco-elastic problem into a linear elastostatic boundary value problem.
• Solve the resulting elastostatic problem using the standard Kelvin fundamental solution of elasticity, avoiding domain integrals and complex time-dependent kernels.
• Reconstruct actual displacements, stresses, and strains from the auxiliary field solution at each time step via explicit algebraic recovery.
• Use collocation BEM for boundary discretization, ensuring no domain variables are required and enhancing computational efficiency.
• Extend the formulation to unilateral contact by incorporating energetic constraints and contact conditions into the boundary integral representation.

Experimental results

Research questions

• RQ1 Can a stable and efficient BEM formulation for quasistatic linear visco-elasticity be developed without requiring inverse Laplace transforms or time-dependent fundamental solutions?
• RQ2 How can the Kelvin-Voigt visco-elastic model be solved efficiently using only the elastostatic fundamental solution via time discretization and variable transformation?
• RQ3 To what extent can the proposed method be generalized to other standard visco-elastic models such as Maxwell, Boltzmann, Jeffreys, and Burgers rheologies?
• RQ4 How does the inclusion of unilateral frictionless contact affect the numerical behavior and stability of the BEM formulation?
• RQ5 Can the method be implemented in existing elastostatic BEM codes with minimal modifications while maintaining accuracy and stability?

Key findings

• The proposed method achieves numerical stability through discrete energy estimates, ensuring robustness for long-time simulations.
• The formulation successfully solves 2D and 3D initial-boundary value problems involving Kelvin-Voigt, Maxwell, Boltzmann, Jeffreys, and Burgers rheologies using only the Kelvin elastostatic fundamental solution.
• The time discretization with implicit Euler and variable transformation enables a recursive, efficient solution process without domain integration or complex transforms.
• Numerical results for a visco-elastic contact problem show that increasing viscosity reduces the contact zone length and the maximum normal traction, with viscous forces exhibiting a finite peak at load removal.
• The elastic resultant force decreases with increasing viscosity, and the viscous force component shows a jump and peak at the time of load removal, indicating energy dissipation and delayed recovery.
• The method is easily implementable in existing BEM codes by introducing a transformed auxiliary field, with minimal code changes required for visco-elastic analysis.