[Paper Review] A finite element implementation of surface elasticity at finite strains using the deal.II library

Andrew McBride, Ali Javili+2 more|arXiv (Cornell University)|Jun 2, 2015

Elasticity and Material Modeling1 references4 citations

TL;DR

This paper presents a fully non-linear finite element implementation of surface elasticity at finite strains using the open-source deal.II library, employing a total Lagrangian formulation with exact tangent contributions from both volume and surface domains. The implementation enables accurate modeling of surface effects in finite deformation regimes, with benchmark validation and full source code availability for nano-scale applications.

ABSTRACT

The potentially significant role of the surface of an elastic body in the overall response of the continuum can be described using the mature theory of surface elasticity. The objective of this contribution is to detail the finite element approximation of the underlying governing equations (both in the volume and on its surface) and their solution using the open-source finite element library deal.II. The fully-nonlinear (geometric and material) setting is considered. The nonlinear problem is solved using a Newton--Raphson procedure wherein the tangent contributions from the volume and surface are computed exactly. The finite element formulation is implemented within the total Lagrangian framework and a Bubnov-Galerkin spatial discretization of the volume and the surface employed. The surface is assumed material. A map between the degrees of freedom on the surface and on the boundary of the volume is used to allocate the contribution from the surface to the global system matrix and residual vector. The deal.II library greatly facilitates the computation of the various surface operators, allowing the numerical implementation to closely match the theory developed in a companion paper. Key features of the theory and the numerical implementation are elucidated using a series of benchmark example problems. The full, documented source code is provided.

Motivation & Objective

• To develop a finite element implementation of surface elasticity in the finite deformation regime, addressing limitations in existing numerical treatments that are restricted to infinitesimal deformations.
• To provide a fully non-linear, geometrically and materially consistent formulation using the total Lagrangian framework and Bubnov–Galerkin spatial discretization.
• To enable accurate modeling of surface effects in nanostructures by integrating surface and volume contributions with exact tangent matrix computation.
• To make the implementation open-source and fully documented, with parallelization support via TBB for performance on complex geometries.
• To bridge theoretical surface elasticity with practical numerical simulation by leveraging deal.II's codimension-one capabilities for curved manifolds.

Proposed method

• The finite element formulation is based on the total Lagrangian framework, with spatial discretization of both volume and surface domains using Bubnov–Galerkin methods.
• Surface contributions are mapped to the boundary of the volume via a degrees-of-freedom mapping, enabling consistent coupling between surface and volume degrees of freedom.
• The nonlinear system is solved using a Newton–Raphson scheme with exactly computed tangent contributions from both volume and surface weak forms.
• The deal.II library’s codimension-one routines are used to handle surface geometry and operators on embedded two-dimensional manifolds in three-dimensional space.
• The surface is modeled as a material membrane with a neo-Hookean-like free energy, extended to include surface tension for fluid-like behavior.
• Parallelization is achieved using the TBB library within a shared-memory paradigm, with support for distributed memory extensions under investigation.

Experimental results

Research questions

• RQ1 How can surface elasticity be accurately implemented in the finite deformation regime using a consistent finite element formulation?
• RQ2 What is the role of exact tangent matrix computation in ensuring convergence and stability in non-linear surface-elastic problems?
• RQ3 How can the deal.II library’s codimension-one capabilities be leveraged to model complex surface geometries in three-dimensional continua?
• RQ4 What are the numerical challenges associated with increasing surface-to-volume ratios upon mesh refinement, and how can they be mitigated?
• RQ5 To what extent can the implementation be parallelized for efficient simulation of realistic nanostructures with complex surface topologies?

Key findings

• The implementation successfully captures finite strain surface effects with high accuracy, validated through benchmark problems involving surface tension and curvature-dependent deformation.
• The use of exact tangent contributions in the Newton–Raphson scheme ensures quadratic convergence and robustness in non-linear iterations.
• The codimension-one feature of deal.II enables precise and efficient computation of surface operators, even on complex, curved surfaces embedded in 3D space.
• The full, documented source code is publicly available, facilitating reproducibility and extension to new material models or applications.
• Parallelization via TBB significantly improves performance for large-scale problems, with potential for further scaling in distributed environments.
• The framework is capable of simulating realistic nanostructures with complex surface geometry, demonstrating its utility beyond academic benchmarks.