Enhancing Biomechanical Simulations Based on A Posteriori Error Estimates: The Potential of Dual Weighted Residual-Driven Adaptive Mesh Refinement

TL;DR

The paper addresses accurate discretization error quantification in patient-specific biomechanical simulations by integrating the Dual Weighted Residual (DWR) method with adaptive mesh refinement. It handles 3D incompressible hyperelastic tissue models with arbitrary quantities of interest by using automatic differentiation to compute dual solutions for $Q(\mathbf{u},\mathbf{p})$. Validation on silicone samples and a patient-specific heel shows that goal-oriented refinement reduces discretization error in the QoI while keeping computational cost acceptable. The framework supports mesh-optimal biomechanical simulations and has potential for integration into clinical workflows.

Abstract

The Finite Element Method (FEM) is a well-established procedure for computing approximate solutions to deterministic engineering problems described by partial differential equations. FEM produces discrete approximations of the solution with a discretisation error that can be an be quantified with \emph{a posteriori} error estimates. The practical relevance of error estimates for biomechanics problems, especially for soft tissue where the response is governed by large strains, is rarely addressed. In this contribution, we propose an implementation of \emph{a posteriori} error estimates targeting a user-defined quantity of interest, using the Dual Weighted Residual (DWR) technique tailored to biomechanics. The proposed method considers a general setting that encompasses three-dimensional geometries and model non-linearities, which appear in hyperelastic soft tissues. We take advantage of the automatic differentiation capabilities embedded in modern finite element software, which allows the error estimates to be computed generically for a large class of models and constitutive laws. First we validate our methodology using experimental measurements from silicone samples, and then illustrate its applicability for patient-specific computations of pressure ulcers on a human heel.

Figures (7)

• Figure 1: Geometry of the silicone sample, position of the holes and initial mesh.
• Figure 2: lar
• Figure 4: First test case (silicone sample). Relative error of discretisation (left) and efficiency of the estimator (right).
• Figure 5: A part of a heel tissue model used in simulations in which its orientation corresponds to the situation when the patient is on bed.
• Figure 6: The heel tissue is considered to be fixed on the surface which has contact with the calcaneum, and on the upper surface, shown by gray colour in (\ref{['fig:heel_fixed_surface']}); the tissue surface where a pressure is applied is shown by red colour, whereas a region of interest is also shown by red colour (\ref{['fig:heel_neumann_and_region_interest']}).