Dual Weighted Residual-driven adaptive mesh refinement to enhance biomechanical simulations

TL;DR

This work develops and validates a Dual Weighted Residual (DWR) based, goal-oriented adaptive mesh refinement framework for biomechanical simulations, spanning linear small-strain elasticity, nonlinear hyperelastic soft tissues, and fluid-structure interaction (FSI). By formulating exact and discrete dual problems and deriving local error indicators, it enables targeted refinement to reduce discretization error in chosen quantities of interest $J$, while handling multi-goal scenarios and complex multiphysics. Numerical experiments on arterial plaque with fiber activation, silicone hyperelasticity, and stationary FSI demonstrate substantial reductions in QoI error and informative refinement patterns, along with discussions on modeling, discretization, and numerical error balancing. The approach supports extensibility to time-space adaptivity and parameter calibration, offering a practical, rigorous tool for efficient and accurate patient-specific biomechanical simulations with potential clinical impact.

Abstract

This chapter describes how a posteriori error estimates targeting a user-defined quantity of interest, using the Dual Weighted Residual (DWR) technique, can be easily applied for biomechanical simulations in current engineering practice. The proposed method considers a very general setting that encompasses complex geometries, model non-linearities (hyperelasticity, fluid-structure interaction) and multi-goal oriented techniques. The developments are substantiated with some numerical tests.

Figures (11)

• Figure 1: Flowchart for adaptive mesh refinement.
• Figure 2: Artery model. Geometry, from floch_vulnerable_2009 (left), fiber orientation (center) and region of interest (right).
• Figure 3: Artery model. Displacement (left) and dual solution for $J$ (right).
• Figure 4: Artery mesh. Refinement driven by the QoI $J$. Initial mesh (left) with 1242 cells and a relative error of $38.3$ %, adapted meshes after 2 iterations (center) with 2079 cells and a relative error of $5.2$ % and after 6 iterations (right) with 15028 cells and a relative error of 3.4 %.
• Figure 5: Artery model. Left: relative error for the quantity of interest $J$vs. the number $N$ of cells in the case of uniform (blue) and adaptive (red) refinement. Right: effectivity indices of $\eta_h$ (blue) and $\sum_K\eta_K$vs. the number of cells $N$ for the quantity of interest $J$.

Theorems & Definitions (1)

• Remark 1