Pre-stressing (Inverse Mechanics)

Pre-stressing (Inverse Mechanics)#

In computational cardiac mechanics, we typically generate computational meshes from medical images (such as MRI or CT scans). These images acquire the heart geometry at a specific point in the cardiac cycle, often at End-Diastole (ED).

At this stage, the heart is not stress-free; it is loaded by the end-diastolic pressure and potentially by active tension. However, standard finite element mechanics formulations usually assume that the input mesh corresponds to a stress-free reference configuration.

Using a loaded geometry as a stress-free reference leads to incorrect stress predictions and unrealistic deformations. To correct this, we must solve an Inverse Problem (often called “pre-stressing” or “unloading”) to find the unknown stress-free configuration that, when loaded, matches the patient-specific geometry acquired from imaging.

The Problem Formulation#

Let:

  • \(\Omega_t\): The Target (Loaded) configuration. This is the geometry known from medical images.

  • \(\Omega_0\): The Reference (Unloaded) configuration. This is unknown.

  • \(\mathbf{t}\): The known traction (pressure) applied to the boundary.

We seek a mapping (or a displacement field) that recovers \(\Omega_0\) such that solving the forward mechanics problem on \(\Omega_0\) with load \(\mathbf{t}\) yields \(\Omega_t\).

fenicsx-pulse provides two distinct methods to solve this problem, demonstrated in the following examples.

Method 1: The Inverse Elasticity Problem (IEP)#

This method formulates the equilibrium equations directly on the known target configuration \(\Omega_t\). We solve for an inverse displacement field \(\mathbf{u}\) defined on the target mesh such that the reference coordinates \(\mathbf{X}\) are given by:

\[ \mathbf{X} = \mathbf{x} + \mathbf{u}(\mathbf{x}) \]

where \(\mathbf{x} \in \Omega_t\). The deformation gradient \(\mathbf{F}\) is computed as the inverse of the gradient of this mapping. This allows us to solve the problem in a single nonlinear solution procedure (typically using load stepping for stability).

Method 2: Fixed-Point Iteration (Backward Displacement)#

This method, often called the Backward Displacement Method, uses an iterative approach. It repeatedly solves the standard forward mechanics problem to update the guess for the reference geometry.

Algorithm:

  1. Initialize the reference guess \(\Omega_0^0 = \Omega_t\).

  2. For iteration \(k = 0, 1, \dots\): a. Solve the forward problem on \(\Omega_0^k\) applying the known pressure to get displacement \(\mathbf{u}_k\). b. Update the reference nodes: \(\mathbf{X}^{k+1} = \mathbf{x}_{\text{target}} - \mathbf{u}_k\). c. Check convergence: \(||\mathbf{X}^{k+1} - \mathbf{X}^k|| < \text{tol}\).

This method is intuitive as it reuses the standard forward solver, but it may require multiple iterations to converge.

Summary of Differences#

Feature

Inverse Elasticity Problem (IEP)

Fixed-Point Iteration

Formulation

Solve equilibrium on \(\Omega_t\)

Iterative forward solves on \(\Omega_0^k\)

Mesh

Mesh remains fixed (\(\Omega_t\))

Mesh coordinates update every iteration

Computational Cost

Generally lower (one system solve)

Higher (multiple forward solves)

Implementation

Requires specific inverse forms

Wraps standard forward solver

Pulse Class

PrestressProblem

FixedPointUnloader

References#

[BRR24]

NA Barnafi, Francesco Regazzoni, and Davide Riccobelli. Reconstructing relaxed configurations in elastic bodies: mathematical formulations and numerical methods for cardiac modeling. Computer Methods in Applied Mechanics and Engineering, 423:116845, 2024. doi:10.1016/j.cma.2024.116845.

[Sel11]

M. Sellier. An iterative method for the inverse elasto-static problem. Journal of Fluids and Structures, 27(8):1461–1470, 2011. URL: https://www.sciencedirect.com/science/article/pii/S088997461100123X, doi:10.1016/j.jfluidstructs.2011.08.002.