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A displacement-based finite element formulation for incompressible and nearly-incompressible cardiac mechanics

DOI:10.1016/j.cma.2014.02.009
Authors:
Myrianthi Hadjicharalambous at King's College London
Myrianthi Hadjicharalambous
Jack Lee at King's College London
Jack Lee
Nicolas P. Smith
Nicolas P. Smith
Nicolas P. Smith
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David Nordsletten at University of Michigan
David Nordsletten
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Abstract and Figures

The Lagrange Multiplier (LM) and penalty methods are commonly used to enforce incompressibility and compressibility in models of cardiac mechanics. In this paper we show how both formulations may be equivalently thought of as a weakly penalized system derived from the statically condensed Perturbed Lagrangian formulation, which may be directly discretized maintaining the simplicity of penalty formulations with the convergence characteristics of LM techniques. A modified Shamanskii–Newton–Raphson scheme is introduced to enhance the nonlinear convergence of the weakly penalized system and, exploiting its equivalence, modifications are developed for the penalty form. Focusing on accuracy, we proceed to study the convergence behavior of these approaches using different interpolation schemes for both a simple test problem and more complex models of cardiac mechanics. Our results illustrate the well-known influence of locking phenomena on the penalty approach (particularly for lower order schemes) and its effect on accuracy for whole-cycle mechanics. Additionally, we verify that direct discretization of the weakly penalized form produces similar convergence behavior to mixed formulations while avoiding the use of an additional variable. Combining a simple structure which allows the solution of computationally challenging problems with good convergence characteristics, the weakly penalized form provides an accurate and efficient alternative to incompressibility and compressibility in cardiac mechanics.
The undeformed and deformed body under consideration. Here X represents the reference state of the body, C ¼ CD [ CN its boundaries subject to Dirichlet and traction conditions, respectively.
The undeformed and deformed body under consideration. Here X represents the reference state of the body, C ¼ CD [ CN its boundaries subject to Dirichlet and traction conditions, respectively.
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Comparison over the cardiac cycle: (a) Display of the pressure-volume loops of the weakly penalized solution of the cardiac cycle on the first three meshes. These pressure-volume loops are converging to the pressure-volume loop of the LM solution on mesh 4. (b) The L 2 norm (top) and H 1 semi-norm (bottom) comparison of the displacement error between the penalty and weakly penalized formulations (k ¼ 10 7 ) in different phases of the cardiac cycle on mesh 2. Letters A, B, C, D map the time in cycle (b) to cardiac phase on the pressure-volume loop (a).
Comparison over the cardiac cycle: (a) Display of the pressure-volume loops of the weakly penalized solution of the cardiac cycle on the first three meshes. These pressure-volume loops are converging to the pressure-volume loop of the LM solution on mesh 4. (b) The L 2 norm (top) and H 1 semi-norm (bottom) comparison of the displacement error between the penalty and weakly penalized formulations (k ¼ 10 7 ) in different phases of the cardiac cycle on mesh 2. Letters A, B, C, D map the time in cycle (b) to cardiac phase on the pressure-volume loop (a).
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Comparison of the convergence behavior of the penalty and weakly penalized formulations when a lower order interpolation scheme is used: The errors between the (red) penalty and (black) weakly penalized forms (u) and the fine grid solution to the incompressible (uinc) elongation problem using linear interpolations are illustrated. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)
Comparison of the convergence behavior of the penalty and weakly penalized formulations when a lower order interpolation scheme is used: The errors between the (red) penalty and (black) weakly penalized forms (u) and the fine grid solution to the incompressible (uinc) elongation problem using linear interpolations are illustrated. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)
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The undeformed and deformed body under consideration. Here Ω represents the reference state of the body, Γ=ΓD∪ΓN its boundaries subject to Dirichlet and traction conditions, respectively.
The undeformed and deformed body under consideration. Here Ω represents the reference state of the body, Γ=ΓD∪ΓN its boundaries subject to Dirichlet and traction conditions, respectively.
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Illustration of the absolute difference (error) in fiber strain (Eff) at 361ms, between the Penalty and LM method (a, d, g) and the weakly penalized and LM method (b, e, h) on the first three meshes (colors representing values of the error between 0 (blue) and 0.025 (red)). Additionally, the fiber strain for the LM method on each mesh is displayed (c, f, i), with colors ranging from blue (-0.1) to red (0.1). Figures created in CMGUI [69]. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)
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Illustration of the absolute difference (error) in fiber strain (Eff) at 361ms, between the Penalty and LM method (a, d, g) and the weakly penalized and LM method (b, e, h) on the first three meshes (colors representing values of the error between 0 (blue) and 0.025 (red)). Additionally, the fiber strain for the LM method on each mesh is displayed (c, f, i), with colors ranging from blue (-0.1) to red (0.1). Figures created in CMGUI [69]. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)
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A displacement-based finite element formulation for incompressible and nearly-incompressible cardiac mechani
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... Alternatively one could use a nearly incompressible formulation, as done in Telle et al. (2021), but with a much higher penalty parameter for the intracellular subdomain. High penalty parameters are, however, associated with locking (Hadjicharalambous et al. 2014;Karabelas et al. 2022), which both can lead to numerical instabilities and underestimation of variables of interest. For these reasons we chose an incompressible formulation. ...
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