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

机译:基于位移的有限元公式,适用于不可压缩和几乎不可压缩的心脏力学

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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.
机译:拉格朗日乘数(LM)和惩罚方法通常用于增强心脏力学模型中的不可压缩性和可压缩性。在本文中,我们展示了如何将这两种公式等效地视为源自静态凝聚的扰动拉格朗日公式的弱惩罚系统,可以将其直接离散化,以保持惩罚公式的简单性和LM技术的收敛特性。引入了改进的Shamanskii-Newton-Raphson方案来增强弱罚系统的非线性收敛性,并利用其等效性对罚分形式进行了修改。关注准确性,我们针对简单的测试问题和更复杂的心脏力学模型,使用不同的插值方案来研究这些方法的收敛行为。我们的结果说明了锁定现象对惩罚方法(特别是对于低阶方案)的众所周知的影响及其对全循环力学精度的影响。此外,我们验证了弱惩罚形式的直接离散化会产生与混合制剂相似的收敛行为,同时避免使用其他变量。结合了简单的结构,可以解决具有良好收敛特性的计算难题,微弱惩罚形式为心脏力学中的不可压缩性和可压缩性提供了一种准确而有效的替代方法。

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