We present a new, stable, mixed finite element (FE) method for linear elastostatics of nearly incompressible solids. The method is the automatic variationally stable FE method of Calo et al., in which we consider a Petrov–Galerkin weak formulation where the stress and displacement variables are in the space H(div) and H¹, respectively. This allows us to employ a fully conforming FE discretization for any elastic solid using classical FE subspaces of H(div) and H¹. Hence, the resulting FE approximation yields both continuous stresses and displacements. To ensure stability of the method, we employ the philosophy of the discontinuous Petrov–Galerkin method of Demkowicz and Gopalakrishnan and use optimal test spaces. Thus, the resulting FE discretization is stable even as the Poisson's ratio $\nu \to 0.5$, and the system of linear algebraic equations is symmetric and positive definite. Our method also comes with a built-in a posteriori error estimator as well as indicators which are used to drive mesh adaptive refinements. We present several numerical verifications of our method including comparisons to existing FE technologies.

中文翻译：

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一种用于近乎不可压缩线性弹性静力学的稳定混合有限元方法

我们提出了一种新的、稳定的、混合有限元 (FE) 方法，用于几乎不可压缩固体的线性弹性静力学。该方法是 Calo 等人的自动变分稳定有限元方法，其中我们考虑 Petrov-Galerkin 弱公式，其中应力和位移变量分别位于空间H ( div ) 和H ^{1 中}。这允许我们使用H ( div ) 和H ^{1 的}经典有限元子空间对任何弹性固体采用完全一致的有限元离散化. 因此，由此产生的有限元近似产生连续应力和位移。为了确保方法的稳定性，我们采用了 Demkowicz 和 Gopalakrishnan 的不连续 Petrov-Galerkin 方法的原理，并使用最佳测试空间。因此，即使泊松比为$\nu \to 0.5$，且线性代数方程组是对称正定的。我们的方法还带有一个内置的后验误差估计器以及用于驱动网格自适应细化的指标。我们提出了我们方法的几种数值验证，包括与现有有限元技术的比较。