The Hilbert Complex References CSFEM for Large Deformation of Solids: From Hilbert Complexes to Numerical Stability Ali Gerami Matin, Arzhang Angoshtari Civil and Environmental Engineering 1.Angoshtari, A., Shojaei, M. F., and Yavari, A. (2017). Compatible-strain mixed finite element methods for 2D compressible nonlinear elasticity. Computer Methods in Applied Mechanics and Engineering , 313, 596-631. 2. Arnold, D., Falk, R., and Winther, R. (2010). Finite element exterior calculus: from Hodge theory to numerical stability. Bulletin of the American mathematical society , 47(2), 281- 354. 3. Bru, J., and Lesch, M. (1992). Hilbert complexes. Journal of Functional Analysis , 108(1), 88-132. 4. Gerami Matin, A., Angoshtari, A. (2018). CSFEMs: From Nonlinear Elasticity Complex to Mixed Finite (In Progress). The trial space of the above FEMs for the displacement gradient K satisfy the classical Hadamard jump condition, which is a necessary condition for the compatibility of K. Thus, these mixed FEMs are called compatible-strain mixed FEMs (CSFEMs) for nonlinear elasticity. Introduction Implementation Figure 1. Preliminary results for 2D nonlinear elasticity [1]: The top row is the standard Cook's membrane problem that suggests good performance of CSFEMs in bending and in the near-incompressible regime. The second row is inhomogeneous compression of a plate; Some other FEMs for this example suffer from the hourglass instability, but CSFEMs perform well. The third row shows tension of a plate with a complex geometry; CSFEMs work well and give accurate approximations of stresses. The bottom row shows deformed configurations of a heterogeneous plate under 100% stretch with different material properties for its inhomogeneity; CSFEMs provide a convenient framework for modeling inhomogeneities. Colors in deformed configurations indicate the distribution of stress, where lighter colors correspond to larger stresses. We will numerically implement (1.1), using the High Performance Computing (HPC). As the first step, we consider the Laplace equation. This elliptic equation can be used to study quasi-linear governing equations of large deformations. FEM Formulation • The governing equations • Mixed formulation By using the underlying spaces of the Hilbert complex of nonlinear elasticity, we introduce the following mixed formulation for nonlinear elasticity. • Existing approaches: Mesh-Free Methods (Belytschko et al, 1994), GFEM (Strouboulis et al, 2000), XFEM (Belytschko et al, 1999), Reduced Integration & Stabilization (Reese & Wriggers, 2000), Enhanced Strain Methods (Simo & Armero, 1992), etc. • Limitations: Bending problems, near-incompressible regime, accurate calculation of stress, heterogeneous materials, domains with complex geometries, inelastic behaviors, etc. The speed up diagram for the parallel implementation of the Laplace equation shows by using the domain decomposition approach. The speed up is defined as the ratio of the uniprocessor execution time over parallel execution time. Speed up figure implies that the domain decomposition approach can significantly decreased run- time near 1/16 (by using 16 processors) in comparison with the sequential implementing. • Objective In comparison with the standard finite element methods for large deformations, CSFEMs will have more degrees of freedom, i.e., more unknowns per element. Thus, Parallel computation schemes will be also developed for an efficient implementation of CSFEMs. plastic deformations fracture deformations of biomaterials Large elastic and inelastic deformations of solids challenge Developing stable numerical methods for such engineering applications SEM image of a micro pillar after plastic deformation (Khalajhedayati, 2015) Fracture in steel bar Mechanical Strength of Cornea (Khalajhedayati, 2015) develop a new class of stable FEMs for large elastic and inelastic deformations of solids and shells. Main tool Hilbert Complexes Some preliminary results for 2D nonlinear elasticity [1] suggest that the proposed numerical methods will potentially have the following features: Optimal convergence rates free from numerical artifacts and instabilities very good performance on domains with complex geometries accurate and mesh- independent approximatio ns of stress a convenient framework for modeling in- homogeneitie s Future works CSFEM has widespread applications in mechanical engineering, bioengineering, robotic problems. For instance; modeling of the soft robots, finding the residual stress in bio organs that will be significant step for prediction of the tumor growth.