# Improvement of Low Order Solid and Solid-Shell Finite Elements with Incompatible Modes / Enhanced Assumed Strains for Explicit Time Integration

Improvements beyond the well established fully underintegrated and selectively underin-tegrated 8-node solid and solid-shell finite elements are highly desirable. A popular element technology to provide better accuracy for low order finite elements in general nonlinear analysis is the method of non-conforming or incompatible modes [1]. Concerning their close connection to the method of enhanced assumed strains (EAS) [2] a considerable amount of research and literature has been devoted to formulate effective 8-node solid and solid-shell finite elements for 3D analysis. A typical drawback regarding computational efficiency is the requirement to condense the incompatible modes respectively the EAS terms via the solution of an extra equation system. While not being dramatic for implicit analysis where solving simultaneous equations dominates, this condensation procedure may considerably affect the computational performance of explicit analyses that are strictly related to element computa-tions. Hence standard treatment of incompatible modes tends to be rather unattractive for the usage in explicit methods, as very small time steps are required. With a modification associating mass terms to the standard incompatible modes [3] an explicit solution instead of the condensation procedure is possible leading to large improvements in computational efficiency. The latter suggestion has been implemented in LS-DYNA as a first step via the user interface and first experiences and results are presented in this contribution. Further discussed are the well-known instabilities related to incompatible mode finite elements, which are a major issue in implicit schemes [4,5] but can be controlled to a certain degree in explicit dynamics [6]. The results and characteristics are examined for linear and nonlinear numerical examples using selected 8-node solid formulations. First results look promising. References [1] Wilson EL, Taylor RL, Doherty WP, and Ghaboussi J. Incompatible Displacement Models. In: Fenves SJ, Perrone N, Robinson AR, Schnobrich WC, editors. Numerical and computer models in structural mechanics. Academic Press, p. 4357, 1973. [2] Simo JC, Rifai MS. A class of mixed assumed strain methods and the method of incompatible modes, Int J Numer Methods Eng, 29(8), 1595-1638, 1990. [3] Mattern S, Schmied C, and Schweizerhof K. Highly efficient solid and solid-shell finite elements with mixed straindisplacement assumptions specifically set up for explicit dynamic simulations using symbolic programming. Comput Struct, 154:210225, 2015. [4] Wriggers P, Reese S. A note on enhanced strain methods for large deformations. Comput Methods Appl Mech Eng, 135:201209, 1996. [5] Sussman T and Bathe K-J. Spurious modes in geometrically nonlinear small displacement finite elements with incompatible modes. Comput Struct, 140:1422, 2014. [6] Schmied C, Mattern S, Schweizerhof K. Enhanced low order solid finite elements using incompatible inertia in explicit time integration for instability reduction. In: 11th world congress on computational mechanics (WCCM XI), Barcelona, 2014.

14_Schmied_KIT.txt — 3.0 KB

## Dateiinhalt

Improvements beyond the well established fully underintegrated and selectively underin-tegrated 8-node solid and solid-shell finite elements are highly desirable. A popular element technology to provide better accuracy for low order finite elements in general nonlinear analysis is the method of non-conforming or incompatible modes [1]. Concerning their close connection to the method of enhanced assumed strains (EAS) [2] a considerable amount of research and literature has been devoted to formulate effective 8-node solid and solid-shell finite elements for 3D analysis. A typical drawback regarding computational efficiency is the requirement to condense the incompatible modes respectively the EAS terms via the solution of an extra equation system. While not being dramatic for implicit analysis where solving simultaneous equations dominates, this condensation procedure may considerably affect the computational performance of explicit analyses that are strictly related to element computa-tions. Hence standard treatment of incompatible modes tends to be rather unattractive for the usage in explicit methods, as very small time steps are required. With a modification associating mass terms to the standard incompatible modes [3] an explicit solution instead of the condensation procedure is possible leading to large improvements in computational efficiency. The latter suggestion has been implemented in LS-DYNA as a first step via the user interface and first experiences and results are presented in this contribution. Further discussed are the well-known instabilities related to incompatible mode finite elements, which are a major issue in implicit schemes [4,5] but can be controlled to a certain degree in explicit dynamics [6]. The results and characteristics are examined for linear and nonlinear numerical examples using selected 8-node solid formulations. First results look promising. References [1] Wilson EL, Taylor RL, Doherty WP, and Ghaboussi J. Incompatible Displacement Models. In: Fenves SJ, Perrone N, Robinson AR, Schnobrich WC, editors. Numerical and computer models in structural mechanics. Academic Press, p. 4357, 1973. [2] Simo JC, Rifai MS. A class of mixed assumed strain methods and the method of incompatible modes, Int J Numer Methods Eng, 29(8), 1595-1638, 1990. [3] Mattern S, Schmied C, and Schweizerhof K. Highly efficient solid and solid-shell finite elements with mixed straindisplacement assumptions specifically set up for explicit dynamic simulations using symbolic programming. Comput Struct, 154:210225, 2015. [4] Wriggers P, Reese S. A note on enhanced strain methods for large deformations. Comput Methods Appl Mech Eng, 135:201209, 1996. [5] Sussman T and Bathe K-J. Spurious modes in geometrically nonlinear small displacement finite elements with incompatible modes. Comput Struct, 140:1422, 2014. [6] Schmied C, Mattern S, Schweizerhof K. Enhanced low order solid finite elements using incompatible inertia in explicit time integration for instability reduction. In: 11th world congress on computational mechanics (WCCM XI), Barcelona, 2014.