A Unified Finite Element Matrix Condensation Method for Arbitrary Nodal Releases Across All Element Types

Abstract

This paper introduces a novel, completely generic matrix condensation method to model arbitrary degree-of-freedom (DOF) releases directly within individual finite element stiffness matrices. While classical static condensation is a staple for handling end releases in one-dimensional beam and frame elements, extending this capability to higher-dimensional continuum elements such as 2D plates/shells and 3D solids has traditionally been obstructed by boundary continuity issues and rotational DOF limitations. Consequently, commercial solvers rely heavily on computationally expensive multipoint constraints (MPCs) or mesh-altering coincident nodes with discrete springs.

To resolve this limitation, we present a unified, element-independent mathematical operator that modifies the isolated element stiffness matrix prior to global assembly. The proposed method makes no underlying assumptions about the element formulation, spatial dimensions, or kinematic fields. It is capable of cleanly releasing any combination of translational or rotational degrees of freedom, including multiple simultaneous releases on a single element, while preserving the structural symmetry and rank stability of the system.