Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity.(English)Zbl 1086.74038

Summary: A new finite element method for fourth-order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem. The method combines concepts from the continuous Galerkin (CG) method, the discontinuous Galerkin (DG) method and stabilization techniques.
A brief review of the CG method, the DG method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth-order elliptic problems. A continuous/discontinuous Galerkin (C/DG) method is proposed which uses $$C^0$$-continuous interpolation functions and is formulated in the primary variable only. The advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided. In the context of thin bending theory, the C/DG method leads to a formulation where displacements are the only degrees of freedom, and no rotational degrees of freedom need to be considered.
The main feature of the C/DG method is the weak enforcement of continuity of first and higher-order derivatives through stabilizing terms on interior boundaries. Consistency, stability and convergence of the method are shown analytically. Numerical experiments verify the theoretical results, and applications are presented for Bernoulli-Euler beam bending, Poisson-Kirchhoff plate bending and a shear layer problem using Toupin-Mindlin strain gradient theory.

MSC:

 74S05 Finite element methods applied to problems in solid mechanics 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 74K20 Plates 74B99 Elastic materials

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