BROM: An extension of beam theory through model order reduction

Abstract

Beam analysis played a crucial role in the design of structures throughout modern history. Classical beam theories rely on analytical derivations under certain a priori kinematic assumptions (and are accurate with low computational cost), but they are not amenable to, for example, orthotropic materials and arbitrary cross sections. On the other hand, full 3D linear elasticity theory solved via the Finite Element (FE) method provides solutions for a wide range of engineering problems at the expense of a high computational cost. Reduced-order models try to provide a theory as general as 3D linear elasticity but as efficient as classical beam theories by assuming the solution to be represented in a low-dimensional basis. Among the several options available in the literature, the domain decomposition and hyperreduction via Empirical Cubature Method (ECM), which we name ddROM [1], exhibits this compelling feature. However, ddROM applied to beam structures, may provide more Degrees of Freedom (DoF) per node than most standard commercial FE codes which only consider rigid body motion of the cross section. This paper presents a methodology, called beam Reduced Order Model (bROM), to integrate ddROM in standard FE codes by means of a condensation process followed by a regression procedure. Condensation gives the stiffness matrix, required by FE commercial codes, of a super-element of a given length. This is achieved by assembling several elements and finding the relationship between applied displacements and reactions (Lagrange multipliers). The condensation process is then used to create a database of stiffness matrices as a function of the beam length over which a regression is performed. Some numerical examples are first presented to validate the proposed methodology against the classical beam theory for the case of isotropic beams, and finally, two orthotropic beam showing the utility of the method in comparison with FE.