Weak-form Quadrature Element Analysis Of Frames Based On Geometrically Exact Beam Theory

The weak form quadrature element method (QEM) is a recently proposednumerical method based on variational principles. The essence of the QEM is toincorporate appropriate numerical quadrature rules and differential quadratureanalogs into the weak form description of a problem. The physical domain is firstdivided into a number of integrable subdomains, termed as quadrature elements, eachbeing mapped onto a standard computational domain. Then the solution of a set ofalgebraic equations at the sampling points is obtained.As a result of the high-order characteristics of the numerical quadrature rule usedin the integrals of the problem and the differential quadrature analog used in theapproximation of the field variables, the QEM has the advantage of high efficiencyand accuracy in handling complex geometry, complicated boundary conditions,inhomogeneous materials and strong nonlinearities. Furthermore, the accuracy ofquadrature element solution at the nodes, due to the coincidence of sampling pointsand grid points, leads to simpler post process of the results.Geometrically exact beam theory was first proposed by Reissner, and laterextended by Simo and Vu-Quoc. As it provides an objective strain-configurationrelationship which remains valid under large deformations and finite rotations, it hasbeen the subject of extensive research during the last decades. The essential idea ofgeometrically exact beam theory is to introduce section rotation into the configurationspace of the beam, and to measure the deformation of the beam in the section frame.As a result,3D finite rotation theory is introduced. Among a multitude ofrepresentations of3D finite rotations, rotational tensor, rotational vector androtational quaternion are commonly used in geometrically exact beam theory. Sincethe rotational quaternion is the least parameter representation without singularities, itis chosen in the present investigation as a basis for a quadrature element formulation.The present work aims at a series of benchmark problems in the field ofgeometric nonlinearity, using the QEM and geometrically exact beam theory. The high efficiency and accuracy of the QEM as well as its robustness are corroborated inthese numerical simulations and the advantages of the QEM in geometric nonlinearanalysis are highlighted.