Unbiased higher-order frictional contact using midplane and patch based segment-to-segment penalty method

A highly accurate, single-pass, unbiased frictional contact algorithm for higher-order elements based on the concept of midplane is presented. Higher-order elements offer a lucrative choice for contact problems as they can better represent the curvature of original geometries compared to the first-order elements. Compressive and frictional contact constraints are applied over the contact pairs of sub-segments obtained by the subdivision of higher-order segments. The normal traction depends upon the penalisation of true interpenetration, and frictional traction depends upon relative sliding between sub-segments over their shared patches. The midplane constructed by linearised subfacets can be corrected to account for the local curvature of interacting physical surfaces. Demonstrated through multiple tests, the use of higher-order elements surpasses the accuracy of first-order elements for curved geometries. Its versatility extends from static to dynamic conditions for flat and curved interfaces, including frictional contact. The presented examples include contact patch test, Hertzian contact, elastic collision, rotation of concentric surfaces, frictional sliding, self-contact and inelastic collision problems. Here, the contact patch test matches the accuracy of finite elements, and Hertzian contact shows a smoother solution compared to first-order meshes. The elastic collision problem highlights the utility of the algorithm in accurate prediction of configuration changes in multibody systems. The frictional sliding demonstrates the ability to represent the expected nonlinear distribution of nodal forces for the higher-order elements. The large deformation problems, e.g. self-contact and inelastic collision, specifically benefit from the accuracy in surface representation using higher-order discretisation and continuous contact constraint imposition on such surfaces during deformation.