A scalable hybridized mixed finite element formulation for 3D crack propagation under large strain and contact

This article presents a recently developed mixed finite element method for modeling brittle and quasi-brittle fracture in large structures, involving contact and large deformations. The proposed formulation includes four independently approximated fields: stresses, logarithmic stretches, rotation vectors, and displacements. The first two are associated with conserving linear and angular momentum, respectively. While, the other two fields are associated with the constitutive and the consistency equation between displacements and deformation. The relationship between the rotation vectors and the rotation tensor is established by an exponential map. The stresses are approximated in the https://www.w3.org/1998/Math/MathML" display="inline"> H div space, and the remaining three fields are approximated in the https://www.w3.org/1998/Math/MathML" display="inline"> L 2 space. This formulation results in a very sparse system of equations that can be efficiently solved in parallel using hybridization with a block solver, thereby enabling highly scalable and robust solvers. The system hybridization involves ’breaking’ the https://www.w3.org/1998/Math/MathML" display="inline"> H d i v space, such that continuity of normal fluxes, i.e. tractions, is no longer enforced apriori. A hybridized field is also introduced on the mesh skeleton faces and contact surfaces. Such a field acts as a kinematic Lagrange multiplier enforcing continuity of normal tractions. Moreover, the hybridized Lagrange multiplier field is conveniently used to enforce contact conditions, and introduce displacements discontinuities. Furthermore, since the https://www.w3.org/1998/Math/MathML" display="inline"> H d i v space provides traces of fluxes on faces, the mixed element enables efficient calculation of face crack release energy. Since this methodology enables crack propagation on fixed mesh technology, along with being based on an energy-based crack propagation criterion, it inherits the robustness of the phase field while being as efficient as methods which resolves crack discreetly. In an algebraic sense, since all fields in the interior become fields in the https://www.w3.org/1998/Math/MathML" display="inline"> L 2 space, and fields on skeleton faces are in https://www.w3.org/1998/Math/MathML" display="inline"> H 1 / 2 space, the system of equations is extremely sparse and can be solved using a Schur complement, eliminating interior fields.