Variational Multiscale Methods for Stabilized Enforcement of Constraints
Open Access
- Author:
- Groeneveld, Andrew
- Graduate Program:
- Civil Engineering
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- September 13, 2024
- Committee Members:
- Farshad Rajabipour, Program Head/Chair
Christian Peco, Outside Unit & Field Member
Jesse Sherburn, Special Member
Aleksandra Radli¿ska, Major Field Member
Pinlei Chen, Chair & Dissertation Advisor
Kostas Papakonstantinou, Major Field Member - Keywords:
- Composite
Debonding
Embedded mesh
Finite element method
Finite strain
Immersed mesh
Interfacial stabilization
Meshfree method
Stabilized method
Variational multiscale method - Abstract:
- This work is concerned with the development of efficient methods for simulating the deformation and failure of composites with complex microstructures, such as ultra-high performance concrete (UHPC). Because of the geometric complexity of the microstructures, it is difficult to create meshes that conform to the various phases of the composite. Thus, we seek to develop immersed methods, in which the matrix and reinforcement phases may be discretized independently. To this end, we apply the variational multiscale (VMS) method to weakly enforce displacement compatibility at the interface. This results in the variational multiscale discontinuous Galerkin (VMDG) method, which has previously been used for body-fitted meshes but which is extended to an immersed setting in this work. General interfacial constitutive behavior, including perfect bond, progressive damage and debonding, or (small-slip) frictional contact, can be treated within VMDG as a unifying framework by the introduction of the interfacial displacement gap as a state variable. We show that the VMS method is a versatile technique for weak enforcement of equality constraints in a Galerkin setting, employing both meshfree (RKPM) and finite element approximations. VMS results in a primal formulation, as do penalty or Nitsche methods, but unlike those methods, no penalty parameter needs to be selected in VMS. Instead, VMS provides variationally derived stabilization that naturally incorporates information about the local geometry and material properties --- and which can evolve during the course of a nonlinear problem. No heuristics need to be introduced. We also derive a simplification of the stabilization tensor that retains the variational character of VMDG while reducing the cost of forming the tensor by 95%. Physically correct treatment of volume terms is achieved by using cut-element integration for FEM or subtraction of the overlapped integrand in meshfree methods. In 2D, cut-element integration is performed on triangular integration cells; in 3D, moment-fitting integration is used with a robust voxel-based algorithm for determining the physical portion of a cut element. VMS methods are derived and demonstrated for meshfree boundary conditions, immersed meshfree perfect bond and debonding, and finite element debonding at finite strains in 2D and 3D. Results are compared to experimental data, numerical computations, or analytical solutions as appropriate.