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Numerical analysis of reproducing kernel collocation method for linear nonlocal models
Hydraulic fracturing has played a major role in north America's âshale revolutionâ over the past decades. Modeling of the hydraulic fracture propagation is challenging. Peridynamics, a nonlocal theory of continuum mechanics, has been used to model complex hydraulic fracturing processes in recent years. While the peridynamics-based hydraulic fracturing model has shown promising simulations results, its current numerical discretization lacks any mathematical analysis. This dissertation is motivated by the numerical solution of the peridynamics-based hydraulic fracturing model. The major objective is to develop a robust numerical method, under the change of the modeling parameters, for linear nonlocal diffusion models and peridynamic Navier equation, which are decoupled models of the peridynamics-based hydraulic fracturing model. Reproducing kernel (RK) collocation method is of our interest due to its mesh-free nature. By choosing special RK support sizes, we have developed a RK collocation method for nonlocal models and numerical solutions converge to the nonlocal solution and also to the corresponding local limit independent of the modeling parameters as the nonlocal interactions vanish. Accurate evaluation of the stiffness matrix for nonlocal models is computationally prohibitive even for collocation method. To save computational costs, the concept of RK approximation is generalized to approximate integrals and the quasi-discrete nonlocal operator, which uses a finite number of symmetric quadrature points to evaluate the integral, is proposed. We have shown RK collocation on the quasi-discrete nonlocal diffusion and peridynamic Navier equation converge to their classical counterparts. Finally, for the pure displacement form of the classical linear elasticity model, finite element solutions deteriorate when the material is nearly incompressible. A common remedy is to introduce an additional variable, pressure, and rewrite the equation in a mixed formulation, but the discrete functional spaces need to satisfy the famous inf-sup condition. For the the mixed form of the quasi-discrete peridynamic Navier equation, the discretization obtained using RK collocation with equal order interpolation for displacements and pressure passes the inf-sup test; the solution does not suffer from instability. Hence, with the use of penalty techniques or artificial compressibility, the proposed RK collocation method is promising in solving the peridynamics-based hydraulic fracturing model, which has an embedded saddle-point problemPetroleum and Geosystems Engineerin