Galerkin Finite Element Method by Using Bivariate Splines for Parabolic PDEs

A Galerkin finite element method by using bivariate splines (GB method) is proposed for solving parabolic partial differential equations (PPDEs). Bivariate spline proper subspace of $S_4^{2,3}(\Delta_{mn}^{(2)})$ satisfying homogeneous boundary conditions on type-2 triangulations and quadratic B-spline interpolating boundary functions are primarily constructed. PPDEs are solved by the GB method

Finite-Rank Multivariate-Basis Expansions of the Resolvent Operator as a Means of Solving the Multivariable Lippmann-Schwinger Equation for Two-Particle Scattering

Cataloged from PDF version of article.Finite-rank expansions of the two-body resolvent operator are explored as a tool for calculating the full three-dimensional two-body T-matrix without invoking the partial-wave decomposition. The separable expansions of the full resolvent that follow from finite-rank approximations of the free resolvent are employed in the Low equation to calculate the T-matrix elements. The finite-rank expansions of the free resolvent are generated via projections onto certain finite-dimensional approximation subspaces. Types of operator approximations considered include one-sided projections (right or left projections), tensor-product (or outer) projection and inner projection. Boolean combination of projections is explored as a means of going beyond tensor-product projection. Two types of multivariate basis functions are employed to construct the finite-dimensional approximation spaces and their projectors: (i) Tensor-product bases built from univariate local piecewise polynomials, and (ii) multivariate radial functions. Various combinations of approximation schemes and expansion bases are applied to the nucleon-nucleon scattering employing a model two-nucleon potential. The inner-projection approximation to the free resolvent is found to exhibit the best convergence with respect to the basis size. Our calculations indicate that radial function bases are very promising in the context of multivariable integral equations

Application of smoothed point interpolation methods to numerical modelling of saturated and unsaturated porous media

This study aims to develop an efficient computational framework for a rigorous coupled flow and deformation analysis of saturated and unsaturated porous media. The governing equations are derived based on equations of equilibrium, and conservation equations of mass and momentum for each phase. For numerical solution of the governing equations, the edge-based smoothed point interpolation method (ESPIM) is employed due to its numerous advantages over the classical techniques. The ESPIM was originally introduced for problems in single phase media. The extension of the technique to multiphase media is not trivial, and therefore as the first development step, ESPIM is extended for the solution of the coupled hydro-mechanical problems in saturated porous media through a novel approach for evaluation of the coupling matrix. Verification of the proposed ESPIM formulation is performed using several benchmark numerical examples. Subsequently, the method of manufactured solutions (MMS) is introduced, for the first time in geomechanics, for a systematic and more rigorous verification of the computational scheme. The proposed numerical framework is then extended to include material nonlinearity. For this purpose, a non-associative Mohr-Coulomb constitutive model is adopted and an algorithm is developed based on the modified Newton-Raphson technique to address the nonlinearities arisen from the elasto-plastic constitutive model. Stress integration is performed using the substepping method. The computational framework is then further extended to include the problems in unsaturated soil mechanics, taking account of coupling among different phases, and the hydraulic hysteresis observed in the behaviour of unsaturated soils. A framework based on the effective stress principle is followed in the formulation and a hysteretic water retention model is taken into account which includes the evolution of water retention curve (WRC) with changes of void ratio. An elasto-plastic constitutive model is employed within the context of bounding surface plasticity theory for predicting the nonlinear behaviour of soil skeleton in saturated and unsaturated porous media. The model is validated by comparing the numerical predictions with experimental or numerical data from the literature for fully and partially saturated soils. The results demonstrate the capability of the proposed numerical framework to predict essential characteristics of variably saturated soils