Abstract: In this paper, we present a high-performance framework for solving partial differential equations using Isogeometric Analysis. It is called PetIGA, and in this work we show how it can be used to solve phase-field problems. We specifically chose the Cahn-Hilliard equation, and the phase-field crystal equation as study-problems. These two models allow us to highlight some of the main advantages that we have access to while using PetIGA for scientific computing.

Abstract: We discretized the two-dimensional linear momentum, microrotation, energy and mass conservation equations from the microrotational theory, with the finite element method, using B-spline basis to create divergence conforming spaces to obtain pointwise divergence free solutions [8]. Weak boundary conditions impositions was handled using Niche’s method for tangential conditions, while normal conditions were imposed strongly.We solved the heat driven cavity problem as a test case, including a variation of the parameters that differentiate micropolar fluids from conventional fluids under different Rayleigh numbers, for a better understanding of the system.

Abstract: In this paper we present a hypergraph grammar model for transformations of two and three dimensional grids. The hypergraph grammar describes the proces for generating uniform grids with two or three dimensional rectangular or hexahedral elements, followed by the proces of h refinements, which involves breaking selected elements into four or eight son elements, in two or three dimensions, respectively. We also provide graph grammar productions for two projection algorithms we use to pre-process material data. The first one is the projection based interpolation solver algorithm used for computing H1 or L2 projections of MRI scan of human head, in two and three dimensions. The second one is utilized for solving the non-stationary problem modeling the three dimensional heat transport in the human head generated by the cellphone usage.

Abstract: After applying the Finite Element Method (FEM) to the diffusion-type and wave-type Partial Differential Equations (PDEs), a first order and a second order Ordinary Differential Equation (ODE) systems are obtained respectively. These ODE systems usually present high stiffness, so numerical methods with good stability properties are required in their resolution. MATLAB offers a set of open source adaptive step functions for solving ODEs. One of these functions is the ode15s recommended for stiff problems and which is based on the Backward Differentiation Formulae (BDF). We describe the error estimation and the step size control implemented in this function. The ode15s is a variable order algorithm, and even though it has an adaptive step size implementation, the advancing formula and the local error estimation that uses correspond to the constant step size formula. We have focused on the second order accurate and unconditionally stable BDF (BDF2) and we have implemented a real adaptive step size BDF2 algorithm using the same strategy as the BDF2 implemented in the ode15s, resulting the new algorithm more efficient than the one implemented in MATLAB.

Abstract: This paper presents a graph-transformation-based multi-frontal direct solver with an optimization technique that allows for a significant decrease of time complexity in some multi-scale simulations of the Step and Flash Imprint Lithography (SFIL). The multi-scale simulation consists of a macro-scale linear elasticity model with thermal expansion coefficient and a nano-scale molecular statics model. The algorithm is exemplified with a photopolimerization simulation that involves densification of a polymer inside a feature followed by shrinkage of the feature after removal of the template. The solver is optimized thanks to a mechanism of reusing sub-domains with similar geometries and similar material properties. The graph transformation formalism is used to describe the algorithm - such an approach helps automatically localize sub-domains that can be reused.