Applications of Alternating Direction Solver for simulations of time-dependent problems

This paper deals with application of Alternating Direction solver (ADS) to nonstationarylinear elasticity problem solved with isogeometric FEM. Employingtensor product B-spline basis in isogeometric analysis under some restrictionsleads to system of linear equations with matrix possessing tensor product structure.Alternating Direction Implicit algorithm is a direct method that exploitsthis structure to solve the system in O (N ), where N is a number of degreesof freedom (basis functions). This is asymptotically faster than state-of-theartgeneral purpose multi-frontal direct solvers. In this paper we also presentthe complexity analysis of ADS incorporating dependence on order of B-splinebasis

Linear computational cost implicit solver for parabolic problems

In this paper, we use the alternating direction method for isogeometric finite elements to simulate implicit dynamics. Namely, we focus on a parabolic problem and use B-spline basis functions in space and an implicit marching method to fully discretize the problem. We introduce intermediate time steps and separate our differential operator into a summation of the blocks, acting along a particular coordinate axis in the intermediate time steps. We show that the resulting stiffness matrix can be represented as a multiplication of two (in 2D) or three (in 3D) multi-diagonal matrices, each one with B-spline basis functions along the particular axis of the spatial system of coordinates. As a result of this algebraic transformations, we get a system of linear equations that can be factorized in linear $O(N)$ computational cost in every time step of the implicit method. We use our method to simulate the heat transfer problem. We demonstrate theoretically and verify numerically that our implicit method is unconditionally stable for heat transfer problems (i.e., parabolic). We conclude our presentation with a discussion on the limitations of the method

Applications of alternating direction solver for simulations of time-dependent problems

This paper deals with the application of an Alternating Direction Solver (ADS) to a non-stationary linear elasticity problem solved with the isogeometric finite element method (IGA-FEM). Employing a tensor product B-spline basis in isogeometric analysis under some restrictions leads to a system of linear equations with a matrix possessing a tensor product structure. The ADI algorithm is a direct method that exploits this Kronecker product structure to solve the system in O (N), where N is the number of degrees of freedom (basis functions). This is asymptotically faster than state-of-the-art, general-purpose, multi-frontal direct solvers when applied to explicit dynamics. In this paper, we also present a complexity analysis of the ADS incorporating dependence on the B-spline basis of order p

Applications of alternating direction solver for simulations of time-dependent problems

This paper deals with the application of an Alternating Direction Solver (ADS) to a non-stationary linear elasticity problem solved with the isogeometric finite element method (IGA-FEM). Employing a tensor product B-spline basis in isogeometric analysis under some restrictions leads to a system of linear equations with a matrix possessing a tensor product structure. The ADI algorithm is a direct method that exploits this Kronecker product structure to solve the system in O (N), where N is the number of degrees of freedom (basis functions). This is asymptotically faster than state-of-the-art, general-purpose, multi-frontal direct solvers when applied to explicit dynamics. In this paper, we also present a complexity analysis of the ADS incorporating dependence on the B-spline basis of order p