High-Order Isogeometric Methods for Compressible Flows

Abstract

Isogeometric analysis was applied very successfully to many problem classes like linear elasticity, heat transfer and incompressible flow problems but its application to compressible flows is very rare. However, its ability to accurately represent complex geometries used in industrial applications makes IGA a suitable tool for the analysis of compressible flow problems that require the accurate resolution of boundary layers. The convection-diffusion solver presented in this chapter, is an indispensable step on the way to developing a compressible solver for complex viscous industrial flows. It is well known that the standard Galerkin finite element method and its isogeometric counterpart suffer from spurious oscillatory behaviour in the presence of shocks and steep solution gradients. As a remedy, the algebraic flux correction paradigm is generalized to B-Spline basis functions to suppress the creation of oscillations and occurrence of non-physical values in the solution. This work provides early results for scalar conservation laws and lays the foundation for extending this approach to the compressible Euler equations in [1].

High-Order Isogeometric Methods for Compressible Flows

“…In this work, we have extended the high-resolution isogeometric scheme presented in [1] to systems of conservation laws, namely, to the compressible Euler equations. The main contribution is the positivity proof of the linearized FCT algorithm for B-Spline based discretizations, which provides the theoretical justification of our IGA-AFC approach.…”

“…The semi-discrete system (10) is discretized in time by an explicit strong stability preserving (SSP) Runge-Kutta time integration schemes of order three [11] MU (1)…”

High-Order Isogeometric Methods for Compressible Flows. II. Compressible Euler Equations

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This work extends the high-resolution isogeometric analysis approach established in [1] to the equations of gas dynamics. The group finite element formulation is adopted to obtain an efficient assembly procedure for the standard Galerkin approximation, which is stabilized by adding artificial viscosities proportional to the spectral radius of the Roe-averaged flux-Jacobian matrix.

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Application of Non-polynomial Splines to Solving Differential Equations

“…Let us have a uniform grid of nodes with a step ℎ. Using the method published in [9] it is easy to show that the next estimate of the error of approximation is valid: | − | ≤ ℎ 4 . Therefore, to estimate the approximation of the second derivative, we obtain the inequality | ′′ − ′′| ≤ ℎ 2 .…”

“…Among the variety of books on splines, we should first of all mention De Boor's book. Among the variety of splines researchers prefer to use the polynomial splines, mostly the B-splines (see, [4]- [6]). In paper [6], two types of basis functions are considered: B-spline and expo-rational B-spline combined with Bernstein polynomials.…”