First Order Error Correction for Trimmed Quadrature in Isogeometric Analysis

International audienceIn this work, we develop a specialized quadrature rule for trimmed domains , where the trimming curve is given implicitly by a real-valued function on the whole domain. We follow an error correction approach: In a first step, we obtain an adaptive subdivision of the domain in such a way that each cell falls in a pre-defined base case. We then extend the classical approach of linear approximation of the trimming curve by adding an error correction term based on a Taylor expansion of the blending between the linearized implicit trimming curve and the original one. This approach leads to an accurate method which improves the convergence of the quadrature error by one order compared to piecewise linear approximation of the trimming curve. It is at the same time efficient, since essentially the computation of one extra one-dimensional integral on each trimmed cell is required. Finally, the method is easy to implement, since it only involves one additional line integral and refrains from any point inversion or optimization operations. The convergence is analyzed theoretically and numerical experiments confirm that the accuracy is improved without compromising the computational complexity

Overlapping Multi-Patch Structures in Isogeometric Analysis

International audienceIn isogeometric analysis (IGA) the domain of interest is usually represented by B-spline or NURBS patches, as they are present in standard CAD models. Complex domains can often be represented as a union of simple overlapping subdomains, parameterized by (tensor-product) spline patches. Numerical simulation on such overlapping multi-patch domains is a serious challenge in IGA. To obtain non-overlapping subdomains one would usually reparameterize the domain or trim some of the patches. Alternatively, one may use methods that can handle overlapping subdomains. In this paper, we propose a non-iterative, robust and efficient method defined directly on overlapping multi-patch domains. Consequently, the problem is divided into several sub-problems, which are coupled in an appropriate way. The resulting system can be solved directly in a single step. We compare the proposed method with iterative Schwarz domain decomposition approaches and observe that our method reduces the computational cost significantly, especially when handling subdomains with small overlaps. Summing up, our method significantly simplifies the domain parameterization problem, since we can represent any domain of interest as a union of overlapping patches without the need to introduce trimming curves/surfaces. The performance of the proposed method is demonstrated by several numerical experiments for the Poisson problem and linear elasticity in two and three dimensions

On the application of isogeometric finite volume method in numerical analysis of wet-steam flow through turbine cascades

The isogeometric finite volume analysis is utilized in this research to numerically simulate the two-dimensional viscous wet-steam flow between stationary cascades of a steam turbine for the first time. In this approach, the analysis-suitable computational mesh with ‘‘curved’’ boundaries is generated for the fluid flow by employing a non- uniform rational B-spline (NURBS) surface that describes the cascade geometry, and the governing equations are then discretized by the NURBS representation. Thanks to smooth and accurate geometry representation of the NURBS formulation, the employed isogeometric framework not only resolves issues concerning the conventional mesh generation techniques of the finite volume method in steam turbine problems, but also, as validated against well-established experiments, significantly improves the accuracy of the numerical solution. In addition, the shock location in the cascade is predicted and tracked with a sufficient accuracy