barycentric interpolation
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AbstractAs the use of spectral/hp element methods, and high-order finite element methods in general, continues to spread, community efforts to create efficient, optimized algorithms associated with fundamental high-order operations have grown. Core tasks such as solution expansion evaluation at quadrature points, stiffness and mass matrix generation, and matrix assembly have received tremendous attention. With the expansion of the types of problems to which high-order methods are applied, and correspondingly the growth in types of numerical tasks accomplished through high-order methods, the number and types of these core operations broaden. This work focuses on solution expansion evaluation at arbitrary points within an element. This operation is core to many postprocessing applications such as evaluation of streamlines and pathlines, as well as to field projection techniques such as mortaring. We expand barycentric interpolation techniques developed on an interval to 2D (triangles and quadrilaterals) and 3D (tetrahedra, prisms, pyramids, and hexahedra) spectral/hp element methods. We provide efficient algorithms for their implementations, and demonstrate their effectiveness using the spectral/hp element library Nektar++ by running a series of baseline evaluations against the ‘standard’ Lagrangian method, where an interpolation matrix is generated and matrix-multiplication applied to evaluate a point at a given location. We present results from a rigorous series of benchmarking tests for a variety of element shapes, polynomial orders and dimensions. We show that when the point of interest is to be repeatedly evaluated, the barycentric method performs at worst $$50\%$$ 50 % slower, when compared to a cached matrix evaluation. However, when the point of interest changes repeatedly so that the interpolation matrix must be regenerated in the ‘standard’ approach, the barycentric method yields far greater performance, with a minimum speedup factor of $$7\times $$ 7 × . Furthermore, when derivatives of the solution evaluation are also required, the barycentric method in general slightly outperforms the cached interpolation matrix method across all elements and orders, with an up to $$30\%$$ 30 % speedup. Finally we investigate a real-world example of scalar transport using a non-conformal discontinuous Galerkin simulation, in which we observe around $$6\times $$ 6 × speedup in computational time for the barycentric method compared to the matrix-based approach. We also explore the complexity of both interpolation methods and show that the barycentric interpolation method requires $${\mathcal {O}}(k)$$ O ( k ) storage compared to a best case space complexity of $${\mathcal {O}}(k^2)$$ O ( k 2 ) for the Lagrangian interpolation matrix method.

A linear barycentric rational collocation method (LBRCM) for solving Schrodinger equation (SDE) is proposed. According to the barycentric interpolation method (BIM) of rational polynomial and Chebyshev polynomial, the matrix form of the collocation method (CM) that is easy to program is obtained. The convergence rate of the LBRCM for solving the Schrodinger equation is proved from the convergence rate of linear barycentric rational interpolation. Finally, a numerical example verifies the correctness of the theoretical analysis.

Purpose The purpose of this paper is to obtain accurate numerical solutions of two-dimensional (2-D) and 3-dimensional (3-D) Klein–Gordon–Schrödinger (KGS) equations. Design/methodology/approach The use of linear barycentric interpolation differentiation matrices facilitates the computation of numerical solutions both in 2-D and 3-D space within reasonable central processing unit times. Findings Numerical simulations corroborate the efficiency and accuracy of the proposed method. Originality/value Linear barycentric interpolation method is applied to 2-D and 3-D KGS equations for the first time, and good results are obtained.

To achieve numerical simulation of large deformation evolution processes in underground engineering, the barycentric interpolation test function is established in this paper based on the manifold cover idea. A large-deformation numerical simulation method is proposed by the double discrete method with the fixed Euler background mesh and moving material points, with discontinuous damage processes implemented by continuous simulation. The material particles are also the integration points. This method is called the manifold cover Lagrangian integral point method based on barycentric interpolation. The method uses the Euler mesh as the background integral mesh and describes the deformation behavior of macroscopic objects through the motion of particles between meshes. Therefore, this method can avoid the problem of computation termination caused by the distortion of the mesh in the calculation process. In addition, this method can keep material particles moving without limits in the set region, which makes it suitable for simulating large deformation and collapse problems in geotechnical engineering. Taking a typical slope as an example, the results of a slope slip surface obtained using the manifold cover Lagrangian integral point method based on barycentric interpolation proposed in this paper were basically consistent with the theoretical analytical method. Hence, the correctness of the method was verified. The method was then applied for simulating the collapse process of the side slope, thereby confirming the feasibility of the method for computing large deformations.

AbstractIn mass customization, software configurators enable novice end-users to design customized products and services according to their needs and preferences. However, traditional configurators hardly provide an engaging experience while avoiding the burden of choice. We propose a Design Participation Model to facilitate navigating the design space, based on two modules. Modeler enables designers to create customizable designs as parametric models, and Navigator subsequently permits novice end-users to explore these designs. While most parametric designs support direct manipulation of low-level features, we propose interpolation features to give customers more flexibility. In this paper, we focus on the implementation of such interpolation features into Navigator and its user interface. To assess our approach, we designed and performed user experiments to test and compare Modeler and Navigator, thus providing insights for further developments of our approach. Our results suggest that barycentric interpolation between qualitative parameters provides a more easily understandable interface that empowers novice customers to explore the design space expeditiously.