As the use of spectral/
 Editors:
 Taylor And Francis Online
 Award ID(s):
 2011615
 Publication Date:
 NSFPAR ID:
 10347006
 Journal Name:
 Applicable analysis
 ISSN:
 00036811
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract hp element methods, and highorder finite element methods in general, continues to spread, community efforts to create efficient, optimized algorithms associated with fundamental highorder 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 highorder methods are applied, and correspondingly the growth in types of numerical tasks accomplished through highorder 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 libraryNektar++ by running a series of baseline evaluations against the ‘standard’ Lagrangian method, where an interpolation matrix is generated and matrixmultiplication applied to evaluate a point at a given location. We present results from a rigorous seriesmore » 
We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions–Mercier type iterative method and next a classical fixedpoint algorithm to solve the finitedimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustratemore »

In order to achieve a more accurate finite element (FE) model for an asbuilt structure, experimental data collected from the actual structure can be used to update selected parameters of the FE model. The process is known as FE model updating. This research compares the performance of two frequencydomain model updating approaches. The first approach minimizes the difference between experimental and simulated modal properties, such as natural frequencies and mode shapes. The second approach minimizes modal dynamic residuals from the generalized eigenvalue equation involving stiffness and mass matrices. Both model updating approaches are formulated as an optimization problem with selected updating parameters as optimization variables. This research also compares the performance of different optimization procedures, including a nonlinear leastsquare, an interiorpoint and an iterative linearization procedure. The comparison is conducted using a numerical example of a space frame structure. The modal dynamic residual approach shows better performance than the modal property difference approach in updating model parameters of the space frame structure.

The classical continuous finite element method with Lagrangian Q^k basis reduces to a finite difference scheme when all the integrals are replaced by the (𝑘+1)×(𝑘+1) Gauss–Lobatto quadrature. We prove that this finite difference scheme is (𝑘+2)th order accurate in the discrete 2norm for an elliptic equation with Dirichlet boundary conditions, which is a superconvergence result of function values. We also give a convenient implementation for the case 𝑘=2, which is a simple fourth order accurate elliptic solver on a rectangular domain.

Abstract The scalability and efficiency of numerical methods on parallel computer architectures is of prime importance as we march towards exascale computing. Classical methods like finite difference schemes and finite volume methods have inherent roadblocks in their mathematical construction to achieve good scalability. These methods are popularly used to solve the NavierStokes equations for fluid flow simulations. The discontinuous Galerkin family of methods for solving continuum partial differential equations has shown promise in realizing parallel efficiency and scalability when approaching petascale computations. In this paper an explicit modal discontinuous Galerkin (DG) method utilizing Implicit Large Eddy Simulation (ILES) is proposed for unsteady turbulent flow simulations involving the threedimensional NavierStokes equations. A study of the method was performed for the TaylorGreen vortex case at a Reynolds number ranging from 100 to 1600. The polynomial order
P = 2 (third order accurate) was found to closely match the Direct NavierStokes (DNS) results for all Reynolds numbers tested outside of Re = 1600, which had a normalized RMS error of 3.43 × 10^{−4}in the dissipation rate for a 60^{3}element mesh. The scalability and performance study of the method was then conducted for a Reynolds number of 1600 for polynomials orders fromP = 2 toP = 6. The highest order polynomial that was tested (P = 6)more »