In this work we compare different families of nested quadrature points, i.e. the classic Clenshaw-Curtis and various kinds of Leja points, in the context of the quasi-optimal sparse grid approximation of random ellipt...
ISBN:
(纸本)9783319198002;9783319197999
In this work we compare different families of nested quadrature points, i.e. the classic Clenshaw-Curtis and various kinds of Leja points, in the context of the quasi-optimal sparse grid approximation of random elliptic PDEs. Numerical evidence suggests that both families perform comparably within such framework.
In this work we discuss a newly developed class of robust and high-order accurate upwind schemes for wave equations in second-order form on curvilinear and overlapping grids. The schemes are based on embedding d'A...
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ISBN:
(纸本)9783319198002;9783319197999
In this work we discuss a newly developed class of robust and high-order accurate upwind schemes for wave equations in second-order form on curvilinear and overlapping grids. The schemes are based on embedding d'Alembert's exact solution for a local Riemann-type problem directly into the discretization (Banks and Henshaw, J Comput Phys 231(17):5854-5889, 2012). High-order accuracy is obtained using a single-step space-time scheme. Overlapping grids are used to represent geometric complexity. The method of manufactured solutions is used to demonstrate that the dissipation introduced through upwinding is sufficient to stabilize the wave equation in the presence of overlapping grid interpolation.
We propose a reduced shell element for Reissner-Mindlin geometric non-linear analysis within the context of T-spline analysis. The shell formulation is based on the displacements and a first order kinematic in the thi...
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ISBN:
(数字)9783319233154
ISBN:
(纸本)9783319233154;9783319233147
We propose a reduced shell element for Reissner-Mindlin geometric non-linear analysis within the context of T-spline analysis. The shell formulation is based on the displacements and a first order kinematic in the thickness is used for the transverse shear strains. A total Lagrangian formulation is considered for the finite transformations. The update of the incremental rotations is made using the quaternion algebra. The standard two-dimensional reduced quadrature rules for structured B-spline and NURBS basis functions are extended to the more flexible T-meshes. The non-uniform Gauss-Legendre and patchwise reduced integrations for quadratic shape functions are both presented and compared to the standard full Gauss-Legendre scheme. The performance of the element is assessed with linear and geometric non-linear two-dimensional problems in structural analysis. The effects of mesh distortion and local refinement, using both full and reduced numerical quadratures, are evaluated.
Recent developments in the SBP-SAT method have made available high-order interpolation operators (Mattsson and Carpenter, SIAM J Sci Comput 32(4):2298-2320, 2010). Such operators allow the coupling of different SBP me...
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ISBN:
(纸本)9783319198002;9783319197999
Recent developments in the SBP-SAT method have made available high-order interpolation operators (Mattsson and Carpenter, SIAM J Sci Comput 32(4):2298-2320, 2010). Such operators allow the coupling of different SBP methods across nonconforming interfaces of multiblock grids while retaining the three fundamental properties of the SBP-SAT method: strict stability, accuracy, and conservation. As these interpolation operators allow a more flexible computational mesh, they are appealing for complex geometries. Moreover, they are well suited for problems involving sliding meshes, like rotor/ stator interactions, wind turbines, helicopters, and turbomachinery simulations in general, since sliding interfaces are (almost) always nonconforming. With such applications in mind, this paper presents an accuracy analysis of these interpolation operators when applied to fluid dynamics problems on moving grids. The classical problem of an inviscid vortex transported by a uniform flow is analyzed: the flow is governed by the unsteady Euler equations and the vortex crosses a sliding interface. Furthermore, preliminary studies on a rotor/stator interaction are also presented.
In this work we present the numerical analysis and study the performance of a finite element projection-based Variational MultiScale (VMS) turbulence model that includes general non-linear wall laws. We introduce Lagr...
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ISBN:
(纸本)9783319257273;9783319257259
In this work we present the numerical analysis and study the performance of a finite element projection-based Variational MultiScale (VMS) turbulence model that includes general non-linear wall laws. We introduce Lagrange finite element spaces adapted to approximate the slip condition. The sub-grid effects are modeled by an eddy diffusion term that acts only on a range of small resolved scales. Moreover, high-order stabilization terms are considered, with the double aim to guarantee stability for coarse meshes, and help to counter-balance the accumulation of sub-grid energy together with the sub-grid eddy viscosity term. We prove stability and convergence for solutions that only need to bear the natural minimal regularity, in unsteady regime. We also study the asymptotic energy balance of the system. We finally include some numerical tests to assess the performance of the model described in this work.
The core of numerical simulations of coupled incompressible flow problems consists of a robust, accurate and fast solver for the time-dependent, incompressible Navier-Stokes equations. We consider inf-sup stable finit...
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ISBN:
(纸本)9783319257273;9783319257259
The core of numerical simulations of coupled incompressible flow problems consists of a robust, accurate and fast solver for the time-dependent, incompressible Navier-Stokes equations. We consider inf-sup stable finite element methods with grad-div stabilization and symmetric stabilization of local projection type. The approach is based on a proper scale separation and only the small unresolved scales are modeled. Error estimates for the spatially discretized problem with reasonable growth of the Gronwall constant for large Reynolds numbers are given together with a critical discussion of the choice of stabilization parameters. The fast solution of the fully discretized problems (using BDF(2) in time) is accomplished via unconditionally stable velocity-pressure segregation.
We consider the C-0 interior penalty Galerkin method for biharmonic eigenvalue problems with the boundary conditions of the clamped plate, the simply supported plate and the Cahn-Hilliard type. We establish the conver...
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ISBN:
(纸本)9783319198002;9783319197999
We consider the C-0 interior penalty Galerkin method for biharmonic eigenvalue problems with the boundary conditions of the clamped plate, the simply supported plate and the Cahn-Hilliard type. We establish the convergence of the method and present numerical results to illustrate its performance. We also compare it with the Argyris C-1 finite element method, the Ciarlet-Raviart mixed finite element method, and the Morley nonconforming finite element method.
We consider the augmented mixed finite element method proposed in Barrios et al. (Comput Methods Appl Mech Eng 283:909-922, 2015) for Darcy flow. We develop the a priori and a posteriori error analyses taking into acc...
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ISBN:
(纸本)9783319257273;9783319257259
We consider the augmented mixed finite element method proposed in Barrios et al. (Comput Methods Appl Mech Eng 283:909-922, 2015) for Darcy flow. We develop the a priori and a posteriori error analyses taking into account the approximation of the Neumann boundary condition. We derive an a posteriori error indicator that consists of two residual terms on interior elements and an additional term that accounts for the error in the boundary condition on boundary elements. We prove that the error indicator is reliable and locally efficient on interior elements. Numerical experiments illustrate the good performance of the adaptive algorithm.
This work deals with the stabilisation of mixed methods for the Stokes problem on anisotropic meshes. For this, we extend a method proposed previously in Liao and Silvester (IMA J Numer Anal 33(2):413-431, 2013), to c...
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ISBN:
(纸本)9783319257273;9783319257259
This work deals with the stabilisation of mixed methods for the Stokes problem on anisotropic meshes. For this, we extend a method proposed previously in Liao and Silvester (IMA J Numer Anal 33(2):413-431, 2013), to cover the case in which the mesh contains anisotropically refined corners. This modification consists of adding extra jump terms in selected edges connecting small shape regular with large anisotropic elements. We prove stability and convergence of the proposed method, and provide numerical evidence for the fact that our approach successfully removes the dependence on the anisotropy.
We consider the solution of large linear systems of equations that arise when two-dimensional singularly perturbed reaction-diffusion equations are discretized. Standard methods for these problems, such as central fin...
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ISBN:
(纸本)9783319257273;9783319257259
We consider the solution of large linear systems of equations that arise when two-dimensional singularly perturbed reaction-diffusion equations are discretized. Standard methods for these problems, such as central finite differences, lead to system matrices that are positive definite. The direct solvers of choice for such systems are based on Cholesky factorisation. However, as observed in MacLachlan and Madden (SIAM J Sci Comput 35:A2225-A2254, 2013), these solvers may exhibit poor performance for singularly perturbed problems. We provide an analysis of the distribution of entries in the factors based on their magnitude that explains this phenomenon, and give bounds on the ranges of the perturbation and discretization parameters where poor performance is to be expected.
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