scholarly journals Analytical solution for viscous incompressible Stokes flow in a spherical shell

Solid Earth ◽  
2017 ◽  
Vol 8 (6) ◽  
pp. 1181-1191 ◽  
Author(s):  
Cedric Thieulot
Keyword(s):  
Finite Element ◽  
Spherical Shell ◽  
Stokes Flow ◽  
Body Force ◽  
Convergence Rates ◽  
Mean Square ◽  
Root Mean Square Velocity ◽  
New Family ◽  
Incompressible Stokes Flow ◽  
Velocity Pressure

Abstract. I present a new family of analytical flow solutions to the incompressible Stokes equation in a spherical shell. The velocity is tangential to both inner and outer boundaries, the viscosity is radial and of the power-law type, and the solution has been designed so that the expressions for velocity, pressure, and body force are simple polynomials and therefore simple to implement in (geodynamics) codes. Various flow average values, e.g., the root mean square velocity, are analytically computed. This forms the basis of a numerical benchmark for convection codes and I have implemented it in two finite-element codes: ASPECT and ELEFANT. I report error convergence rates for velocity and pressure.

scholarly journals Analytical solution for viscous incompressible Stokes flow in a spherical shell

2017 ◽  
Author(s):  
Cedric Thieulot
Keyword(s):  
Finite Element ◽  
Spherical Shell ◽  
Stokes Flow ◽  
Body Force ◽  
Convergence Rates ◽  
Mean Square ◽  
Root Mean Square Velocity ◽  
New Family ◽  
Incompressible Stokes Flow ◽  
Velocity Pressure

Abstract. I present a new family of analytical flow solutions to the incompressible Stokes equation in a spherical shell. The velocity is tangential to both inner and outer boundaries, the viscosity is radial and of power-law type, and the solution has been designed so that the expressions for velocity, pressure, and body force are simple polynomials and therefore simple to implement in (geodynamics) codes. Various flow average values, e.g. the root mean square velocity, are analytically computed. This forms the basis for a numerical benchmark for convection codes and I have implemented it in two finite element codes ASPECT and ELEFANT. I report on error convergence rates for velocity and pressure.

scholarly journals Evaluating the accuracy of hybrid finite element/particle-in-cell methods for modelling incompressible Stokes flow

2019 ◽  
Vol 219 (3) ◽  
pp. 1915-1938 ◽  
Author(s):  
Rene Gassmöller ◽  
Harsha Lokavarapu ◽  
Wolfgang Bangerth ◽  
Elbridge Gerry Puckett
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Stokes Flow ◽  
Time Dependent ◽  
Particle Methods ◽  
Chemical Compositions ◽  
Element Formulation ◽  
Particle In Cell ◽  
Incompressible Stokes Flow ◽  
Number Of Particles

SUMMARY Combining finite element methods for the incompressible Stokes equations with particle-in-cell methods is an important technique in computational geodynamics that has been widely applied in mantle convection, lithosphere dynamics and crustal-scale modelling. In these applications, particles are used to transport along properties of the medium such as the temperature, chemical compositions or other material properties; the particle methods are therefore used to reduce the advection equation to an ordinary differential equation for each particle, resulting in a problem that is simpler to solve than the original equation for which stabilization techniques are necessary to avoid oscillations. On the other hand, replacing field-based descriptions by quantities only defined at the locations of particles introduces numerical errors. These errors have previously been investigated, but a complete understanding from both the theoretical and practical sides was so far lacking. In addition, we are not aware of systematic guidance regarding the question of how many particles one needs to choose per mesh cell to achieve a certain accuracy. In this paper we modify two existing instantaneous benchmarks and present two new analytic benchmarks for time-dependent incompressible Stokes flow in order to compare the convergence rate and accuracy of various combinations of finite elements, particle advection and particle interpolation methods. Using these benchmarks, we find that in order to retain the optimal accuracy of the finite element formulation, one needs to use a sufficiently accurate particle interpolation algorithm. Additionally, we observe and explain that for our higher-order finite-element methods it is necessary to increase the number of particles per cell as the mesh resolution increases (i.e. as the grid cell size decreases) to avoid a reduction in convergence order. Our methods and results allow designing new particle-in-cell methods with specific convergence rates, and also provide guidance for the choice of common building blocks and parameters such as the number of particles per cell. In addition, our new time-dependent benchmark provides a simple test that can be used to compare different implementations, algorithms and for the assessment of new numerical methods for particle interpolation and advection. We provide a reference implementation of this benchmark in aspect (the ‘Advanced Solver for Problems in Earth’s ConvecTion’), an open source code for geodynamic modelling.

Research of the Connecting Form about the Spherical Shell and the Nozzle

2011 ◽  
Vol 413 ◽  
pp. 520-523
Author(s):  
Cai Xia Luo
Keyword(s):  
Finite Element ◽  
Stress Distribution ◽  
Finite Element Model ◽  
Spherical Shell ◽  
Maximum Stress ◽  
Peak Stress ◽  
Element Model ◽  
Small Opening ◽  
Large Opening ◽  
Reducing Stress

The Stress Distribution in the Connection of the Spherical Shell and the Opening Nozzle Is Very Complex. Sharp-Angled Transition and Round Transition Are Used Respectively in the Connection in the Light of the Spherical Shell with the Small Opening and the Large One. the Influence of the Two Connecting Forms on Stress Distribution Is Analyzed by Establishing Finite Element Model and Solving it. the Result Shows there Is Obvious Stress Concentration in the Connection. Round Transition Can Reduce the Maximum Stress in Comparison with Sharp-Angled Transition in both Cases of the Small Opening and the Large Opening, Mainly Reducing the Bending Stress and the Peak Stress, but Not the Membrane Stress. the Effect of Round Transition on Reducing Stress Was Not Significant. so Sharp-Angled Transition Should Be Adopted in the Connection when a Finite Element Model Is Built for Simplification in the Future.

scholarly journals MDFEM: Multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM

2021 ◽  
Author(s):  
Dong T.P. Nguyen ◽  
Dirk Nuyens
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Monte Carlo ◽  
Convergence Rates ◽  
Higher Order ◽  
Elliptic Pdes ◽  
Infinite Dimensional ◽  
Quasi Monte Carlo ◽  
Higher Order Convergence ◽  
Element Method

We introduce the \emph{multivariate decomposition finite element method} (MDFEM) for elliptic PDEs with lognormal diffusion coefficients, that is, when the diffusion coefficient has the form $a=\exp(Z)$ where $Z$ is a Gaussian random field defined by an infinite series expansion $Z(\bsy) = \sum_{j \ge 1} y_j \, \phi_j$ with $y_j \sim \calN(0,1)$ and a given sequence of functions $\{\phi_j\}_{j \ge 1}$. We use the MDFEM to approximate the expected value of a linear functional of the solution of the PDE which is an infinite-dimensional integral over the parameter space. The proposed algorithm uses the \emph{multivariate decomposition method} (MDM) to compute the infinite-dimensional integral by a decomposition into finite-dimensional integrals, which we resolve using \emph{quasi-Monte Carlo} (QMC) methods, and for which we use the \emph{finite element method} (FEM) to solve different instances of the PDE.   We develop higher-order quasi-Monte Carlo rules for integration over the finite-di\-men\-si\-onal Euclidean space with respect to the Gaussian distribution by use of a truncation strategy. By linear transformations of interlaced polynomial lattice rules from the unit cube to a multivariate box of the Euclidean space we achieve higher-order convergence rates for functions belonging to a class of \emph{anchored Gaussian Sobolev spaces} while taking into account the truncation error. These cubature rules are then used in the MDFEM algorithm.   Under appropriate conditions, the MDFEM achieves higher-order convergence rates in term of error versus cost, i.e., to achieve an accuracy of $O(\epsilon)$ the computational cost is $O(\epsilon^{-1/\lambda-\dd/\lambda}) = O(\epsilon^{-(p^* + \dd/\tau)/(1-p^*)})$ where $\epsilon^{-1/\lambda}$ and $\epsilon^{-\dd/\lambda}$ are respectively the cost of the quasi-Monte Carlo cubature and the finite element approximations, with $\dd = d \, (1+\ddelta)$ for some $\ddelta \ge 0$ and $d$ the physical dimension, and $0 < p^* \le (2 + \dd/\tau)^{-1}$ is a parameter representing the sparsity of $\{\phi_j\}_{j \ge 1}$.

Agglomeration-based physical frame dG discretizations: An attempt to be mesh free

2014 ◽  
Vol 24 (08) ◽  
pp. 1495-1539 ◽  
Author(s):  
Francesco Bassi ◽  
Lorenzo Botti ◽  
Alessandro Colombo
Keyword(s):  
Finite Element ◽  
Convergence Rates ◽  
Numerical Test ◽  
High Order ◽  
Test Cases ◽  
Computational Domain ◽  
Element Discretization ◽  
Mesh Free ◽  
Domain Boundaries ◽  
Coarse Grids

In this work we consider agglomeration-based physical frame discontinuous Galerkin (dG) discretization as an effective way to increase the flexibility of high-order finite element methods. The mesh free concept is pursued in the following (broad) sense: the computational domain is still discretized using a mesh but the computational grid should not be a constraint for the finite element discretization. In particular the discrete space choice, its convergence properties, and even the complexity of solving the global system of equations resulting from the dG discretization should not be influenced by the grid choice. Physical frame dG discretization allows to obtain mesh-independent h-convergence rates. Thanks to mesh agglomeration, high-order accurate discretizations can be performed on arbitrarily coarse grids, without resorting to very high-order approximations of domain boundaries. Agglomeration-based h-multigrid techniques are the obvious choice to obtain fast and grid-independent solvers. These features (attractive for any mesh free discretization) are demonstrated in practice with numerical test cases.

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