Arbitrary Lagrangian Eulerian-Type Finite Element Methods Formulation for PDEs on Time-Dependent Domains with Vanishing Discrete Space Conservation Law

2019 ◽  
Vol 41 (3) ◽  
pp. A1548-A1573 ◽  
Author(s):  
Filip Ivančić ◽  
Tony W.-H. Sheu ◽  
Maxim Solovchuk
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Conservation Law ◽  
Discrete Space ◽  
Time Dependent ◽  
Arbitrary Lagrangian Eulerian

Three-Dimensional Computations of Transport and Growth for Crystal Growth Systems

2000 ◽  
Author(s):  
Jeffrey J. Derby ◽  
Andrew Yeckel
Keyword(s):  
Finite Element ◽  
Crystal Growth ◽  
Finite Element Methods ◽  
Three Dimensional ◽  
Bulk Crystal ◽  
Time Dependent ◽  
Engineering Systems ◽  
Bulk Crystal Growth ◽  
Strongly Nonlinear ◽  
Parallel Supercomputers

Abstract Modern finite element methods implemented on parallel supercomputers promise to allow the study of three-dimensional, time-dependent continuum phenomena in many engineering systems. This paper shows several examples of the fruitful application of these approaches to bulk crystal growth systems, where strongly nonlinear coupled phenomena are important.

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.

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