Implementing and using high-order finite element methods

Iterative solution of Helmholtz problem with high-order isogeometric analysis and finite element method at mid-range frequencies

In this paper, we develop an adaptive high-order surface finite element method (FEM) incorporating spectral deferred correction method for chain contour discretization to solve polymeric self-consistent field equations on general curved surfaces. The high-order surface FEM is obtained by the high-order surface geometrical approximation and the high-order function space approximation. Numerical results demonstrate that the precision order of these methods is consistent with theoretical prediction. In order to describe the sharp interface in the strongly segregated system more accurately, an adaptive FEM equipped with a new $Log$ marking strategy is proposed. The $Log$ marking strategy can not only label the elements that need to be refined or coarsened, but also give the refined or coarsened times, which can make full use of the information of a posterior error estimator and improve the efficiency of the adaptive algorithm. To demonstrate the power of our approach, we investigate the self-assembled patterns of diblock copolymers on several distinct curved surfaces. Numerical results illustrate the efficiency of the proposed method, especially for strongly segregated systems with economical discretization nodes.

This document presents a new electromagnetic in-house parallel EM simulator named HOFEM (Higher Order Finite Element Method). The simulator makes use of some of the research developments on the area made within the Group of Radiofrequency, Electromagnetism, Microwaves and Antennas (GREMA) of the Universidad Carlos III de Madrid to which the authors belong to. HOFEM makes use of a weak formulation based on double curl vector wave equation discretized with the higher-order isoparametric curl-conforming tetrahedral (and prisms) finite element rigorous implementations of Nédélec's first family of elements. A non-standard mesh truncation technique, FE-IIEE (Finite Element - Iterative Integral Equation Evaluation) for open region problems provides an arbitrarily exact absorbing boundary condition while retaining the original sparse structure of the FEM matrices. The simulator provides a multi-platform (Linux & Windows) user friendly graphical user interface. Remote job submission to HPC clusters is supported by integrating an in-house software tool developed for this purpose.

An electromagnetic in-house parallel solver for the analysis of microwave devices is presented in this paper. The solver makes use of a weak formulation based on double curl vector wave equation discretized with a rigorous implementation of higher-order isoparametric curl-conforming finite elements of Nedelec first family in tetrahedral and triangular prismatic shapes. A non-standard iterative domain decomposition paradigm is used providing an arbitrarily exact radiation boundary condition for the analysis of radiation problems. This methodology retains the original sparse structure of the FEM matrices while minimizing the size of the computational domain. A numerical example is presented showing the reliability and performance of the solver for microwave devices analysis.

In this paper, we present a high-order weak Galerkin finite element method (WG-FEM) for solving the stationary Stokes interface problems with discontinuous velocity and pressure in ℝ d , d = 2, 3. This WG method is equipped with stable finite elements consisting of usual polynomials of degree k ⩾ 1 for the velocity and polynomials of degree k − 1 for the pressure, both are discontinuous. Optimal convergence rates of order k + 1 for the velocity and order k for the pressure are established in L 2-norm on hybrid meshes. Numerical experiments verify the expected order of accuracy for both two-dimensional and three-dimensional examples. Moreover, numerically it is shown that the proposed WG algorithm is able to accommodate geometrically complicated and very irregular interfaces having sharp edges, cusps, and tips.

High-order accurate finite element methods provide unique benefits for problems that have strong anisotropies and complicated geometries and for stiff equation systems that are coupled through large source terms, e.g. Lorentz force, collisions, or atomic reactions. Magnetized plasma simulations of realistic devices using the kinetic or the multi-fluid plasma models are examples that benefit from high-order accuracy. The multi-fluid plasma model only assumes local thermodynamic equilibrium within each fluid, e.g. ion and electron fluids for the two-fluid plasma model. The algorithm implements a discontinuous Galerkin method with an approximate Riemann solver to compute the fluxes of the fluids and electromagnetic fields at the computational cell interfaces. The multi-fluid plasma model has time scales on the order of the electron and ion cyclotron frequencies, the electron and ion plasma frequencies, the electron and ion sound speeds, and the speed of light. A general model for atomic reactions has been developed and is incorporated in the multi-fluid plasma model. The multi-fluid plasma algorithm is implemented in a flexible code framework (WARPX) that allows easy extension of the physical model to include multiple fluids and additional physics. The code runs on multi-processor machines and is being adapted with OpenCL to many-core systems, characteristic of the next generation of high performance computers. The algorithm is applicable to study advanced physics calculations of plasma dynamics including magnetic plasma confinement and astrophysical plasmas. The discontinuous Galerkin method has also been applied to solve the Vlasov-Poisson kinetic model. Recently, a blended finite element algorithm has been developed and implemented which exploits the expected physical behavior to apply either a discontinuous or continuous finite element representation, which improves computational efficiency without sacrificing accuracy.

• Adoption of object-oriented programming in gyrokinetic particle simulations, supporting scalable studies involving multiple species. • The implementation of a high-order piecewise field-aligned finite element method and a mixed particle-in-cell-particle-in-Fourier method within an electrostatic gyrokinetic particle simulation model, enabling efficient 3D and 2D simulations of tokamak plasmas. • A rigorous strong-form treatment of the gyrocenter equations of motion, enhancing accuracy and enabling filter- and buffer-free simulations. • A novel cache optimization strategy, delivering up to a tenfold speedup in specific simulation cases. • Support for unstructured meshes, facilitating whole-device simulations with complex geometries. The features of the TRIMEG-GKX code are described with emphasis on the exploration using novel/different schemes compared to other gyrokinetic codes, particularly the use of object-oriented programming, filter/buffer-free treatment, and a high-order piecewise field-aligned finite element method. The TRIMEG-GKX code solves the electromagnetic gyrokinetic equation using the particle-in-cell scheme, taking into account multi-species effects and shear Alfvén physics. The mixed-variable/pullback scheme has been implemented to enable electromagnetic studies. This code is parallelized using particle decomposition and domain cloning among computing nodes, replacing traditional domain decomposition techniques. The applications to study the micro- and macro-instabilities are demonstrated, including the energetic-particle-driven Alfvén eigenmode, ion temperature gradient mode, and kinetic ballooning mode. Good performance is achieved in both ad hoc and experimentally reconstructed equilibria, such as those of the ASDEX Upgrade (AUG), Tokamak á configuration variable (TCV), and the Joint European Torus (JET). Future studies of edge physics using the high-order C 1 finite element method for triangular meshes in the TRIMEG-C1 code will be built upon the same numerical methods.

A high-order source removal finite element method for a class of elliptic interface problems

In this paper, a new high-order multiscale finite element method (MsFEM) is developed for elliptic problems with highly oscillating coefficients. The method is inspired by the MsFEM developed in [G. Allaire and R. Brizzi, Multiscale Model. Simul., 4 (2005), pp. 790--812], but a more explicit multiscale finite element space is constructed. The approximation space is nonconforming when an oversampling technique is used. We use a Petrov--Galerkin formulation suggested in [T. Y. Hou, X.-H. Wu, and Y. Zhang, Commun. Math. Sci., 2 (2004), pp. 185--205] to simplify the implementation and to improve the accuracy. The method is natural for high-order finite element methods used with the advantage of solving the coarse grained problem. We prove optimal error estimates in the case of periodically oscillating coefficients and support the findings by various numerical experiments.

A non-overlapping Schwarz domain decomposition method with high-order finite elements for flow acoustics

A new higher order finite element method for elliptic partial differential equations on a stationary smooth surface $\Gamma$ is introduced and analyzed. We assume that $\Gamma$ is characterized as the zero level of a level set function $\phi$ and only a finite element approximation $\phi_h$ (of degree $k \geq 1$) of $\phi$ is known. For the discretization of the partial differential equation, finite elements (of degree $m\geq 1$) on a piecewise linear approximation of $\Gamma$ are used. The discretization is lifted to $\Gamma_h$, which denotes the zero level of $\phi_h$, using a quasi-orthogonal coordinate system that is constructed by applying a gradient recovery technique to $\phi_h$. A complete discretization error analysis is presented in which the error is split into a geometric error, a quadrature error, and a finite element approximation error. The main result is a $H^1(\Gamma)$-error bound of the form $c(h^m + h^{k+1})$. Results of numerical experiments illustrate the higher order convergence of t...

This thesis presents an object oriented architecture for the High-Order Finite Element Method (HO-FEM) to solve problems of large deformations in structural mechanics. The software hp 2 fem was implemented using the Matlab programming language and is fully modular and easily extensible to any problem. The formulation of large deformation considered the kinematic, stress-strain operators and constitutive equations. The solution of nonlinear system equations is performed by the Newton-Raphson method. Results are presented for the simulation of large deformation problem using neo-Hookean compressible material in 2D and 3D problems and the high order methods. Also, contact problems and shape optimization are performed here. It is concluded that the architecture works very well and that the use of the HO-FEM brings great benefits in terms of convergence rate.

Fast direct and inverse algorithms for expansion in terms of eigenvectors of one-dimensional eigenvalue problems for a high-order finite element method (FEM) are proposed based on the fast discrete Fourier transform. They generalize logarithmically optimal Fourier algorithms for solving boundary value problems for Poisson-type equations on rectangular meshes to high-order FEM. The algorithms can be extended to the multidimensional case and can be applied to nonstationary problems.

An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation

Over the past years,SLAC's Advanced Computations Department (ACD) has developed the parallel finite element (FE) particle-in-cell code Pic3P (Pic2P) for simulations of beam-cavity interactions dominated by space- charge effects. As opposed to standard space-charge dominated beam transport codes, which are based on the electrostatic approximation,Pic3P (Pic2P) includes space-charge, retardation and boundary effects as it self-consistently solves the complete set of Maxwell-Lorentz equations using higher-order FE methods on conformal meshes. Use of efficient, large-scale parallel processing allows for the modeling of photoinjectors with unprecedented accuracy, aiding the design and operation of the next-generation of accelerator facilities. Applications to the Linac Coherent Light Source (LCLS) RF gun are presented.