Finite element multigrid framework for mimetic finite di erence discretizations xiaozhe hu tufts university polytopal element methods in mathematics and engineering, october 26 28, 2015 joint work with. Multigrid finite element methods for electromagnetic field. Illconditioning of the system matrix is a wellknown complication in immersed finite element methods and trimmed isogeometric analysis. It can be used for obtaining the numerical solutions of the partial differential. The finitevolume method is a method for representing and evaluating partial differential equations in the form of algebraic equations leveque, 2002.
High level implementation of geometric multigrid solvers. A peacemanrachford type iteration is used as a smoother to decouple the nonlinearity from the divergence constraint. The finite element method fem is the most widely used method for solving problems of engineering and mathematical models. Thus, we solve many local problems, on each coarselevelelement, in parallel, on the high level. Freefem is a free and opensource parallel fea software for multiphysics simulations. This paper investigates one of such iterative solvers, the geometric multigrid solver, to find the. The multigrid method described above belongs to a class of optimal order methods for solving linear systems emanating from the discretization techniques like the finite element method. The hyteg finiteelement software framework for scalable. Based on a finite element discretization with continuous finite elements, the taylorgalerkin approach is globally mass. Efficient multigrid solvers for mixed finite element.
The software is available freely with source f77 and the site has a lot of further information available online. The nonlinear equation can be solved element wise with a closed formulae. Cangellaris this is the first comprehensive monograph that features stateoftheart multigrid methods for enhancing the modeling versatility, numerical robustness, and computational efficiency of one of the most popular classes of numerical electromagnetic. Parallel multigrid solver for 3d unstructured finite element problems mark adams yjames w. Distributed algebraic multigrid for finite element. Parallel multigrid solver for 3d unstructured finite. Immersed finite element method for interface problems with. The problems are defined in terms of their variational formulation and can be easily implemented using freefem language. Which is the best method to iterate the required vector so that the computation time taken is minimized. Citeseerx nonnested multigrid methods in finite element. The hyteg finiteelement software framework for scalable multigrid solvers.
Its known convergence speed does not deteriorate when the discretization is refined whereas classical iterative solvers slow down for the decreasing mesh size. Ansys structural analysis software enables you to solve complex structural engineering problems and make better, faster design decisions. The nonlinear equation can be solved elementwise with a closed formulae. Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. Multigrid methods can also be applied to integral equations, or for problems in statistical physics. This does not necessarily mean that companies and research institutes save money by adopting fea. Multigrid solvers for immersed finite element methods and. Design and data structure of fully adaptive, multigrid. These discretizations are described in detail in the lecture notes of numerical mathematics iii. Which is the best method to iterate the required v. Multigrid analysis of finite element methods 411 that depend on the accuracy of the quadrature scheme. Diffpack numerical solution of pdes spcpm po 3d chemnitz parallel f. This is a list of software packages that implement the finite element method for solving partial differential equations.
The hyteg finiteelement software framework for scalable multigrid. On the software and, more importantly in the scope of this paper, the algorithmic. Multigrid methods are an established concept in fluid mechanics as well, however, and similar schwarztype methods have successfully been applied to flow problems with both immersed finite element methods and meshfitting multigrid solvers, i. Therefore, it is anticipated that the presented preconditioning technique. This is the first comprehensive monograph that features stateoftheart multigrid methods for enhancing the modeling versatility, numerical robustness, and computational efficiency of one of the most popular classes of numerical electromagnetic field modeling methods.
They do, however, get more development for their dollar. A multigrid finite element solver for the cahnhilliard equation is presented that has meshindependent convergence rates for any timestep size, including in the important limit. Another set of multiresolution methods is based upon wavelets. Multigrid methods for finite elements combines two rapidly developing fields. In this paper, we present an objectoriented software framework for finite element based tes simulations. The implementation of efficient multigrid preconditioners for elliptic partial differential equations pdes is a challenge due to the complexity of the resulting algorithms and corresponding computer code. With the finite element analysis fea solvers available in the suite, you can customize and automate solutions for your structural mechanics problems and parameterize them to analyze multiple design scenarios. Multigrid methods are solvers for linear system of equations that arise, e. An objectoriented framework for versatile finite element. A finite element multigridframework to solve the sea ice momentum equation. This problem can be circumvented by closely coupling the multigrid method with the mesh generation and finite element analysis software 3,4. Multigrid methods for mortar finite elements springerlink.
We consider linear finite element or equivalently 5point stencil discretization of the poisson equation on a uniform grid of 0,12 with size h. For poisson or convectiondiffusion equation as follows. This free software is compatible with the windows platform and is an open source software that can be used for finite element analysis and for many multiphysical problems. For simplicity, we assume h 12l and zero dirichlet bounary condition. Multigrid finite element methods for electromagnetic field modeling yu zhu, andreas c. The system includes a lot of tools and an incompressible fluidflow package called featflow. The modular software architecture is carefully designed to exploit the resources of modern and future supercomputers. Yes, yes, yes up to 147k processes, test for 4k processes and geometric multigrid for 147k, strong and weak scaling, no, yes, demonstrated. Prometheus is a multigrid solver for finite element matrices on unstructured meshes in solid mechanics. By means of the biorthogonality between the nodal basis functions of our new lagrange multiplier space and the finite element trace space, we derive a symmetric positive definite mortar formulation on the unconstrained product space. On the numerical modeling of convectiondiffusion problems. The hierarchical hybrid grids hhg software framework 3, 17 is designed to close this gap between finite element flexibility and geometric multigrid performance by using a compromise between. Second, geometric mg has problems with discontinuous material properties which limit the degree of coarsening in the mesh. A finite element multigridframework to solve the sea ice.
Home acm journals acm transactions on mathematical software vol. An efficient nonlinear multigrid method for a mixed finite element method of the darcyforchheimer model is constructed in this paper. The algebraic approach also has the advantage that strainlocalizations, in nonlinearmaterial problems, in. Currently, such linear elastic finite element analysis capabilities with the ebe pcg solver are incorporated into microct scanner software scanco medical. The mixed finite element multigrid method for stokes equations. Efficient finite element geometric multigrid solvers for unstructured. Though some sections are dated, the contents of this book remain a solid foundation for understanding the behavior of finite element techniques in theory and in practice. Local modification of meshes for adaptive andor multigrid finite. The author indicates how this data structure is used to support the multigrid iterative method for solving finite element equations on the series of grids and discusses. Pdf the mixed finite element multigrid method for stokes.
A massively parallel multigrid method for finite elements. Multiple features of objectoriented design and programming are utilized to create a modular software architecture that encapsulates medical, mathematical, and computational attributes of tes. Many other papers on the use of the oop for numerical analysis were published, and. Fast and accurate finiteelement multigrid solvers for pde. Its known convergence speed does not deteriorate when the discretization is refined whereas classical iterative solvers slow down for the decreasing mesh size 1, 2, 5, 6. There are many methods like multigrid and conjugate gradient methods. Multigrid methods for a mixed finite element method of the. In this article, a new generic higherorder finiteelement framework for massively parallel simulations is presented.
A multigrid finite element solver for the cahnhilliard. Finite element analysis fea software comsol multiphysics. The hyteg finiteelement software framework for scalable multigrid solvers article in international journal of parallel emergent and distributed systems august 2018 with 70 reads. It supports trilinear finite element discretizations constructed using octees. This result was obtained under the assumption that there was no approximation of the boundary. This allows the user full control over all components of the algorithm and multigrid solvers for finite element discretisations have been implemented for example in dune and deal. Immersed finite element method for interface problems with algebraic multigrid solver by wenqiang feng a thesis presented to the faculty of the graduate school of missouri university of science and technology in partial ful. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.
Algebraic multigrid amg is one of the most effective iterative. It includes models of structural mechanics, fluid dynamics, heat transfer, electromagnetics etc. Finite element multigrid framework for mimetic finite. Finite element analysis software uses iteration to find displacement vector. For sophisticated finite element discretisations on unstructured grids an efficient implementation can be very time consuming and requires the programmer to have indepth knowledge of the. What is the best iteration method used in finite element. Local modification of meshes for adaptive andor multigrid finiteelement methods. For example, one use of wavelets is to reformulate the finite element approach in terms of a multilevel method. Open source finite element software for multiphysical problems. Finite element approximate inverses in conjunction with richardons iterative method could be used as smoothers in the multigrid method. Multigrid solvers for immersed finite element methods and immersed. Numerical methods for partial differential equations. This is a list of software packages that implement the finite element method for solving partial. Applications of algebraic multigrid to largescale finite element.
The fem is a particular numerical method for solving. Feat is a general purpose software system for solving pdes with the finite element method. If you have no interest in finite elements but enjoy wellwritten mathematical literature, this book still belongs on your shelf. Multigrid solvers for immersed finite element methods and immersed isogeometric analysis article pdf available in computational mechanics 65. Multigrid methods for finite elements mathematics and its. The purpose of finite element analysis fea software is to reduce the number of prototypes and experiments that have to be run when designing, optimizing, or controlling a device or process.
The effectiveness of the multigrid method can be also exploited by using the finite element method. Elmer is a finite element software for numerical solution of partial differential equations and multiphysical problems. These wavelet methods can be combined with multigrid methods. On the multigrid solution of finite element equations with.
Design and data structure of fully adaptive, multigrid, finiteelement. Prometheus uses maximal independent sets on modified matrix graphs. At the theoretical level, shaidurov justifies the rate of convergence of various multigrid algorithms for selfadjoint and nonselfadjoint problems, positive definite and indefinite problems, and singular and spectral problems. List of finite element software packages wikipedia. International journal of parallel, emergent and distributed systems.