Guide Advanced finite element methods and applications

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The first method, introduced in Harizanov et al. Thus, the two BURA methods have mutually complementary advantages. A three-level extension of the GDSW overlapping Schwarz preconditioner in two dimensions is presented, constructed by recursively applying the GDSW preconditioner to the coarse problem.

Numerical results, obtained for a parallel implementation using the Trilinos software library, are presented for up to 90, cores of the JUQUEEN supercomputer. The superior weak parallel scalability of the three-level method is verified.


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  • FEEG | Advanced Finite Element Analysis | University of Southampton.
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For large problems and a large number of cores, the three-level method is faster by more than a factor of two, compared to the standard two-level method. The three-level method can also be expected to scale when the classical method will already be out-of-memory. Isogeometric Analysis IgA is a framework for setting up spline-based discretizations of partial differential equations, which has been introduced around a decade ago and has gained much attention since then.

If large spline degrees are considered, one obtains the approximation power of a high-order method , but the number of degrees of freedom behaves like for a low-order method. One important ingredient to use a discretization with large spline degree, is a robust and preferably parallelizable solver. While numerical evidence shows that multigrid solvers with standard smoothers like Gauss Seidel does not perform well if the spline degree is increased, the multigrid solvers proposed by the authors and their co-workers proved to behave optimal both in the grid size and the spline degree.

In the present paper, the authors want to show that those solvers are parallelizable and that they scale well in a parallel environment. The consistency of a posteriori error bounds for solutions by the finite element methods assumes in this paper that their orders of accuracy in respect to the mesh size h coincide with those in the corresponding sharp a priori bounds.

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Additionally, it assumes that for such a coincidence it is sufficient that the testing fluxes possess only the standard approximation properties without resorting to the equilibration. As a consequence of these facts, there is a wide range of computationally cheap and efficient procedures for evaluating the test fluxes, making the obtained a posteriori error bounds sharp. The technique of obtaining the consistent a posteriori bounds was exposed in [arXiv NA] 6 Nov ] and very briefly in [ Doklady Mathematics , 96 1 , , —].

We introduce a completely unstructured, conforming space-time finite element method for the numerical solution of parabolic initial-boundary value problems with variable in space and time, possibly discontinuous diffusion coefficients. Discontinuous diffusion coefficients allow the treatment of moving interfaces.

We show stability of the method and an a priori error estimate , including the case of local stabilizations which are important for adaptivity. To study the method in practice, we consider several typical model problems in one, two, and three spatial dimensions. The implementation of our space-time finite element method is fully parallelized with MPI. Extensive numerical tests were performed to study the convergence behavior of the stabilized space-time finite element discretization method and the scaling properties of the parallel AMG-preconditioned GMRES solver that we use to solve the huge system of space-time finite element equations.

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An initial segmentation of the surface elements is shown to be a reasonable tool to prevent problematic blocks which appear on surfaces with edges. It leads to significantly easier control of the Partial ACA algorithm and our numerical results show perfect convergence of all numerical quantities corresponding to the theory of BEM. In particular, third order convergence is reached for the gradient of the solution inside the domain. In this work, we develop a specialized quadrature rule for trimmed domains, where the trimming curve is given implicitly by a real-valued function on the whole domain.

We follow an error correction approach: In a first step, we obtain an adaptive subdivision of the domain in such a way that each cell falls in a predefined base case. We then extend the classical approach of linear approximation of the trimming curve by adding an error correction term based on a Taylor expansion of the blending between the linearized implicit trimming curve and the original one.

The finite element method and applications in engineering using ANSYS

This approach leads to an accurate method which improves the convergence of the quadrature error by one order compared to piecewise linear approximation of the trimming curve. It is at the same time efficient , since essentially the computation of one extra one-dimensional integral on each trimmed cell is required. Finally, the method is easy to implement , since it only involves one additional line integral and refrains from any point inversion or optimization operations. The convergence is analyzed theoretically and numerical experiments confirm that the accuracy is improved without compromising the computational complexity.

In this work, we study a Galerkin—Petrov space—time finite element method for a linear system of parabolic—elliptic equations with in general anisotropic conductivity matrices , which may be considered as a simplified version of the nonlinear bidomain equations. The discretization is based on a stable space—time variational formulation employing continuous and piecewise linear finite elements in both spatial and temporal directions simultaneously.

We show stability of the space—time formulation on both the continuous and discrete level for such a coupled problem under a rather general condition on the conductivity matrices. We further discuss the construction of a monolithic algebraic multigrid AMG method for solving the coupled linear system of algebraic equations globally.

Numerical experiments are performed to demonstrate the convergence of the space—time finite element approximations, and the performance of the AMG method with respect to the mesh discretization parameter. This unit develops an alternate derivation of the weak form, which is applicable to certain physical problems. In this unit, we develop the finite element method for three-dimensional scalar problems, such as the heat conduction or mass diffusion problems.

In this unit, you will complete some details of the three-dimensional formulation that depend on the choice of basis functions, as well as be introduced to the second coding assignment. In this unit, we take a detour to study the two-dimensional formulation for scalar problems, such as the steady state heat or diffusion equations. This unit introduces the problem of three-dimensional, linearized elasticity at steady state, and also develops the finite element method for this problem. Aspects of the code templates are also examined. In this unit, we study the unsteady heat conduction, or mass diffusion, problem, as well as its finite element formulation.

In this unit we study the problem of elastodynamics, and its finite element formulation. It is very well structured and Dr Krishna Garikipati helps me understand the course in very simple manner.

I would like to thank coursera community for making this course available. The course is great and the tutors are very helpful.

Lecture - 1 Advanced Finite Elements Analysis

I just have a suggestion that there should be more coding assignment like one for every week. Peer review assignments can only be submitted and reviewed once your session has begun. If you choose to explore the course without purchasing, you may not be able to access certain assignments. When you purchase a Certificate you get access to all course materials, including graded assignments. Upon completing the course, your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile.

If you only want to read and view the course content, you can audit the course for free. You will need computing resources sufficient to install the code and run it. Additionally, you will need a specific visualization program that we recommend. Altogether, if you have 1GB you should be fine.

Alternately, you could download a Virtual Machine Interface.

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You will be able to write code that simulates some of the most beautiful problems in physics, and visualize that physics. You will need to know about matrices and vectors. Having seen partial differential equations will be very helpful.


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  • EL - Advanced Finite Element Analysis | ASME - ASME;
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More questions? Visit the Learner Help Center. Browse Chevron Right. Physical Science and Engineering Chevron Right. Mechanical Engineering. Offered By. University of Michigan. About this Course 42, recent views. Flexible deadlines. Flexible deadlines Reset deadlines in accordance to your schedule. Intermediate Level.

INF – Advanced Finite Element Methods - University of Oslo

Hours to complete. Apart from the lectures, expect to put in between 3 and 5 hours a week. Available languages. English Subtitles: English. Learners taking this Course are. Chevron Left. Syllabus - What you will learn from this course. Video 11 videos.