# Hp-version fem-

Computational Mechanics. The paper is the first of the series of two which analyses the h-p version of the finite element method in two dimensions. The main result is that the h-p version leads to an exponential rate of convergence when solving problems with piecewise analytic data. Unable to display preview. Download preview PDF.

Added To Cart. Finite Elements pp Cite as. This process Hp-version fem experimental and the keywords may be updated as the learning algorithm improves. RAIRO 21 — Part 1: The Hp-verslon analysis of the p -version, Tech. Jian Ming Zhang Pornography samples, Yong He. Google Scholar. SIAM J. October Continue Shopping To Cart.

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Clough with co-workers at UC BerkeleyO. The h, p and h-p versions of the finite element method in 1 dimension - Part II. Altman z model Scientific Publishers; FEA simulations provide a valuable resource Hp-vwrsion they remove multiple instances of creation and testing of hard prototypes for various high fidelity situations. Demkowicz, W. In the hp-FEM, the polynomial degrees can Hp-version fem from element to element. The convergence graphs show the approximation error as a function of the number of degrees of freedom DOF. By using this site, you agree to the Terms of Use Hp-version fem Privacy Policy. Main article: Discontinuous Galerkin method. In other projects Wikimedia Commons Wikiversity.

Finite Elements pp Cite as.

- The exponential convergence of the hp-FEM was not only predicted theoretically but also observed by numerous independent researchers.
- The finite element method FEM is a numerical method for solving problems of engineering and mathematical physics.

Finite Elements pp Cite as. There are three versions of the finite element method. The p version keeps the mesh fixed, and the accuracy is achieved by increasing the degree p. The h-p version combines both approaches. Unable to display preview. Download preview PDF. Skip to main content. Advertisement Hide. Authors Authors and affiliations I.

Conference paper. This process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, log in to check access. Arnold, I. Methods Appl. Anderson and U. Google Scholar. Aziz, ed.

Gui, and B. Hayduk and A. Noor, eds. Anal , to appear. Vichnevetsky and R. Stepleman, eds. Guo, and J. Methods Engrg. Modelling Numer.

RAIRO 21 — Szabo, and I. SIAM J. Barnhart and J. Delves and C. Anal , 21 , — MathSciNet Google Scholar. Anal , 23 , 58— Dunavant and B. CrossRef Google Scholar. Bathe and R. Owen, eds. Gelfand and G. Shilov, Generalized Functions , Vol. Gottlieb, M. Hussaini, and S. Voigt, D. Gottlieb, and M. Hussaini, eds. Gui and I. Guo and I. Hayes-Roth, D. Waterman, and D. Lenat eds. Louis, Katz, A.

Peano, and M. Katz, and E. Moscow Math. Surveys , 38 , No. Noor and I. Design , 3 , 1— Appl , 2 , — Peano and J. Rank and I. Schiermeier and B. Design , 3 , 93— Appl , 5 , 99— Methods Appl Math. Babuska, J. Gago, E. Oliveira, and D. Zienkiwicz, eds. Appl Numer. Methods , 2 , — Louis, MO, Louis, Google Scholar. Wang, I. Katz and G. Katz, and B. Zienkiewicz, B. Irons, I. Scott, and J. Zienkiewicz, J. Gago, and D. Personalised recommendations.

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The hpk-FEM combines adaptively, elements with variable size h , polynomial degree of the local approximations p and global differentiability of the local approximations k-1 in order to achieve best convergence rates. It is assumed that the reader is familiar with calculus and linear algebra. The paper is the second in the series of three devoted to the detailed analysis of the three, basic versions of the finite element method in one dimensional setting. Please review our privacy policy. The color represents the amplitude of the magnetic flux density , as indicated by the scale in the inset legend, red being high amplitude. Figure 5. We will demonstrate the finite element method using two sample problems from which the general method can be extrapolated.

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In h-FEM, the polynomial order p of the element shape functions is constant and the element size h is decreasing. To have an accurate simulation with the h-FEM, a mesh with large number of nodes and elements is usually needed. In order to overcome this problem, the high order finite element method p-FEM was proposed. In the p-version, the polynomial order is increasing and the mesh size is constant. The hp-FEM needs a smaller number of nodes and consequently, less computational time and less memory to achieve the same or even better accuracy than h-FEM.

The conductivity distribution is to be reconstructed from these potential measurements. The applications of EIT in medical, detection of cancerous tumors from breast tissue[ 1 — 3 ], measuring brain function[ 4 — 5 ], and gastric functions[ 6 — 8 ].

In industry EIT has applications such as imaging of fluid flows in process pipelines[ 9 ], and non-destructive testing of materials[ 10 ]. For a review on EIT see also[ 6 , 9 ]. EIT is a badly posed inverse problem. Small errors in the measurements or in solution of the forward problems can produce large errors in the reconstruction. One of the important approaches to solve forward problem is the finite element approximation.

In the h-FEM, the polynomial order p of the element shape functions is constant and the element size h is decreasing. With decreasing the element size, discretisation error is reduced. In the p-version, the basis functions are higher order Legendre polynomials. In this method the polynomial order is increased and the mesh size is constant.

Improving the algorithm efficiency by optimizing the number of mesh elements[ 18 ]. Considering and carrying out simulations on the h- and p- version of FEM[ 12 ]. In section 2, the high order FEM forward model was described. Some details of the presented method were discussed in section 4. The forward problem involves the solution of a Maxwell equation. In the quasi-static and low frequencies approximation the fields can be described with Laplace equation.

The complete electrode model with the mixed boundary condition Dirichlet and Neumann. In this equation. The detail of numerical implementation of the forward problem based on CEM has been discussed in [ 19 , 20 ]. The set of one-dimensional hierarchic shape functions, introduced by Szabo and Babuska , is given by. In 2D, the standard triangular element, shown in Fig. In this figure:. The standard triangular element[ 13 ].

Nodal shape functions : These shape functions are the same as the shape functions in h-FEM. Side modes : There are 3 p — 1 side modes for each triangle. The side modes associated with side 1 are:. In order to apply the p-FEM to the EIT forward problem, the physical domain is first divided into triangular elements. The global matrix in 8 is assembled by going through each triangular element and by computing the elemental matrices B e , C e and G e in local reference coordinates.

Note that the boundary contribution in 9 is nonzero only for elements on the outer boundary. The mesh used in the PDE Toolbox has triangular cells and nodes. The current patterns are designed in a way that 16 adjacent pairs are allocates as source and sink electrodes and one extra electrode is embedded as the ground electrode.

The mesh used in precise solution has cells and nodes. In Fig. It can be seen that the hp-version is able to achieve better accuracy than the conventional FEM. In order to achieve an accuracy of just 0. In[ 15 ] Pursiainen presented that performance of the p-version is better than that of the h-version. The secondary field is basically caused by the objects. So, it is essential that the forward model is able to simulate the secondary field accurately.

To verify the accuracy of the high order FEM forward solver for the secondary field, we used as phantom shown in Fig. The high order finite element method has been developed and validated for the EIT forward problem. National Center for Biotechnology Information , U. J Med Signals Sens. Saeedizadeh , 1 S. Kermani , 1 and H. Rabbani 1. Author information Article notes Copyright and License information Disclaimer.

Received Aug 11; Accepted Oct This is an open-access article distributed under the terms of the Creative Commons Attribution-Noncommercial-Share Alike 3. The effectiveness of GFEM has been shown when applied to problems with domains having complicated boundaries, problems with micro-scales, and problems with boundary layers. The mixed finite element method is a type of finite element method in which extra independent variables are introduced as nodal variables during the discretization of a partial differential equation problem.

The hp-FEM combines adaptively, elements with variable size h and polynomial degree p in order to achieve exceptionally fast, exponential convergence rates. The hpk-FEM combines adaptively, elements with variable size h , polynomial degree of the local approximations p and global differentiability of the local approximations k-1 in order to achieve best convergence rates. It extends the classical finite element method by enriching the solution space for solutions to differential equations with discontinuous functions.

Extended finite element methods enrich the approximation space so that it is able to naturally reproduce the challenging feature associated with the problem of interest: the discontinuity, singularity, boundary layer, etc. It was shown that for some problems, such an embedding of the problem's feature into the approximation space can significantly improve convergence rates and accuracy. Several research codes implement this technique to various degrees: 1. However, unlike the boundary element method, no fundamental differential solution is required.

It was developed by combining meshfree methods with the finite element method. Spectral element methods combine the geometric flexibility of finite elements and the acute accuracy of spectral methods. Spectral methods are the approximate solution of weak form partial equations that are based on high-order Lagragian interpolants and used only with certain quadrature rules. Loubignac iteration is an iterative method in finite element methods.

Some types of finite element methods conforming, nonconforming, mixed finite element methods are particular cases of the gradient discretisation method GDM.

Hence the convergence properties of the GDM, which are established for a series of problems linear and non linear elliptic problems, linear, nonlinear and degenerate parabolic problems , hold as well for these particular finite element methods. Generally, FEM is the method of choice in all types of analysis in structural mechanics i. This is especially true for 'external flow' problems, like air flow around the car or airplane, or weather simulation.

A variety of specializations under the umbrella of the mechanical engineering discipline such as aeronautical, biomechanical, and automotive industries commonly use integrated FEM in design and development of their products.

Several modern FEM packages include specific components such as thermal, electromagnetic, fluid, and structural working environments. In a structural simulation, FEM helps tremendously in producing stiffness and strength visualizations and also in minimizing weight, materials, and costs.

FEM allows detailed visualization of where structures bend or twist, and indicates the distribution of stresses and displacements. FEM software provides a wide range of simulation options for controlling the complexity of both modeling and analysis of a system. FEM allows entire designs to be constructed, refined, and optimized before the design is manufactured.

This powerful design tool has significantly improved both the standard of engineering designs and the methodology of the design process in many industrial applications.

In the s FEA was proposed for use in stochastic modelling for numerically solving probability models [24] and later for reliability assessment. From Wikipedia, the free encyclopedia. Numerical method for solving physical or engineering problems. For the elements of a poset , see compact element. Navier—Stokes differential equations used to simulate airflow around an obstruction. Natural sciences Engineering. Order Operator.

Relation to processes. Difference discrete analogue Stochastic Stochastic partial Delay. General topics. Phase portrait Phase space. Numerical integration Dirac delta function. Solution methods. Colours indicate that the analyst has set material properties for each zone, in this case a conducting wire coil in orange; a ferromagnetic component perhaps iron in light blue; and air in grey.

Although the geometry may seem simple, it would be very challenging to calculate the magnetic field for this setup without FEM software, using equations alone. FEM solution to the problem at left, involving a cylindrically shaped magnetic shield. The ferromagnetic cylindrical part is shielding the area inside the cylinder by diverting the magnetic field created by the coil rectangular area on the right.

The color represents the amplitude of the magnetic flux density , as indicated by the scale in the inset legend, red being high amplitude. The area inside the cylinder is low amplitude dark blue, with widely spaced lines of magnetic flux , which suggests that the shield is performing as it was designed to. Interpolation of a Bessel function. The linear combination of basis functions yellow reproduces J 0 blue to any desired accuracy. Main article: Applied element method.

Main article: Mixed finite element method. Main article: Extended finite element method. Main article: Smoothed finite element method.

Main article: Spectral element method. Main article: Meshfree methods. Main article: Discontinuous Galerkin method.

Main article: Finite element limit analysis. Main article: Stretched grid method. This section does not cite any sources. Please help improve this section by adding citations to reliable sources.

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Logan A first course in the finite element method. Cengage Learning. Retrieved Journal of Applied Mechanics. Bulletin of the American Mathematical Society. Retrieved 17 March Prentice Hall. Zhu 31 August Finite Element Procedures.

Programming the Finite Element Method Fifth ed. June International Journal of Computational Methods. Solin, K. Segeth, I. Computer Methods in Applied Mechanics and Engineering. Machine Design.

Coventive Composites. Archived from the original on Journal of Computational and Applied Mathematics. Archives of Computational Methods in Engineering. Journal of Nuclear Materials. Numerical partial differential equations by method. Categories : Continuum mechanics Finite element method Numerical differential equations Partial differential equations Structural analysis Computational electromagnetics.

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In recent three decades, the finite element method FEM has rapidly developed as an important numerical method and used widely to solve large-scale scientific and engineering problems. Zhang and Y. Request Permissions. Numer Anal. Gui and I. Craig, J. Mandel and J. Demokowicz, T. Oden, W. Rachowicz and O. Hardy, Towards a universal hp-adaptive finite element method, I. Constrained approximation and data structure, Comp.

Guo and W. Cao, A preconditioner for the h-p version of the finite element method in two dimensions, Numer. Oden, L. Demokowicz, W. Rachowicz and T. A-posteriori error estimation, Comp. Pavarino and O. Widlund, A polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions, SIAM J.

John Wiley and Sons, Inc. Ben Belgacem, Polynomial extensions of compatible polynomial traces in three dimensions Comput. Methods Appl. Costabel, M. Dauge and L. Guo, Direct and inverse approximation theorems of the p version of finite element method in the framework of weighted Besov spaces, Part 1: Approximability of functions in weighted Besov spaces, SIAM J.

Anal, 39 Guo, Direct and inverse approximation theorems of the p-version of the finite element method in the framework of weighted Besov spaces, Part 2: Optimal convergence of the p-version of the finite element method, Math. Guo and L. Guo and J. All Rights Reserved. Log In. Paper Titles. Article Preview. Abstract: In recent three decades, the finite element method FEM has rapidly developed as an important numerical method and used widely to solve large-scale scientific and engineering problems.

Add to Cart. Applied Mechanics and Materials Volumes Main Theme:. Advances in Computational Modeling and Simulation. Edited by:. Online since:. October Jian Ming Zhang , Yong He. Cited by. Related Articles. Paper Title Pages. Intel Core-Dual 2. The coupling procedure is that, firstly, the three-dimension geometry model is built with ANSYS and the model meshes are divided. Abstract: Reasonable simplification of finite element modeling of complex mechanical systems can reduce the amount of elements and nodes.

The solution efficiency is improved in this way, which is of great economic significance in the practical engineering problem. Based on the modeling and simulation of a Conveyor System between Roving Frame and Spine Frame in automated textile production line, this paper investigates and proves the effectiveness of simplification of the complex mechanical system in the dominant finite element software — ANSYS. Abstract: A new formulation of triangle element is developed.

Based on the concept of unsymmetric finite element formulation, classical linear triangle shape functions are used as test functions, and FE-LSPIM TRI3 element shape functions are used as trial functions.

The former is fulfill the requirements of intra-element and inter-element continuity in displacement field, and the latter is a good alternative for requirements of completeness in displacement field. Typical test problems for static solids are analyzed. Added To Cart. Continue Shopping To Cart.

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