Статті в журналах: "Conforming FEM"

1

Rammerstorfer, Franz, and Martin Schanz. "FEM-BEM coupling with non-conforming interfaces." PAMM 11, no. 1 (December 2011): 487–88. http://dx.doi.org/10.1002/pamm.201110235.

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2

Wilson, Peter, Tobias Teschemacher, Philipp Bucher, and Roland Wüchner. "Non-conforming FEM-FEM coupling approaches and their application to dynamic structural analysis." Engineering Structures 241 (August 2021): 112342. http://dx.doi.org/10.1016/j.engstruct.2021.112342.

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3

Margenov, Svetozar, and Nikola Kosturski. "MIC(0) preconditioning of 3D FEM problems on unstructured grids: Conforming and non-conforming elements." Journal of Computational and Applied Mathematics 226, no. 2 (April 2009): 288–97. http://dx.doi.org/10.1016/j.cam.2008.08.033.

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4

Georgiev, I., J. Kraus, and S. Margenov. "Multilevel preconditioning of rotated bilinear non-conforming FEM problems." Computers & Mathematics with Applications 55, no. 10 (May 2008): 2280–94. http://dx.doi.org/10.1016/j.camwa.2007.11.008.

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5

Rüberg, Thomas, Martin Schanz, and Gernot Beer. "Non-conforming FEM-BEM coupling for wave propagation phenomena." PAMM 8, no. 1 (December 2008): 10333–34. http://dx.doi.org/10.1002/pamm.200810333.

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6

Selzer, Philipp, and Olaf A. Cirpka. "Postprocessing of standard finite element velocity fields for accurate particle tracking applied to groundwater flow." Computational Geosciences 24, no. 4 (June 24, 2020): 1605–24. http://dx.doi.org/10.1007/s10596-020-09969-y.

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Abstract Particle tracking is a computationally advantageous and fast scheme to determine travel times and trajectories in subsurface hydrology. Accurate particle tracking requires element-wise mass-conservative, conforming velocity fields. This condition is not fulfilled by the standard linear Galerkin finite element method (FEM). We present a projection, which maps a non-conforming, element-wise given velocity field, computed on triangles and tetrahedra, onto a conforming velocity field in lowest-order Raviart-Thomas-Nédélec ($\mathcal {RTN}_{0}$ R T N 0 ) space, which meets the requirements of accurate particle tracking. The projection is based on minimizing the difference in the hydraulic gradients at the element centroids between the standard FEM solution and the hydraulic gradients consistent with the $\mathcal {RTN}_{0}$ R T N 0 velocity field imposing element-wise mass conservation. Using the conforming velocity field in $\mathcal {RTN}_{0}$ R T N 0 space on triangles and tetrahedra, we present semi-analytical particle tracking methods for divergent and non-divergent flow. We compare the results with those obtained by a cell-centered finite volume method defined for the same elements, and a test case considering hydraulic anisotropy to an analytical solution. The velocity fields and associated particle trajectories based on the projection of the standard FEM solution are comparable to those resulting from the finite volume method, but the projected fields are smoother within zones of piecewise uniform hydraulic conductivity. While the $\mathcal {RTN}_{0}$ R T N 0 -projected standard FEM solution is thus more accurate, the computational costs of the cell-centered finite volume approach are considerably smaller.

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7

Schedensack, Mira. "A New Generalization of the P1 Non-Conforming FEM to Higher Polynomial Degrees." Computational Methods in Applied Mathematics 17, no. 1 (January 1, 2017): 161–85. http://dx.doi.org/10.1515/cmam-2016-0031.

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AbstractThis paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz spaces of arbitrary polynomial degree and its discretization coincides with the mentioned non-conforming FEM for the lowest polynomial degree. The discretization directly approximates the gradient of the solution instead of the solution itself. Besides the a priori and medius analysis, this paper proves optimal convergence rates for an adaptive algorithm for the new discretization. These are also demonstrated in numerical experiments. Furthermore, this paper focuses on extensions of this new scheme to quadrilateral meshes, mixed FEMs, and three space dimensions.

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8

Margenov, S., and P. Minev. "On a preconditioning of non-conforming mixed FEM elliptic problems." Mathematics and Computers in Simulation 76, no. 1-3 (October 2007): 149–54. http://dx.doi.org/10.1016/j.matcom.2007.01.021.

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9

Kolev, Tzanio V., and Svetozar D. Margenov. "Two-level preconditioning of pure displacement non-conforming FEM systems." Numerical Linear Algebra with Applications 6, no. 7 (October 1999): 533–55. http://dx.doi.org/10.1002/(sici)1099-1506(199910/11)6:7<533::aid-nla175>3.0.co;2-7.

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10

XU, X., G. R. LIU, Y. T. GU, and G. Y. ZHANG. "A CONFORMING POINT INTERPOLATION METHOD (CPIM) BY SHAPE FUNCTION RECONSTRUCTION FOR ELASTICITY PROBLEMS." International Journal of Computational Methods 07, no. 03 (September 2010): 369–95. http://dx.doi.org/10.1142/s0219876210002295.

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A conforming point interpolation method (CPIM) is proposed based on the Galerkin formulation for 2D mechanics problems using triangular background cells. A technique for reconstructing the PIM shape functions is proposed to create a continuous displacement field over the whole problem domain, which guarantees the CPIM passing the standard patch test. We prove theoretically the existence and uniqueness of the CPIM solution, and conduct detailed analyses on the convergence rate; computational efficiency and band width of the stiffness matrix of CPIM. The CPIM does not introduce any additional degrees of freedoms compared to the linear FEM and original PIM; while convergence rate of quadratic CPIM is in between that of linear FEM and quadratic FEM which results in the high computational efficiency. Intensive numerical studies verify the properties of the CPIM.

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11

Besuievsky, G., E. García-Nevado, G. Patow, and B. Beckers. "Procedural modeling buildings for finite element method simulation." Journal of Physics: Conference Series 2042, no. 1 (November 1, 2021): 012074. http://dx.doi.org/10.1088/1742-6596/2042/1/012074.

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Abstract Finite element methods for heat simulation at urban scale require mesh-volume models, where the meshing process requires a special attention in order to satisfy FEM requirements. In this paper we propose a procedural volume modeling approach for automatic creation of mesh-volume buildings, which are suitable for FEM simulations at urban scale. We develop a basic rule-set library and a building generation procedure that guarantee conforming meshes. In this way, urban models can be easily built for energy analysis. Our test-case shows a street created with building prototypes that fulfill all the requirements for being loaded in a FEM numerical platform such as Cast3M (www-cast3m.cea.fr).

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12

Gudi, Thirupathi, and Papri Majumder. "Conforming and discontinuous Galerkin FEM in space for solving parabolic obstacle problem." Computers & Mathematics with Applications 78, no. 12 (December 2019): 3896–915. http://dx.doi.org/10.1016/j.camwa.2019.06.022.

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13

Mantell, S. C., H. Chanda, J. E. Bechtold, and R. F. Kyle. "A Parametric Study of Acetabular Cup Design Variables Using Finite Element Analysis and Statistical Design of Experiments." Journal of Biomechanical Engineering 120, no. 5 (October 1, 1998): 667–75. http://dx.doi.org/10.1115/1.2834760.

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To isolate the primary variables influencing acetabular cup and interface stresses, we performed an evaluation of cup loading and cup support variables, using a Statistical Design of Experiments (SDOE) approach. We developed three-dimensional finite element (FEM) models of the pelvis and adjacent bone. Cup support variables included fixation mechanism (cemented or noncemented), amount of bone support, and presence of metal backing. Cup loading variables included head size and cup thickness, cup/head friction, and conformity between the cup and head. Interaction between and among variables was determined using SDOE techniques. Of the variables tested, conformity, head size, and backing emerged as significant influences on stresses. Since initially nonconforming surfaces would be expected to wear into conforming surfaces, conformity is not expected to be a clinically significant variable. This indicates that head size should be tightly toleranced during manufacturing, and that small changes in head size can have a disproportionate influence on the stress environment. In addition, attention should be paid to the use of nonmetal backed cups, in limiting cup/bone interface stresses. No combination of secondary variables could compensate for, or override the effect of, the primary variables. Based on the results using the SDOE approach, adaptive FEM models simulating the wear process may be able to limit their parameters to head size and cup backing.

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14

Nadal, E., J. J. Ródenas, J. Albelda, M. Tur, J. E. Tarancón, and F. J. Fuenmayor. "Efficient Finite Element Methodology Based on Cartesian Grids: Application to Structural Shape Optimization." Abstract and Applied Analysis 2013 (2013): 1–19. http://dx.doi.org/10.1155/2013/953786.

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This work presents an analysis methodology based on the use of the Finite Element Method (FEM) nowadays considered one of the main numerical tools for solving Boundary Value Problems (BVPs). The proposed methodology, so-called cg-FEM (Cartesian grid FEM), has been implemented for fast and accurate numerical analysis of 2D linear elasticity problems. The traditional FEM uses geometry-conforming meshes; however, in cg-FEM the analysis mesh is not conformal to the geometry. This allows for defining very efficient mesh generation techniques and using a robust integration procedure, to accurately integrate the domain’s geometry. The hierarchical data structure used in cg-FEM together with the Cartesian meshes allow for trivial data sharing between similar entities. The cg-FEM methodology uses advanced recovery techniques to obtain an improved solution of the displacement and stress fields (for which a discretization error estimator in energy norm is available) that will be the output of the analysis. All this results in a substantial increase in accuracy and computational efficiency with respect to the standard FEM. cg-FEM has been applied in structural shape optimization showing robustness and computational efficiency in comparison with FEM solutions obtained with a commercial code, despite the fact that cg-FEM has been fully implemented in MATLAB.

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15

Guillén-Oviedo, Helen, Jeremías Ramírez-Jiménez, Esteban Segura-Ugalde, and Filánder Sequeira-Chavarría. "Description and implementation of an algebraic multigrid preconditioner for H1-conforming finite element schemes." Uniciencia 34, no. 2 (July 31, 2020): 55–81. http://dx.doi.org/10.15359/ru.34-2.4.

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This paper presents detailed aspects regarding the implementation of the Finite Element Method (FEM) to solve a Poisson’s equation with homogeneous boundary conditions. The aim of this paper is to clarify details of this implementation, such as the construction of algorithms, implementation of numerical experiments, and their results. For such purpose, the continuous problem is described, and a classical FEM approach is used to solve it. In addition, a multilevel technique is implemented for an efficient resolution of the corresponding linear system, describing and including some diagrams to explain the process and presenting the implementation codes in MATLAB®. Finally, codes are validated using several numerical experiments. Results show an adequate behavior of the preconditioner since the number of iterations of the PCG method does not increase, even when the mesh size is reduced.

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16

Dond, Asha K., and Amiya K. Pani. "A Priori and A Posteriori Estimates of Conforming and Mixed FEM for a Kirchhoff Equation of Elliptic Type." Computational Methods in Applied Mathematics 17, no. 2 (April 1, 2017): 217–36. http://dx.doi.org/10.1515/cmam-2016-0041.

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AbstractIn this article, a priori and a posteriori estimates of conforming and expanded mixed finite element methods for a Kirchhoff equation of elliptic type are derived. For the expanded mixed finite element method, a variant of Brouwer’s fixed point argument combined with a monotonicity argument yields the well-posedness of the discrete nonlinear system. Further, a use of both Helmholtz decomposition of $L^{2}$-vector valued functions and the discrete Helmholtz decomposition of the Raviart–Thomas finite elements helps in a crucial way to achieve optimal a priori as well as a posteriori error bounds. For both conforming and expanded mixed form, reliable and efficient a posteriori estimators are established. Finally, the numerical experiments are performed to validate the theoretical convergence rates.

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17

Ma, Zhaoyang, Yunwei Xu, Shu Li, Qingda Yang, Xingming Guo, and Xianyue Su. "A Conforming A-FEM for Modeling Arbitrary Crack Propagation and Branching in Solids." International Journal of Applied Mechanics 13, no. 01 (January 2021): 2150010. http://dx.doi.org/10.1142/s1758825121500101.

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In this paper, an improved conforming AFEM (C-AFEM) for efficient modeling of arbitrary crack propagation and branching is proposed and validated. An explicit formulation for branching cracks has been derived within the C-AFEM framework. The conjugate gradient method is integrated into the C-AFEM formulation to solve the local problem that consists of all elements traversed by single or multiple cracks. Multiple numerical evidences show that this new approach can substantially improve the modeling efficiency. The solution accuracy and numerical robustness are also significantly improved.

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18

Din-Kow Sun, L. Vardapetyan, and Z. Cendes. "Two-dimensional curl-conforming singular elements for FEM solutions of dielectric waveguiding structures." IEEE Transactions on Microwave Theory and Techniques 53, no. 3 (March 2005): 984–92. http://dx.doi.org/10.1109/tmtt.2004.842477.

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19

Zhang, Y. F., J. H. Yue, M. Li, and R. P. Niu. "Contact Analysis of Functionally Graded Materials Using Smoothed Finite Element Methods." International Journal of Computational Methods 17, no. 05 (June 14, 2019): 1940012. http://dx.doi.org/10.1142/s0219876219400127.

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In the paper, the smoothed finite element method (S-FEM) based on linear triangular elements is used to solve 2D solid contact problems for functionally graded materials. Both conforming and nonconforming contacts algorithms are developed using modified Coulomb friction contact models including tangential strength and normal adhesion. Based on the smoothed Galerkin weak form, the system stiffness matrices are created using the formulation procedures of node-based S-FEM (NS-FEM) and edge-based S-FEM (ES-FEM), and the contact interface equations are discretized by contact point-pairs. Then these discretized system equations are converted into a form of linear complementarity problems (LCPs), which can be further solved efficiently using the Lemke method. The singular value decomposition method is used to deal with the singularity of the stiffness matrices in the procedure constructing the standard LCP, which can greatly improve the stability and accuracy of the numerical results. Numerical examples are presented to investigate the effects of the various parameters of functionally graded materials and comparisons have been made with reference solutions and the standard FEM. The numerical results demonstrate that the strain energy solutions of ES-FEM have higher convergence rate and accuracy compared with that of NS-FEM and FEM for functionally graded materials through the present contact analysis approach.

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20

ZHANG, G. Y., G. R. LIU, T. T. NGUYEN, C. X. SONG, X. HAN, Z. H. ZHONG, and G. Y. LI. "THE UPPER BOUND PROPERTY FOR SOLID MECHANICS OF THE LINEARLY CONFORMING RADIAL POINT INTERPOLATION METHOD (LC-RPIM)." International Journal of Computational Methods 04, no. 03 (September 2007): 521–41. http://dx.doi.org/10.1142/s0219876207001308.

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It has been proven by the authors that both the upper and lower bounds in energy norm of the exact solution to elasticity problems can now be obtained by using the fully compatible finite element method (FEM) and linearly conforming point interpolation method (LC-PIM). This paper examines the upper bound property of the linearly conforming radial point interpolation method (LC-RPIM), where the Radial Basis Functions (RBFs) are used to construct shape functions and node-based smoothed strains are used to formulate the discrete system equations. It is found that the LC-RPIM also provides the upper bound of the exact solution in energy norm to elasticity problems, and it is much sharper than that of LC-PIM due to the decrease of stiffening effect. An effective procedure is also proposed to determine both upper and lower bounds for the exact solution without knowing it in advance: using the LC-RPIM to compute the upper bound, using the standard fully compatible FEM to compute the lower bound based on the same mesh for the problem domain. Numerical examples of 1D, 2D and 3D problems are presented to demonstrate these important properties of LC-RPIM.

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21

Hu, Jun, and Mira Schedensack. "Two low-order nonconforming finite element methods for the Stokes flow in three dimensions." IMA Journal of Numerical Analysis 39, no. 3 (April 19, 2018): 1447–70. http://dx.doi.org/10.1093/imanum/dry021.

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Abstract In this paper, we propose two low-order nonconforming finite element methods (FEMs) for the three-dimensional Stokes flow that generalize the nonconforming FEM of Kouhia & Stenberg (1995, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Eng, 124, 195–212). The finite element spaces proposed in this paper consist of two globally continuous components (one piecewise affine and one enriched component) and one component that is continuous at the midpoints of interior faces. We prove that the discrete Korn inequality and a discrete inf–sup condition hold uniformly in the mesh size and also for a nonempty Neumann boundary. Based on these two results, we show the well-posedness of the discrete problem. Two counterexamples prove that there is no direct generalization of the Kouhia–Stenberg FEM to three space dimensions: the finite element space with one nonconforming and two conforming piecewise affine components does not satisfy a discrete inf–sup condition with piecewise constant pressure approximations, while finite element functions with two nonconforming and one conforming component do not satisfy a discrete Korn inequality.

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22

Wang, Jianye, and Rui Ma. "Unified a Priori Error Estimate and a Posteriori Error Estimate of CIP-FEM for Elliptic Equations." Advances in Applied Mathematics and Mechanics 8, no. 4 (May 27, 2016): 517–35. http://dx.doi.org/10.4208/aamm.2014.m834.

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AbstractThis paper is devoted to a unified a priori and a posteriori error analysis of CIP-FEM (continuous interior penalty finite element method) for second-order elliptic problems. Compared with the classic a priori error analysis in literature, our technique can easily apply for any type regularity assumption on the exact solution, especially for the case of lowerH1+sweak regularity under consideration, where 0 ≤s≤ 1/2. Because of the penalty term used in the CIP-FEM, Galerkin orthogonality is lost and Céa Lemma for conforming finite element methods can not be applied immediately when 0≤s≤1/2. To overcome this difficulty, our main idea is introducing an auxiliaryC1finite element space in the analysis of the penalty term. The same tool is also utilized in the explicit a posteriori error analysis of CIP-FEM.

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23

LUBE, GERT, and GERD RAPIN. "RESIDUAL-BASED STABILIZED HIGHER-ORDER FEM FOR A GENERALIZED OSEEN PROBLEM." Mathematical Models and Methods in Applied Sciences 16, no. 07 (July 2006): 949–66. http://dx.doi.org/10.1142/s0218202506001418.

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In many numerical schemes for standard turbulence models for the nonstationary, incompressible Navier–Stokes equations, the problem is split into linearized auxiliary problems of advection-diffusion-reaction and of Oseen type. Here we present the numerical analysis of a conforming hp-version for stabilized Galerkin methods of SUPG/PSPG-type of the latter problem whereas the analysis of the former problem is reviewed in Ref. 22. We prove a modified inf–sup condition with a constant, which is independent of the spectral order and the viscosity. Moreover, the analysis of the stabilization parameters highlights the role of grad-div stabilization, in particular in case of div-stable velocity-pressure approximation.

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24

Roos, Hans-Görg, Despo Savvidou, and Christos Xenophontos. "On the Finite Element Approximation of Fourth-Order Singularly Perturbed Eigenvalue Problems." Computational Methods in Applied Mathematics 22, no. 2 (January 23, 2022): 465–76. http://dx.doi.org/10.1515/cmam-2021-0130.

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Abstract We consider fourth-order singularly perturbed eigenvalue problems in one-dimension and the approximation of their solution by the h version of the Finite Element Method (FEM). In particular, we use a C 1 {C^{1}} -conforming FEM with piecewise polynomials of degree p ≥ 3 {p\geq 3} defined on an exponentially graded mesh. We show that the method converges uniformly, with respect to the singular perturbation parameter, at the optimal rate when the error in the eigenvalues is measured in absolute value and the error in the eigenvectors is measured in the energy norm. We also illustrate our theoretical findings through numerical computations for the case p = 3 {p=3} .

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25

Qu, Xin, Lijun Su, Zhijun Liu, Xingqian Xu, Fangfang Diao, and Wei Li. "Bending of Nonconforming Thin Plates Based on the Mixed-Order Manifold Method with Background Cells for Integration." Advances in Materials Science and Engineering 2020 (December 12, 2020): 1–14. http://dx.doi.org/10.1155/2020/6681214.

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As it is very difficult to construct conforming plate elements and the solutions achieved with conforming elements yield inferior accuracy to those achieved with nonconforming elements on many occasions, nonconforming elements, especially Adini’s element (ACM element), are often recommended for practical usage. However, the convergence, good numerical accuracy, and high computing efficiency of ACM element with irregular physical boundaries cannot be achieved using either the finite element method (FEM) or the numerical manifold method (NMM). The mixed-order NMM with background cells for integration was developed to analyze the bending of nonconforming thin plates with irregular physical boundaries. Regular meshes were selected to improve the convergence performance; background cells were used to improve the integration accuracy without increasing the degrees of freedom, retaining the efficiency as well; the mixed-order local displacement function was taken to improve the interpolation accuracy. With the penalized formulation fitted to the NMM for Kirchhoff’s thin plate bending, a new scheme was proposed to deal with irregular domain boundaries. Based on the present computational framework, comparisons with other studies were performed by taking several typical examples. The results indicated that the solutions achieved with the proposed NMM rapidly converged to the analytical solutions and their accuracy was vastly superior to that achieved with the FEM and the traditional NMM.

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26

Carstensen, Carsten, and Stefan A. Funken. "Averaging technique for FE – a posteriori error control in elasticity. Part I: Conforming FEM." Computer Methods in Applied Mechanics and Engineering 190, no. 18-19 (January 2001): 2483–98. http://dx.doi.org/10.1016/s0045-7825(00)00248-6.

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27

LIU, G. R., Y. LI, K. Y. DAI, M. T. LUAN, and W. XUE. "A LINEARLY CONFORMING RADIAL POINT INTERPOLATION METHOD FOR SOLID MECHANICS PROBLEMS." International Journal of Computational Methods 03, no. 04 (December 2006): 401–28. http://dx.doi.org/10.1142/s0219876206001132.

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A linearly conforming radial point interpolation method (LC-RPIM) is presented for stress analysis of two-dimensional solids. In the LC-RPIM method, each field node is enclosed by a Voronoi polygon, and the displacement field function is approximated using RPIM shape functions of Kronecker delta function property created by simple interpolation using local nodes and radial basis functions augmented with linear polynomials to guarantee linear consistency. The system equations are then derived using the Galerkin weak form and nodal integration techniques, and the essential boundary conditions are imposed directly as in the finite element method. The LC-RPIM method is verified via various numerical examples and an extensive comparison study is conducted with the conventional RPIM, analytical approach and FEM. It is found that the presented LC-RPIM is more stable, more accurate in stress and more efficient than the conventional RPIM.

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28

Marques, Eva S. V., António B. Pereira, and Francisco J. G. Silva. "Quality Assessment of Laser Welding Dual Phase Steels." Metals 12, no. 8 (July 26, 2022): 1253. http://dx.doi.org/10.3390/met12081253.

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Since non-conforming parts create waste for industry, generating undesirable costs, it is necessary to set up quality plans that not only guarantee product conformity but also cut the root causes of welding defects by developing the concept of quality at origin. Due to their increasing use in automotive industry, dual phase (DP) steels have been the chosen material for this study. A quality plan for welding DP steel components by laser was developed. This plan is divided into three parts: pre-welding, during and post-welding. A quality assessment regarding mechanical properties, such as hardness, microstructure and tensile strength, was also performed. It was revealed that DP steel does not present considerable weldability problems, except for the usual softening of the heat affected zone (HAZ) and the growth of martensite in the fusion zone (FZ), and the best analysis techniques to avoid failures in these steels are finite element method (FEM), visual techniques during welding procedure and digital image correlation (DIC) for post-weld analysis.

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29

Lafontaine, D., E. A. Spence, and J. Wunsch. "A sharp relative-error bound for the Helmholtz h-FEM at high frequency." Numerische Mathematik 150, no. 1 (November 27, 2021): 137–78. http://dx.doi.org/10.1007/s00211-021-01253-0.

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AbstractFor the h-finite-element method (h-FEM) applied to the Helmholtz equation, the question of how quickly the meshwidth h must decrease with the frequency k to maintain accuracy as k increases has been studied since the mid 80’s. Nevertheless, there still do not exist in the literature any k-explicit bounds on the relative error of the FEM solution (the measure of the FEM error most often used in practical applications), apart from in one dimension. The main result of this paper is the sharp result that, for the lowest fixed-order conforming FEM (with polynomial degree, p, equal to one), the condition “$$h^2 k^3$$ h 2 k 3 sufficiently small" is sufficient for the relative error of the FEM solution in 2 or 3 dimensions to be controllably small (independent of k) for scattering of a plane wave by a nontrapping obstacle and/or a nontrapping inhomogeneous medium. We also prove relative-error bounds on the FEM solution for arbitrary fixed-order methods applied to scattering by a nontrapping obstacle, but these bounds are not sharp for $$p\ge 2$$ p ≥ 2 . A key ingredient in our proofs is a result describing the oscillatory behaviour of the solution of the plane-wave scattering problem, which we prove using semiclassical defect measures.

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30

Curto-Cárdenas, David, Jose Calaf-Chica, Pedro Miguel Bravo Díez, Mónica Preciado Calzada, and Maria-Jose Garcia-Tarrago. "Cold Expansion Process with Multiple Balls—Numerical Simulation and Comparison with Single Ball and Tapered Mandrels." Materials 13, no. 23 (December 4, 2020): 5536. http://dx.doi.org/10.3390/ma13235536.

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Cold expansion technology is an extended method used in aeronautics to increase fatigue life of holes and hence extending inspection intervals. During the cold expansion process, a mechanical mandrel is forced to pass along the hole generating compressive residual hoop stresses. The most widely accepted geometry for this mandrel is the tapered one and simpler options like balls have generally been rejected based on the non-conforming residual hoop stresses derived from their use. In this investigation a novelty process using multiple balls with incremental interference, instead of a single one, was simulated. Experimental tests were performed to validate the finite element method (FEM) models and residual hoop stresses from multiple balls simulation were compared with one ball and tapered mandrel simulations. Results showed that the use of three incremental balls significantly reduced the magnitude of non-conforming residual hoop stresses and the extension of these detrimental zone.

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31

Dong, Zhaonan, Emmanuil H. Georgoulis, and Tristan Pryer. "Recovered finite element methods on polygonal and polyhedral meshes." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 4 (June 18, 2020): 1309–37. http://dx.doi.org/10.1051/m2an/2019047.

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Recovered Finite Element Methods (R-FEM) have been recently introduced in Georgoulis and Pryer [Comput. Methods Appl. Mech. Eng. 332 (2018) 303–324]. for meshes consisting of simplicial and/or box-type elements. Here, utilising the flexibility of the R-FEM framework, we extend their definition to polygonal and polyhedral meshes in two and three spatial dimensions, respectively. An attractive feature of this framework is its ability to produce arbitrary order polynomial conforming discretizations, yet involving only as many degrees of freedom as discontinuous Galerkin methods over general polygonal/polyhedral meshes with potentially many faces per element. A priori error bounds are shown for general linear, possibly degenerate, second order advection-diffusion-reaction boundary value problems. A series of numerical experiments highlight the good practical performance of the proposed numerical framework.

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32

Köster, M., and S. Turek. "The Influence of Higher Order FEM Discretisations on Multigrid Convergence." Computational Methods in Applied Mathematics 6, no. 2 (2006): 221–32. http://dx.doi.org/10.2478/cmam-2006-0011.

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AbstractQuadratic and even higher order finite elements are interesting candidates for the numerical solution of partial differential equations (PDEs) due to their improved approximation properties in comparison to linear approaches. The systems of equations that arise from the discretisation of the underlying (elliptic) PDEs are often solved by iterative solvers like preconditioned Krylow-space methods, while multigrid solvers are still rarely used – which might be caused by the high effort that is associated with the realisation of the necessary data structures as well as smoothing and intergrid transfer operators. In this note, we discuss the numerical analysis of quadratic conforming finite elements in a multigrid solver. Using the “correct” grid transfer operators in conjunction with a quadratic finite element approximation allows to formulate an improved approximation property which enhances the (asymptotic) behaviour of multigrid: If m denotes the number of smoothing steps, the convergence rates behave asymptotically like O(1/m2) in contrast to O(1/m) for linear FEM.

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33

Carstensen, Carsten, and Sören Bartels. "Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM." Mathematics of Computation 71, no. 239 (February 4, 2002): 945–69. http://dx.doi.org/10.1090/s0025-5718-02-01402-3.

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34

Blaheta, R., S. Margenov, and M. Neytcheva. "Uniform estimate of the constant in the strengthened CBS inequality for anisotropic non-conforming FEM systems." Numerical Linear Algebra with Applications 11, no. 4 (April 19, 2004): 309–26. http://dx.doi.org/10.1002/nla.350.

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35

LIU, G. R., G. Y. ZHANG, K. Y. DAI, Y. Y. WANG, Z. H. ZHONG, G. Y. LI, and X. HAN. "A LINEARLY CONFORMING POINT INTERPOLATION METHOD (LC-PIM) FOR 2D SOLID MECHANICS PROBLEMS." International Journal of Computational Methods 02, no. 04 (December 2005): 645–65. http://dx.doi.org/10.1142/s0219876205000661.

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A linearly conforming point interpolation method (LC-PIM) is developed for 2D solid problems. In this method, shape functions are generated using the polynomial basis functions and a scheme for the selection of local supporting nodes based on background cells is suggested, which can always ensure the moment matrix is invertible as long as there are no coincide nodes. Galerkin weak form is adopted for creating discretized system equations, and a nodal integration scheme with strain smoothing operation is used to perform the numerical integration. The present LC-PIM can guarantee linear exactness and monotonic convergence for the numerical results. Numerical examples are used to examine the present method in terms of accuracy, convergence, and efficiency. Compared with the finite element method (FEM) using linear triangle elements and the radial point interpolation method (RPIM) using Gauss integration, the LC-PIM can achieve higher convergence rate and better efficiency.

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36

Beirão da Veiga, L., F. Brezzi, L. D. Marini, and A. Russo. "Polynomial preserving virtual elements with curved edges." Mathematical Models and Methods in Applied Sciences 30, no. 08 (July 2020): 1555–90. http://dx.doi.org/10.1142/s0218202520500311.

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In this paper, we tackle the problem of constructing conforming Virtual Element spaces on polygons with curved edges. Unlike previous VEM approaches for curvilinear elements, the present construction ensures that the local VEM spaces contain all the polynomials of a given degree, thus providing the full satisfaction of the patch test. Moreover, unlike standard isoparametric FEM, this approach allows to deal with curved edges at an intermediate scale, between the small scale (treatable by homogenization) and the bigger one (where a finer mesh would make the curve flatter and flatter). The proposed method is supported by theoretical analysis and numerical tests.

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37

Vanselow, Reiner. "About Delaunay Triangulations and Discrete Maximum Principles for the Linear Conforming FEM Applied to the Poisson Equation." Applications of Mathematics 46, no. 1 (February 2001): 13–28. http://dx.doi.org/10.1023/a:1013775420323.

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38

Moezzibadi, Mohammad, Isabelle Charpentier, Adrien Wanko, and Robert Mosé. "Sensitivity of groundwater flow with respect to the drain–aquifer leakage coefficient." Journal of Hydroinformatics 20, no. 1 (October 23, 2017): 177–90. http://dx.doi.org/10.2166/hydro.2017.026.

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Abstract Mitigation measures may be used to prevent soil and water pollution from waste disposal, landfill sites, septic or chemical storage tanks. Among them, drains and impervious barriers may be set up. The efficiency of this technique can be evaluated by means of groundwater modeling tools. The groundwater flow and the leakage drain–aquifer interactions are implemented in a conforming finite element method (FEM) and a mixed hybrid FEM (MHFEM) in a horizontal two-dimensional domain modeling regional aquifer below chemical storage tanks. Considering the influence of uncertainties in the drain–aquifer exchange rate parameter and using an automatic differentiation (AD) tool, the aim of this paper is to carry out a sensitivity analysis with respect to the leakage coefficient for the piezometric head, velocity field, and streamlines to provide a new insight into groundwater waterbody exchanges. Computations are performed with both an ideal homogeneous hydraulic conductivity and a realistic heterogeneous one. The tangent linear codes are validated using Taylor tests performed on the head and the velocity field. The streamlines computed using AD are well approximated in comparison with the nondifferentiated codes. Piezometric head computed by the MHFEM is the more sensitive, particularly near to the drain, than the FEM one.

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39

Führer, Thomas, Norbert Heuer, Michael Karkulik, and Rodolfo Rodríguez. "Combining the DPG Method with Finite Elements." Computational Methods in Applied Mathematics 18, no. 4 (October 1, 2018): 639–52. http://dx.doi.org/10.1515/cmam-2017-0041.

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AbstractWe propose and analyze a discretization scheme that combines the discontinuous Petrov–Galerkin and finite element methods. The underlying model problem is of general diffusion-advection-reaction type on bounded domains, with decomposition into two sub-domains. We propose a heterogeneous variational formulation that is of the ultra-weak (Petrov–Galerkin) form with broken test space in one part, and of Bubnov–Galerkin form in the other. A standard discretization with conforming approximation spaces and appropriate test spaces (optimal test functions for the ultra-weak part and standard test functions for the Bubnov–Galerkin part) gives rise to a coupled DPG-FEM scheme. We prove its well-posedness and quasi-optimal convergence. Numerical results confirm expected convergence orders.

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40

Santini, E., D. Vigilante, and F. dell'Isola. "Purely electrical damping of vibrations in arbitrary PEM plates: a mixed non-conforming FEM-Runge-Kutta time evolution analysis." Archive of Applied Mechanics (Ingenieur Archiv) 73, no. 1-2 (August 1, 2003): 26–48. http://dx.doi.org/10.1007/s00419-002-0251-8.

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41

Wang, Weilong, Jilian Wu, and Xinlong Feng. "A novel characteristic variational multiscale FEM for incompressible natural convection problem with variable density." International Journal of Numerical Methods for Heat & Fluid Flow 29, no. 2 (February 4, 2019): 580–601. http://dx.doi.org/10.1108/hff-06-2018-0265.

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Purpose The purpose of this paper is to propose a new method to solve the incompressible natural convection problem with variable density. The main novel ideas of this work are to overcome the stability issue due to the nonlinear inertial term and the hyperbolic term for conventional finite element methods and to deal with high Rayleigh number for the natural convection problem. Design/methodology/approach The paper introduces a novel characteristic variational multiscale (C-VMS) finite element method which combines advantages of both the characteristic and variational multiscale methods within a variational framework for solving the incompressible natural convection problem with variable density. The authors chose the conforming finite element pair (P2, P2, P1, P2) to approximate the density, velocity, pressure and temperature field. Findings The paper gives the stability analysis of the C-VMS method. Extensive two-dimensional/three-dimensional numerical tests demonstrated that the C-VMS method not only can deal with the incompressible natural convection problem with variable density but also with high Rayleigh number very well. Originality/value Extensive 2D/3D numerical tests demonstrated that the C-VMS method not only can deal with the incompressible natural convection problem with variable density but also with high Rayleigh number very well.

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42

Song, Yi Jie, Chi On Ho, and Zi Fei Qing. "A Study of New Deployable Structure." Advanced Materials Research 1049-1050 (October 2014): 1083–89. http://dx.doi.org/10.4028/www.scientific.net/amr.1049-1050.1083.

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Deployable structures are new prefabricated frames that can be transformed from a closed stage or compact configuration to a predetermined, stable expanded form. The structure is very convenient for transportation and recycling because it can be stretched out, drawn back and disassembled into pieces easily. This paper describes a new deployable structure composed of scissor composite members, each of which consists of universal scissor components, connected by bolts, and braced by pre-tensioned ropes out-of-plane, conforming a stable system. An aluminum-alloy deployable model was fabricated and a test program was carried out under vertical load to evaluate the capacity of the structure. Numerical analysis using FEM was conducted for validation purpose. By studying the stability and capacity of the structure, comprehensive evaluations of the structure were made. Possessing several advantages stated in this paper, deployable structures can be used as semi-permanent and temporary large spatial buildings.

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43

Radcliffe, A. J. "FEM-BEM coupling for the exterior Stokes problem with non-conforming finite elements and an application to small droplet deformation dynamics." International Journal for Numerical Methods in Fluids 68, no. 4 (March 7, 2011): 522–36. http://dx.doi.org/10.1002/fld.2518.

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44

Guo, Hong Wei, Hong Zheng, and Wei Li. "Implement of Ameliorated ACM Element in Numerical Manifold Space for Tackling Kirchhoff Plate Bending Problems." Applied Mechanics and Materials 638-640 (September 2014): 1710–15. http://dx.doi.org/10.4028/www.scientific.net/amm.638-640.1710.

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Ab ridging the chasm between the prevalent ly employed continuum methods (e.g. FEM) and discontinuum methods (e.g. DDA) ,the numerical manifold (NNM) ,which utilizes two covers, namely the mathematical cover and physical cover , has evinced various advantages in solving solid mechanic al issues. The forth-order partial elliptic differential equation governing Kirchhoff plate bending makes it arduous to establish the -regular Lagrangian partition of unity ,nevertheless, this study renders a modified conforming ACM manifold element , irrespective of accreting its cover degrees, to resolve the fourth-order problems. In tandem with the forming of the finite element cover system that erected on r ectangular mesh es , a succession of n umerical manifold formulas are derived on grounds of the minimum potential energy principle and the displacement boundary conditions are executed by penalty function methods. The numerical example elucidates that , compared with the orthodox ACM element , the proposed methods bespeak the accuracy and precipitating convergence of the NMM .

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45

Jiang, Ying, Minghui Nian, and Qinghui Zhang. "A Stable Generalized Finite Element Method Coupled with Deep Neural Network for Interface Problems with Discontinuities." Axioms 11, no. 8 (August 5, 2022): 384. http://dx.doi.org/10.3390/axioms11080384.

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The stable generalized finite element method (SGFEM) is an improved version of generalized or extended FEM (GFEM/XFEM), which (i) uses simple and unfitted meshes, (ii) reaches optimal convergence orders, and (iii) is stable and robust in the sense that conditioning is of the same order as that of FEM and does not get bad as interfaces approach boundaries of elements. This paper designs the SGFEM for the discontinuous interface problem (DIP) by coupling a deep neural network (DNN). The main idea is to construct a function using the DNN, which captures the discontinuous interface condition, and transform the DIP to an (approximately) equivalent continuous interface problem (CIP) based on the DNN function such that the SGFEM for CIPs can be applied. The SGFEM for the DIP is a conforming method that maintains the features (i)–(iii) of SGFEM and is free from penalty terms. The approximation error of the proposed SGFEM is analyzed mathematically, which is split into an error of SGFEM of the CIP and a learning error of the DNN. The learning dimension of DNN is one dimension less than that of the domain and can be implemented efficiently. It is known that the DNN enjoys advantages in nonlinear approximations and high-dimensional problems. Therefore, the proposed SGFEM coupled with the DNN has great potential in the high-dimensional interface problem with interfaces of complex geometries. Numerical experiments verify the efficiency and optimal convergence of the proposed method.

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46

Xiao, Yingxiong, and Zhenyou Li. "Preconditioned Conjugate Gradient Methods for the Refined FEM Discretizations of Nearly Incompressible Elasticity Problems in Three Dimensions." International Journal of Computational Methods 17, no. 03 (November 20, 2019): 1850136. http://dx.doi.org/10.1142/s0219876218501360.

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Nearly incompressible problems in three dimensions are the important problems in practical engineering computation. The volume-locking phenomenon will appear when the commonly used finite elements such as linear elements are applied to the solution of these problems. There are many efficient approaches to overcome this locking phenomenon, one of which is the higher-order conforming finite element method. However, we often use the lower-order nonconforming elements as Wilson elements by considering the computational complexity for three-dimensional (3D) problems considered. In general, the convergence of Wilson elements will heavily rely on the quality of the meshes. It will greatly deteriorate or no longer converge when the mesh distortion is very large. In this paper, the refined element method based on Wilson element is first applied to solve nearly incompressible elasticity problems, and the influence of mesh quality on the refined element is tested numerically. Its validity is verified by some numerical examples. By using the internal condensation method, the refined element discrete system of equations is deduced into the one which is spectrally equivalent to an 8-node hexahedral element discrete system of equations. And then, a type of efficient algebraic multigrid (AMG) preconditioner is presented by combining both the coarsening techniques based on the distance matrix and the effective smoothing operators. The resulting preconditioned conjugate gradient (PCG) method is efficient for 3D nearly incompressible problems. The numerical results verify the efficiency and robustness of the proposed method.

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47

Li, Zhe, Guillaume Oger, and David Le Touzé. "A partitioned framework for coupling LBM and FEM through an implicit IBM allowing non-conforming time-steps: Application to fluid-structure interaction in biomechanics." Journal of Computational Physics 449 (January 2022): 110786. http://dx.doi.org/10.1016/j.jcp.2021.110786.

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48

Shang, Yan, Song Cen, Zheng-Hua Qian, and Chenfeng Li. "High-performance unsymmetric 3-node triangular membrane element with drilling DOFs can correctly undertake in-plane moments." Engineering Computations 35, no. 7 (October 1, 2018): 2543–56. http://dx.doi.org/10.1108/ec-04-2018-0200.

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PurposeThis paper aims to propose a simple but robust three-node triangular membrane element with rational drilling DOFs for efficiently analyzing plane problems.Design/methodology/approachThis new element is developed within the general framework of unsymmetric FEM. The element test functions are determined by using a conforming displacement field which is slightly different with the classical Allman’s interpolations, while a self-equilibrated stress field formulated based on the analytical airy stress solutions is adopted as the trial functions. To ensure the correctness between the drilling DOFs and the true rotations in elasticity, reasonable constraints are introduced through the penalty function method. Moreover, the special quadrature strategy is used for operating related integrations for future enrichment of element behavior.FindingsNumerical benchmark tests reveal that this new triangular membrane element has exceptional prediction capabilities. In particular, this element can correctly reproduce a rigid body rotation motion and correctly undertake the external in-plane twisting moments; thus, it is a reasonable choice for being used to formulate flat shell elements or to be connected with other kind of elements with physical rotational DOFs.Originality/valueThis work provides a new approach for developing high-performance lower-order elements with simple formulations and good numerical accuracies.

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49

LIU, G. R., and G. Y. ZHANG. "A NOVEL SCHEME OF STRAIN-CONSTRUCTED POINT INTERPOLATION METHOD FOR STATIC AND DYNAMIC MECHANICS PROBLEMS." International Journal of Applied Mechanics 01, no. 01 (March 2009): 233–58. http://dx.doi.org/10.1142/s1758825109000083.

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This paper presents a new scheme of strain-constructed point interpolation method (SC-PIM) for static, free and forced vibration analysis of solids and structures using triangular cells. In the present scheme, displacement fields are assumed using shape functions created via the point interpolation method (PIM), which possess the Kronecker delta property facilitating the straightforward enforcement of displacement boundary conditions. Using the generalized gradient smoothing technique, the "smoothed" strains at the middle points of the cells edges are first obtained using the corresponding edge-based smoothing domains and the assumed displacement field. In each triangular background cell, the strains at the vertices are assigned using these smoothed strains in a proper manner, and then piecewisely linear strain fields are constructed by the linear interpolation for each sub-triangular cell using the edge-based "smoothed" strains. With the assumed displacements and constructed linear strain fields, the discretized system equations are created using the Strain Constructed Galerkin (SC-Galerkin) weak form. A number of benchmark numerical examples, including the standard patch test, static, free and forced vibration problems, have been studied and intensive numerical results have demonstrated that the present method possesses the following properties: (1) it works well with the simplest triangular mesh, no additional degrees of freedom and parameters are introduced and very easy to implement; (2) it is at least linearly conforming; (3) it possesses a close-to-exact stiffness: it is much stiffer than the "overly-soft" node-based smoothed point interpolation method (NS-PIM) and much softer than the "overly-stiff" FEM model; (4) the results of the present method are of superconvergence and ultra-accuracy: about one order of magnitude more accurate than those of the linear FEM; (5) there are no spurious non-zeros energy modes found and it is also temporally stable, hence the present method works well for dynamic problems.

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50

Devaud, Denis. "Petrov–Galerkin space-time hp-approximation of parabolic equations in H1/2." IMA Journal of Numerical Analysis 40, no. 4 (October 16, 2019): 2717–45. http://dx.doi.org/10.1093/imanum/drz036.

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Abstract We analyse a class of variational space-time discretizations for a broad class of initial boundary value problems for linear, parabolic evolution equations. The space-time variational formulation is based on fractional Sobolev spaces of order $1/2$ and the Riemann–Liouville derivative of order $1/2$ with respect to the temporal variable. It accommodates general, conforming space discretizations and naturally accommodates discretization of infinite horizon evolution problems. We prove an inf-sup condition for $hp$-time semidiscretizations with an explicit expression of stable test functions given in terms of Hilbert transforms of the corresponding trial functions; inf-sup constants are independent of temporal order and the time-step sequences, allowing quasi-optimal, high-order discretizations on graded time-step sequences, and also $hp$-time discretizations. For solutions exhibiting Gevrey regularity in time and taking values in certain weighted Bochner spaces, we establish novel exponential convergence estimates in terms of $N_t$, the number of (elliptic) spatial problems to be solved. The space-time variational setting allows general space discretizations and, in particular, for spatial $hp$-FEM discretizations. We report numerical tests of the method for model problems in one space dimension with typical singular solutions in the spatial and temporal variable. $hp$-discretizations in both spatial and temporal variables are used without any loss of stability, resulting in overall exponential convergence of the space-time discretization.

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