Journal articles: 'Element matriciel'

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Suárez P, Santiago A. "Estudio de elemento estructural con apoyo deslizante." Ciencia y Sociedad 31, no. 4 (December 1, 2006): 520–32. http://dx.doi.org/10.22206/cys.2006.v31i4.pp520-532.

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Con el advenimiento de las computadoras digitales y la masificación de su uso fue posible viabilizar el análisis matricial de las estructuras. Se han desarrollado matrices de rigidez para elementos con diferentes tipos de restricciones. En este trabajo se desarrollará la matriz de rigidez de un elemento con apoyo deslizante en uno de sus nodos y se utilizarán estos resultados para analizar una estructura que tiene una barra apoyada de esta manera. Se realizará una comprobación de los resultados, analizando la estructura usando un método clásico no matricial y se compararán los resultados. Al final se llega a conclusiones que conducirán a una definición más exacta del término rigidez.

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Armstrong, John T. "Problem Elements and Spectrometry Problems in X-Ray Microanalysis: the Black Holes of the Periodic Table." Microscopy and Microanalysis 4, S2 (July 1998): 216–17. http://dx.doi.org/10.1017/s1431927600021206.

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Few quantitative analysis techniques attempt as large an extrapolation between the compositions of standards and samples than is attempted in electron microbeam x-ray emission analysis. In-situ x-ray microanalysis can be performed for essentially all elements in the periodic table in complex matricies that may contain, in extreme cases, thirty or more detectable elements. Analyses are attempted for the same elements, using the same standards, in various matrices whose average atomic numbers might range from 4 to 94. Unlike most analytical techniques, where suites of standards are synthesized having similar bulk compositions as the samples and bracketing the concentrations of the elements of interest, the standards employed in microbeam analysis are most commonly pure elements, simple oxides, or other binary element compounds. This is true even though matrix effects on electron retardation and scattering, x-ray absorption, and secondary x-ray fluorescence can cause major variations in the differences between relative intensity and relative concentration.

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Zingoni, Alphose. "Truss and Beam Finite Elements Revisited: A Derivation Based on Displacement-Field Decomposition." International Journal of Space Structures 11, no. 4 (December 1996): 371–80. http://dx.doi.org/10.1177/026635119601100404.

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Where a finite element possesses symmetry properties, derivation of fundamental element matrices can be achieved more efficiently by decomposing the general displacement field into subspaces of the symmetry group describing the configuration of the element. In this paper, the procedure is illustrated by reference to the simple truss and beam elements, whose well-known consistent-mass matrices are obtained via the proposed method. However, the procedure is applicable to all one-, two- and three-dimensional finite elements, as long as the shape and node configuration of the element can be described by a specific symmetry group.

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Pan, Wen Jun, and Zhi Wu Wei. "Reckoning for Element Characteristics Matrix of Space Timoshenko-Beam Based on Energy Variational Principle." Applied Mechanics and Materials 66-68 (July 2011): 1356–61. http://dx.doi.org/10.4028/www.scientific.net/amm.66-68.1356.

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To analyze and calculate the element characteristics matrices of space Timoshenko-beam, research work were carried out on the basis of energy variational principle. Displacement function for the space Timoshenko-beam were put forward, the expressions for element mass matrix, stiffness matrix and load array were deduced by energy functional extremum, and the explicit forms of element mass and stiffness matrices were integrated finally. Results show that the element mass and stiffness matrices computed by this method are consistent with those in related references. It has a good theoretical and practical value in the calculation for characteristics matrices of other elements.

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KIDGER, D. J., and I. M. SMITH. "EIGENVALUES OF ELEMENT STIFFNESS MATRICES. PART I: 2‐D PLANE ELEMENTS." Engineering Computations 9, no. 3 (March 1992): 307–16. http://dx.doi.org/10.1108/eb023868.

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KIDGER, D. J., and I. M. SMITH. "EIGENVALUES OF ELEMENT STIFFNESS MATRICES. PART II: 3‐D SOLID ELEMENTS." Engineering Computations 9, no. 3 (March 1992): 317–28. http://dx.doi.org/10.1108/eb023869.

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Fazio, P., K. Gowri, and K. H. Ha. "Rectangular hybrid elements for the analysis of sandwich plate structures." Canadian Journal of Civil Engineering 14, no. 4 (August 1, 1987): 455–60. http://dx.doi.org/10.1139/l87-069.

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The structural behaviour of sandwich plate structures are characterized by transverse shear deformations in the core. The assumed stress hybrid finite element technique is particularly suitable for developing sandwich plate bending elements. In the present study, rectangular three-layer sandwich plate elements have been formulated using simple assumed stress functions. Numerical test problems have been solved to examine the convergence property and suitability of these elements. The results are compared with that of a complete quadratic stress mode element and with analytical solutions. Six degrees of freedom per node shell elements are formulated by combining the plate bending elements with membrane elements. A folded plate sandwich panel roof has been analyzed using these elements and the results are compared with the experimental values. The use of simple stress function gives satisfactory results and reduces the size of the matrices to be used, the length of the program, and the computation time for the formulation of element stiffness matrices. Key words: sandwich panel, structural analysis, finite element method, stress hybrid approach, folded plates.

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Zhou, Ling Yuan, and Qiao Li. "Analysis of Reinforced Concrete Column Using a Beam-Column Element with a Meshed Section." Advanced Materials Research 255-260 (May 2011): 1954–58. http://dx.doi.org/10.4028/www.scientific.net/amr.255-260.1954.

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A efficient 3D reinforced-concrete beam element based on the flexibility method and distributed nonlinearity theory is proposed, The sections of the beam element are divided into the plane isoparametric elements in this formulation, the section stiffness matrices are calculated through the integration of stress-strain relations of concrete including reinforcing steel effect in the section. The flexibility matrices of the sections are calculated by inverting the stiffness matrices, and the element flexibility matrix is formed through the force interpolation functions. The element stiffness matrix is evaluated through the element flexibility matrix. Finally, the buckling behaviors of a reinforced concrete beam under various eccentric loads are analyzed with the proposed formulation to illustrate its accuracy and computational efficiency.

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Farenick, D. R. "C*-Convexity and Matricial Ranges." Canadian Journal of Mathematics 44, no. 2 (April 1, 1992): 280–97. http://dx.doi.org/10.4153/cjm-1992-019-1.

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AbstractC* -convex sets in matrix algebras are convex sets of matrices in which matrix-valued convex coefficients are admitted along with the usual scalar-valued convex coefficients. A Carathéodory-type theorem is developed for C*-convex hulls of compact sets of matrices, and applications of this theorem are given to the theory of matricial ranges. If T is an element in a unital C*-algebra , then for every n ∈ N, the n x n matricial range Wn(T) of T is a compact C* -convex set of n x n matrices. The basic relation W1(T) = conv σ-(T) is well known to hold if T exhibits the normal-like quality of having the spectral radius of β T + μ 1 coincide with the norm ||β T + μ 1|| for every pair of complex numbers β and μ. An extension of this relation to the matrix spaces is given by Theorem 2.6: Wn (T) is the C*-convex hull of the n x n matricial spectrum σn(T) of T if, for every B,M ∈ ℳn, the norm of T ⊗ B + 1 ⊗ M in ⊗ ℳn is the maximum value in {||∧⊗B + 1 ⊗M|| : Λ ∈ σn (T)}. The spatial matricial range of a Hilbert space operator is the analogue of the classical numerical range, although it can fail to be convex if n > 1. It is shown in § 3 that if T has a normal dilation N with σ (N) ⊂ σ (T), then the closure of the spatial matricial range of T is convex if and only if it is C*-convex.

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Leu, L. J., and C. W. Huang. "Stiffness Matrices for Linear and Buckling Analyses of Composite Beams with Partial Shear Connection." Journal of Mechanics 22, no. 4 (December 2006): 299–309. http://dx.doi.org/10.1017/s1727719100000952.

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AbstractThis paper is concerned with linear and buckling analyses of composite beams with partial shear connection (partial composite beams) using the finite element method. Two elements derived from different types of shape functions are proposed in this study. The first element, referred to as exact, is based on the exact shape functions obtained by solving the differential equations governing the transverse displacement and the slip of the shear connector layer of a partial composite beam. The second element, referred to as approximate, is based on the conventional linear and cubic shape functions used in conventional axial and beam elements. By making use of these two types of shape functions, the elastic and geometric stiffness matrices can be derived explicitly from the strain energy and the load potential, respectively. Both types of elements can be used to carry out linear static and buckling analyses. As expected, the exact element is more accurate than the approximate element if the same discretization is adopted. However, the approximate element has the advantage of easy implementation since the expressions of its elastic and geometric stiffness matrices are very simple. Also, the solutions obtained from the approximate element converge very fast; with reasonable discretization, say 8 elements per member, very accurate solutions can be obtained.

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XING, YUFENG, BO LIU, and GUANG LIU. "A DIFFERENTIAL QUADRATURE FINITE ELEMENT METHOD." International Journal of Applied Mechanics 02, no. 01 (March 2010): 207–27. http://dx.doi.org/10.1142/s1758825110000470.

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This paper studies the differential quadrature finite element method (DQFEM) systematically, as a combination of differential quadrature method (DQM) and standard finite element method (FEM), and formulates one- to three-dimensional (1-D to 3-D) element matrices of DQFEM. It is shown that the mass matrices of C 0 finite element in DQFEM are diagonal, which can reduce the computational cost for dynamic problems. The Lagrange polynomials are used as the trial functions for both C 0 and C 1 differential quadrature finite elements (DQFE) with regular and/or irregular shapes, this unifies the selection of trial functions of FEM. The DQFE matrices are simply computed by algebraic operations of the given weighting coefficient matrices of the differential quadrature (DQ) rules and Gauss-Lobatto quadrature rules, which greatly simplifies the constructions of higher order finite elements. The inter-element compatibility requirements for problems with C 1 continuity are implemented through modifying the nodal parameters using DQ rules. The reformulated DQ rules for curvilinear quadrilateral domain and its implementation are also presented due to the requirements of application. Numerical comparison studies of 2-D and 3-D static and dynamic problems demonstrate the high accuracy and rapid convergence of the DQFEM.

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Abdurrazzaq, Achmad, Ari Wardayani, and Suroto Suroto. "RING MATRIKS ATAS RING KOMUTATIF." Jurnal Ilmiah Matematika dan Pendidikan Matematika 7, no. 1 (June 26, 2015): 11. http://dx.doi.org/10.20884/1.jmp.2015.7.1.2895.

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This paper discusses a matrices over a commutative ring. A matrices over commutative rings is a matrices whose entries are the elements of the commutative ring. We investigates the structure of the set of the matrices over the commutative ring. We obtain that the set of the matrices over the commutative ring equipped with an addition and a multiplication operation of matrices is a ring with a unit element.

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Gilewski, W., and M. Sitek. "The Inf-Sup Condition Tests for Shell/Plate Finite Elements / Testy Warunku Inf-Sup Do Oceny Płytowych I Powłokowych Elementów Skonczonych." Archives of Civil Engineering 57, no. 4 (December 1, 2011): 425–47. http://dx.doi.org/10.2478/v.10169-011-0030-4.

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Abstract Development of high-performance finite elements for thick, moderately thick, as well as thin shells and plates, was one of the active areas of the finite element technology for 40 years, followed by hundreds of publications. A variety of shell elements exist in the FE codes, but “the best” finite element is still to be discovered. The paper deals with an evaluation of some existing shell finite elements, from the point of view of the third of three requirements to be satisfied by the element: ellipticity, consistency and inf-sup condition. It is difficult to prove the inf-sup condition analytically, so, a numerical verification is proposed. A set of numerical tests is considered for shell and plate problems. Two norm matrices and a selection of the stiffness matrices (bending, shear and membrane dominated) are analysed. Finite elements from various computer systems can be evaluated and compared with the use of the proposed tests.

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Ortigoza Capetillo, Gerardo Mario, Alberto Pedro Lorandi Medina, and Alfonso Cuauhtemoc García Reynoso. "Reordering edges and elements in unstructured meshes to reduce execution time in Finite Element Computations." Nova Scientia 10, no. 20 (May 25, 2018): 263–79. http://dx.doi.org/10.21640/ns.v10i20.1317.

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Reverse Cuthill McKee (RCM) reordering can be applied to either edges or elements of unstructured meshes (triangular/tetrahedral) , in accordance to the respective finite element formulation, to reduce the bandwidth of stiffness matrices . Grid generators are mainly designed for nodal based finite elements. Their output is a list of nodes (2d or 3d) and an array describing element connectivity, be it triangles or tetrahedra. However, for edge-defined finite element formulations a numbering of the edges is required. Observations are reported for Triangle/Tetgen Delaunay grid generators and for the sparse structure of the assembled matrices in both edge- and element-defined formulations. The RCM is a renumbering algorithm traditionally applied to the nodal graph of the mesh. Thus, in order to apply this renumbering to either the edges or the elements of the respective finite element formulation, graphs of the mesh were generated. Significant bandwidth reduction was obtained. This translates to reduction in the execution effort of the sparse-matrix-times-vector product. Compressed Sparse Row format was adopted and the matrix-times-vector product was implemented in an OpenMp parallel routine.

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Gao, Yuefeng, Jianlong Chen, and Yuanyuan Ke. "*-DMP elements in *-semigroups and *-rings." Filomat 32, no. 9 (2018): 3073–85. http://dx.doi.org/10.2298/fil1809073g.

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In this paper, we investigate *-DMP elements in *-semigroups and *-rings. The notion of *-DMP element was introduced by Patr?cio and Puystjens in 2004. An element a is *-DMP if there exists a positive integer m such that am is EP. We first characterize *-DMP elements in terms of the {1,3}-inverse, Drazin inverse and pseudo core inverse, respectively. Then, we characterize the core-EP decomposition utilizing the pseudo core inverse, which extends the core-EP decomposition introduced by Wang for complex matrices to an arbitrary *-ring; and this decomposition turns to be a useful tool to characterize *-DMP elements. Further, we extend Wang?s core-EP order from complex matrices to *-rings and use it to investigate *-DMP elements. Finally, we give necessary and sufficient conditions for two elements a,b in *-rings to have aaD = bbD, which contribute to study *-DMP elements.

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Silva, P. B., A. L. Goldstein, and J. R. F. Arruda. "Building Spectral Element Dynamic Matrices Using Finite Element Models of Waveguide Slices and Elastodynamic Equations." Shock and Vibration 20, no. 3 (2013): 439–58. http://dx.doi.org/10.1155/2013/306437.

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Structural spectral elements are formulated using the analytical solution of the applicable elastodynamic equations and, therefore, mesh refinement is not needed to analyze high frequency behavior provided the elastodynamic equations used remain valid. However, for modeling complex structures, standard spectral elements require long and cumbersome analytical formulation. In this work, a method to build spectral finite elements from a finite element model of a slice of a structural waveguide (a structure with one dimension much larger than the other two) is proposed. First, the transfer matrix of the structural waveguide is obtained from the finite element model of a thin slice. Then, the wavenumbers and wave propagation modes are obtained from the transfer matrix and used to build the spectral element matrix. These spectral elements can be used to model homogeneous waveguides with constant cross section over long spans without the need of refining the finite element mesh along the waveguide. As an illustrating example, spectral elements are derived for straight uniform rods and beams and used to calculate the forced response in the longitudinal and transverse directions. Results obtained with the spectral element formulation are shown to agree well with results obtained with a finite element model of the whole beam. The proposed approach can be used to generate spectral elements of waveguides of arbitrary cross section and, potentially, of arbitrary order.

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McTavish, D. J., and P. C. Hughes. "Modeling of Linear Viscoelastic Space Structures." Journal of Vibration and Acoustics 115, no. 1 (January 1, 1993): 103–10. http://dx.doi.org/10.1115/1.2930302.

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The GHM Method provides viscoelastic finite elements derived from the commonly used elastic finite elements. Moreover, these GHM elements are used directly and conveniently in second-order structural models jut like their elastic counterparts. The forms of the GHM element matrices preserve the definiteness properties usually associated with finite element matrices—namely, the mass matrix is positive definite, the stiffness matrix is nonnegative definite, and the damping matrix is positive semi-definite. In the Laplace domain, material properties are modeled phenomenologically as a sum of second-order rational functions dubbed mini-oscillator terms. Developed originally as a tool for the analysis of damping in large flexible space structures, the GHM method is applicable to any structure which incorporates viscoelastic materials.

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Jeyakarthikeyan, P. V., R. Yogeshwaran, and Karthikk Sridharan. "An Efficient Method to Generate Element Stiffness Matrix of Quadrilateral Elements in Closed Form on Application of Vehicle Analysis." Applied Mechanics and Materials 852 (September 2016): 582–87. http://dx.doi.org/10.4028/www.scientific.net/amm.852.582.

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This paper presents about generating elemental stiffness matrix for quadrilateral elements in closed form solution method for application on vehicle analysis which is convenient and simple as long as Jacobian is matrix of constant. The interpolation function of the field variable to be found can integrate explicitly once for all, which gives the constant universal matrices A, B and C. Therefore, stiffness matrix is no longer integration of the given functional, it is simple calculation of universal matrices and local co-ordinates of the element. So time taken for generation of element stiffness can be reduced considerably compared to Gauss numerical integration method. For effective use of quadrilateral elements hybrid grid generation is recommended that contains all interior element edges are parallel to each other (rectangle or square elements) and outer boundary elements are quadrilaterals with distortion. So in the Proposed method, the closed form and Gauss numerical method is used explicitly for interior elements and outer elements respectively. The time efficiency of proposed method is compared with conventional Gauss quadrature that is used for entire domain. It is found that the proposed method is much efficient than Gauss Quadrature.

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EVERSMAN, WALTER. "A REFLECTION FREE BOUNDARY CONDITION FOR PROPAGATION IN UNIFORM FLOW USING MAPPED INFINITE WAVE ENVELOPE ELEMENTS." Journal of Computational Acoustics 08, no. 01 (March 2000): 25–41. http://dx.doi.org/10.1142/s0218396x00000030.

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Variable order mapped infinite wave envelope elements are developed for finite element modeling of acoustic radiation in a uniformly moving medium. These elements are used as a nonreflecting boundary condition for computations on an infinite domain in which a radiating body is immersed in a moving medium which is essentially undisturbed outside of the near field. The mapped elements provide a boundary condition equivalent to element stiffness, mass, and damping matrices appended to an inner standard FEM mesh. A demonstration of the performance of mapped elements as influenced by element order is given in the context of acoustic radiation from a turbofan inlet and exhaust.

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Wang, Yu, and Guangyu Shi. "Simple and Accurate Eight-Node and Six-Node Solid-Shell Elements with Explicit Element Stiffness Matrix Based on Quasi-Conforming Element Technique." International Journal of Applied Mechanics 09, no. 01 (January 2017): 1750012. http://dx.doi.org/10.1142/s1758825117500120.

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Based on the quasi-conforming (QC) element technique, accurate and reliable eight-node and six-node solid-shell elements are presented in this paper. These QC solid-shell elements can alleviate shear and Poisson thickness locking by appropriately interpolating the strain fields over the element domain, and they are completely free from hourglass modes by ensuring the rank sufficiency of the element stiffness matrix a priori. Furthermore, the element stiffness matrices of the present elements are evaluated explicitly rather than resorting to the numerical integration, which leads to a high computational efficiency. The QC solid-shell elements with the properly interpolated element strain fields can rigorously pass both membrane and bending patch tests. The popular benchmark problems are used to evaluate the performance of the QC solid-shell elements. The numerical results show that the present QC solid-shell elements yield not only accurate displacements but also good stress results for all the stress components. Particularly, the present QC solid-shell elements are capable of giving quite accurate results even with very coarse mesh.

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Saada, A., and P. Velex. "An Extended Model for the Analysis of the Dynamic Behavior of Planetary Trains." Journal of Mechanical Design 117, no. 2A (June 1, 1995): 241–47. http://dx.doi.org/10.1115/1.2826129.

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An extended model for determining critical frequencies for tooth loading on spur and helical gear planetary trains is proposed. Torsional, flexural and axial generalized displacements of all the components are considered and a finite element procedure is used for generality. In order to avoid modulations between meshing pulsations and the carrier angular velocity, equations are written relative to rotating frames fixed to planet centers. Depending on their architectures, complex drives are broken down in basic 12 degree of freedom elements, namely: external gear element (sun gear-planet element); internal gear element (ring gear-planet element); pin-carrier element; classical elements for shafts, bearings, couplings, etc. Details are given for elementary stiffness matrices. Due to contact conditions between mating teeth, these matrices are full and torsional, bending and traction effects are coupled. State equations point to parametrically excited differential systems with gyroscopic contributions. A first application of the model is conducted on a 3 planet epicyclic drive whose gyroscopic terms are neglected. Potentially dangerous frequencies for sun gear-planet and planet-ring gear contacts are determined and contributions of some components of the planetary train are discussed.

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Asubiojo, O. I., F. M. Adebiyi, E. I. Obiajunwa, and J. A. Ajao. "Elemental Studies of Soil and Food Flour for Risk Assessment of Highway Pollution Using Particle-Induced X-Ray Emission (PIXE) Spectrometry." ISRN Analytical Chemistry 2012 (March 29, 2012): 1–8. http://dx.doi.org/10.5402/2012/594598.

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The study investigated potential toxic elements in soils and food flours for highway pollution using PIXE spectrometry. The contaminated soils and cassava food flours contained higher levels of the elements than their control samples, while comparison with their standard permissible limits followed similar trend which was attributable to anthropogenic influences. These were corroborated by their elevated Enrichment factor, Pollution index and Geoaccumulation index values for the elements, suggesting significant anthropogenically—derived contaminations of the soils. T-test value (0.038) for the elemental composition of the contaminated soils & cassava flours was significant due to considerable higher concentrations of the elements in the soils than the cassava flours. Cross plot analysis result for the contaminated soils and cassava fours showed moderate positive correlation (R2 = 0.426), indicating inter-element relationship between them. Cluster analysis results for the analyzed elements in the contaminated soil samples indicated that Mn, Fe, V, Cr, Zn, Cl, Ti and S showed closest inter-element clustering and was corroborated by the results of Pearson correlation matrices, while inter-element clustering in the food flour followed the same trend and was also supported by their results of Pearson correlation matrices, validating that the soils and cassava flours were contaminated via similar sources.

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Hua, X., and C. W. S. To. "Simple and Efficient Tetrahedral Finite Elements With Rotational Degrees of Freedom for Solid Modeling." Journal of Computing and Information Science in Engineering 7, no. 4 (July 29, 2007): 382–93. http://dx.doi.org/10.1115/1.2798120.

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A mixed variational principle and derivation of two simple and efficient tetrahedral finite elements with rotational degrees of freedom (DOF) are presented. Each element has four nodes. Every node has six DOF, which include three translational and three rotational DOF. Each element is capable of providing six rigid-body modes. The rotational DOF are based on the displacement formulation, while the translational DOF are hinged on the hybrid strain Hellinger–Reissner functional. Explicit expressions for stiffness matrices are obtained. Element performance has been evaluated with benchmark problems, indicating that they have superior accuracy compared with other lower-order tetrahedral elements.

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Xin, Jianguo, Wei Cai, and Nailong Guo. "On the Construction of Well-Conditioned Hierarchical Bases for (div)-Conforming ℝn Simplicial Elements." Communications in Computational Physics 14, no. 3 (September 2013): 621–38. http://dx.doi.org/10.4208/cicp.100412.041112a.

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AbstractHierarchical bases of arbitrary order for (div)-conforming triangular and tetrahedral elements are constructed with the goal of improving the conditioning of the mass and stiffness matrices. For the basis with the triangular element, it is found numerically that the conditioning is acceptable up to the approximation of order four, and is better than a corresponding basis in the dissertation by Sabine Zaglmayr [High Order Finite Element Methods for Electromagnetic Field Computation, Johannes Kepler Universität, Linz, 2006]. The sparsity of the mass matrices from the newly constructed basis and from the one by Zaglmayr is similar for approximations up to order four. The stiffness matrix with the new basis is much sparser than that with the basis by Zaglmayr for approximations up to order four. For the tetrahedral element, it is identified numerically that the conditioning is acceptable only up to the approximation of order three. Compared with the newly constructed basis for the triangular element, the sparsity of the mass matrices from the basis for the tetrahedral element is relatively sparser.

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Rucka, M., W. Witkowski, J. Chróscielewski, and K. Wilde. "Damage Detection of A T-Shaped Panel by Wave Propagation Analysis in the Plane Stress / Wykrywanie Uszkodzen W Tarczy Typu T Z Uzyciem Analizy Propagacji Fal W Płaskim Stanie Naprezenia." Archives of Civil Engineering 58, no. 1 (March 1, 2012): 3–24. http://dx.doi.org/10.2478/v.10169-012-0001-4.

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Abstract A computational approach to analysis of wave propagation in plane stress problems is presented. The initial-boundary value problem is spatially approximated by the multi-node C0 displacement-based isoparametric quadrilateral finite elements. To integrate the element matrices the multi-node Gauss-Legendre-Lobatto quadrature rule is employed. The temporal discretization is carried out by the Newmark type algorithm reformulated to accommodate the structure of local element matrices. Numerical simulations are conducted for a T-shaped steel panel for different cases of initial excitation. For diagnostic purposes, the uniformly distributed loads subjected to an edge of the T-joint are found to be the most appropriate for design of ultrasonic devices for monitoring the structural element integrity

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Korneev, V. G. "Solving Discrete Dirichlet Problems on Spectral Finite Elements by Fast Domain Decomposition Algorithm." Applied Mechanics and Materials 587-589 (July 2014): 2312–29. http://dx.doi.org/10.4028/www.scientific.net/amm.587-589.2312.

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A key component of DD (domain decomposition) solvers for $hp$ discretizations of elliptic equations is the solver for the internal stiffness matrices of $p$-elements. We consider the algorithm of the linear complexity for solving such problems on spectral $p$-elements, which, therefore, in the leading DD solver plays the role of the second stage DD solver. It is based on the first order finite element preconditioning of the Orszag type for the reference element stiffness matrices. Earlier, for spectral elements, only fast solvers obtained with the use of special preconditioners in factored form were known. The most intricate part of the algorithm is the inter-subdomain Schur complement preconditioning by inexact iterative solver employing two preconditioners -- preconditioner-solver and preconditioner-multiplicator. From general point of view, the solver developed in the paper, is the DD solver for the discretization on a strongly variable in size and shape deteriorating mesh with the number of subdomains growing with the growth of the number of degrees of freedom.

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Soares, Delfim. "A Simple Explicit–Implicit Time-Marching Technique for Wave Propagation Analysis." International Journal of Computational Methods 16, no. 01 (November 21, 2018): 1850082. http://dx.doi.org/10.1142/s0219876218500822.

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A new explicit–implicit time integration technique is proposed here for wave propagation analysis. In the present formulation, the time integrators of the model are selected at the element level, allowing each element to be considered as explicit or implicit. In the implicit elements, controllable algorithm dissipation is provided, enabling an [Formula: see text]-stable formulation. In the explicit elements, null amplitude decay is considered, enabling maximal critical time-step values. The new methodology renders a very simple and effective time-marching algorithm. Here, just displacement–velocity relations are considered, and no computation of accelerations is required. Moreover, explicit/implicit analyses can be carried out just by the tuning of local effective matrices, inputting or not stiffness matrices into their computations. At the end of the paper, numerical results are presented, illustrating the performance and potentialities of the new method.

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Yawen, Liu, Fan Qinmin, and Li Daolun. "X-Ray Fluorescence Determination of Trace Elements in Complicated Matrices." Advances in X-ray Analysis 30 (1986): 309–14. http://dx.doi.org/10.1154/s0376030800021431.

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AbstractA method for the direct determination of trace elements in light element matrices is described. The intensity of Compton scatter of the incident X-rays from the specimen was used to evaluate the apparent absorption factor for the direct correction for matrix effects. This permits the use of single element standards for samples of varied and complicated matrix composition. Relative errors of approximately 5% were obtained for results in the concentration range of 10-200 ppm.

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Chappell, Bruce W. "Trace Element Analysis of Rocks by X-ray Spectrometry." Advances in X-ray Analysis 34 (1990): 263–76. http://dx.doi.org/10.1154/s0376030800014555.

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Undoubtedly the most important applications of X-ray fluorescence spectrometry (XRF) have been in the analysis of major elements where the technique provides a unique method of measuring the concentration of all elements having Z > 10 with extremely good precision in a wide range of matrices. However, XRF is in addition a powerful method for trace element analysis. In this discussion, the principles of the method for the trace element analysis of rocks are outlined, its capabilities are summarized, and the advantages and disadvantages of the technique are pointed out.

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Wang, Jing Ling, Zhong Yr Cai, Mine Zhe Li, and Hui Yang. "Influence of Element Number on Shape Accuracy in Multi-Point Stretch Forming of Air Craft." Applied Mechanics and Materials 130-134 (October 2011): 2240–44. http://dx.doi.org/10.4028/www.scientific.net/amm.130-134.2240.

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Multi-point stretch forming is a flexible manufacturing technique for three-dimensional shape forming of craft skin. Its die surface is constructed by many pairs of matrices of elements whose height is controlled by computer. It uses the curved surface of elements instead of the die surface. The element numberis an important parameter because it has great influence on the part quality. This paper simulates the forming process of paraboloid part and saddle-shaped part with different number of elements and studies the influence of element number on the shape accuracy of the part .That will provides guidance for the application of multi-point stretch forming.

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Shi, G., Y. Liu, and X. Wang. "Accurate, Efficient, and Robust Q4-Like Membrane Elements Formulated in Cartesian Coordinates Using the Quasi-Conforming Element Technique." Mathematical Problems in Engineering 2015 (2015): 1–14. http://dx.doi.org/10.1155/2015/198390.

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By using the quasi-conforming element technique, two four-node quadrilateral membrane elements with 2 degrees of freedom at each node (Q4-like membrane element) are formulated in rectangular Cartesian coordinates. One of the four-node quadrilateral membrane elements is based on the assumed strain field with only five independent strain parameters and accounting for the Poisson effect explicitly. There are no independent internal parameters and numerical integration involved in the evaluation of the strain parameters in these four-node quadrilateral membrane elements, and their element stiffness matrices are computed explicitly in Cartesian coordinates. Consequently, the formulation of these four-node quadrilateral membrane elements is extremely simple, and the resulting elements are very computationally efficient. These two quasi-conforming quadrilateral membrane elements pass the patch test and are free from shear locking and insensitive to the element distortion in the range of practical application. The numerical result comparison with other four-node quadrilateral membrane elements, including Q4-like plane elements with drilling degrees of freedom and the Q6-type isoparametric elements with very complicated nonconforming modes, shows that the present quasi-conforming quadrilateral membrane elements are not only reliable and robust, but also very accurate in both displacement and stress evaluations in the analysis of practical plane elasticity problems.

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Abrahamsson, T. J. S., and J. H. Sa¨llstro¨m. "A Spinning Finite Beam Element of General Orientation Analyzed With Rayleigh/Timoshenko/Saint-Venant Theory." Journal of Engineering for Gas Turbines and Power 118, no. 1 (January 1, 1996): 86–94. http://dx.doi.org/10.1115/1.2816554.

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Linear vibrations are studied for a straight uniform finite beam element of general orientation spinning at a constant angular speed about a fixed axis in the inertial space. The gyroscopic and circulatory matrices and also the geometric stiffness matrix of the beam element are presented. The effect of the centrifugal static axial load on the bending and torsional dynamic stiffnesses is thereby accounted for. The Rayleigh/Timoshenko/Saint-Venant theory is applied, and polynomial shape functions are used in the construction of the deformation fields. Nonzero off-diagonal elements in the gyroscopic and circulatory matrices indicate coupled bending/shearing/torsional/tensional free and forced modes of a generally oriented spinning beam. Two numerical examples demonstrate the use and performance of the beam element.

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Shahba, Ahmad, Reza Attarnejad, and Shahin Hajilar. "Free Vibration and Stability of Axially Functionally Graded Tapered Euler-Bernoulli Beams." Shock and Vibration 18, no. 5 (2011): 683–96. http://dx.doi.org/10.1155/2011/591716.

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Structural analysis of axially functionally graded tapered Euler-Bernoulli beams is studied using finite element method. A beam element is proposed which takes advantage of the shape functions of homogeneous uniform beam elements. The effects of varying cross-sectional dimensions and mechanical properties of the functionally graded material are included in the evaluation of structural matrices. This method could be used for beam elements with any distributions of mass density and modulus of elasticity with arbitrarily varying cross-sectional area. Assuming polynomial distributions of modulus of elasticity and mass density, the competency of the element is examined in stability analysis, free longitudinal vibration and free transverse vibration of double tapered beams with different boundary conditions and the convergence rate of the element is then investigated.

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Xiao, Xiang, and Wei-Xin Ren. "A Versatile 3D Vehicle-Track-Bridge Element for Dynamic Analysis of the Railway Bridges under Moving Train Loads." International Journal of Structural Stability and Dynamics 19, no. 04 (April 2019): 1950050. http://dx.doi.org/10.1142/s0219455419500500.

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There has been a growing interest to carry out the vehicle–track–bridge (VTB) dynamic interaction analysis using 2D or 3D finite elements based on simplified wheel–rail relationships. The simplified or elastic wheel–rail contact relationships, however, cannot consider the lateral contact forces and geometric shapes of the wheel and rails, and even the occasional jump of wheels from the rails. This does not guarantee a reliable analysis for the safety running of trains over bridges. To consider the wheel–rail constraint and contact forces, this paper proposes a versatile 3D VTB element, consisting of a vehicle, eight rail beam elements, four bridge beam elements, and continuous springs as well as the dampers between the rail and bridge girder. With the 3D VTB element matrices formulated, a procedure for assembling the interaction matrices of the 3D VTB element is presented based on the virtual work principle. The global equations of motion of the VTB interaction system are established accordingly, which can be solved by time integration methods to obtain the dynamic responses of the vehicle, track and bridge, as well as the stability and safety indices of the moving train. Finally, an illustrative example is used to verify the proposed the versatile 3D VTB element for the dynamic interactive analysis of railway bridges under moving train loads.

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BARBÉ, ANDRÉ M. "ON A CLASS OF FRACTAL MATRICES (I) EXCESS-MATRICES AND THEIR SELF-SIMILAR PROPERTIES." International Journal of Bifurcation and Chaos 02, no. 04 (December 1992): 841–60. http://dx.doi.org/10.1142/s0218127492000471.

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An infinite class of integer valued matrices of unbounded size and of arbitrary dimensions is presented here as a generalization of the so-called excess-matrix associated with a particular cellular automaton. These matrices are constructed through a recursive arithmetical procedure, and show an intricate fractal-like distribution of integers. The elements in these matrices have a number theoretical interpretation which is directly related to a proper number-base representation of the coordinates of the element. These excess-matrices exhibit numerical and symbolical self-similarities under agglomerating and decimating coarse-graining operations, and under so-called disguising transformations.

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Aidemirov, K. R., V. P. Agapov, and G. M. Murtazaliev. "Application of multilayer finite elements of variable thickness in the calculation of reinforced concrete slabs in the "Prince" computer complex." Herald of Dagestan State Technical University. Technical Sciences 47, no. 4 (January 21, 2021): 112–21. http://dx.doi.org/10.21822/2073-6185-2020-47-4-112-121.

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Objective. A family of multilayer finite elements designed for calculating reinforced concrete slabs and shells of variable thickness is described. The features of the formation of stiffness matrices associated with the variability of the cross-section of elements are considered.Methods. The family is based on the simplest planar triangular element constructed using the Kirchhoff hypothesis. The transverse displacements in this element are approximated by an incomplete cubic polynomial. This element is not suitable for practical use, but it is based on improved elements of three- and foursided shape in the plan. Special attention is paid to the consideration of cross-section variability.Results. The results of testing the developed elements are presented and the advantages of their use in the practice of design and calculation of structures are shown.Conclusion. The developed PRINCE software package can be useful in the design and calculation of structures containing plates of variable thicknesses.

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Rathod, H. T., Md Shafiqul Islam, Bharath Rathod, and K. Sugantha Devi. "Finit element solution of Poisson Equation over Polygonal Domains using a novel auto mesh generation technique and an explicit integration scheme for linear convex quadrilaterals of cubic order Serendipity and Lagrange families." International Journal Of Engineering And Computer Science 7, no. 01 (January 6, 2018): 23329–482. http://dx.doi.org/10.18535/ijecs/v7i1.01.

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This paper presents an explicit integration scheme to compute the stiffness matrix of twelve node and sixteen node linear convex quadrilateral finite elements of Serendipity and Lagrange families using an explicit integration scheme and discretisation of polygonal domain by such finite elements using a novel auto mesh generation technique, In finite element analysis, the boundary value problems governed by second order linear partial differential equations, the element stiffness matrices are expressed as integrals of the product of global derivatives over the linear convex quadrilateral region. These matrices can be shown to depend on the material properties matrices and the matrix of integrals with integrands as rational functions with polynomial numerator and the linear denominator (4+) in the bivariates over a 2-square (-1 ) with the nodes on the boundary and in the interior of this simple domain. The finite elements up to cubic order have nodes only on the boundary for Serendipity family and the finite elements with boundary as well as some interior nodes belong to the Lagrange family. The first order element is the bilinear convex quadrilateral finite element which is an exception and it belongs to both the families. We have for the present ,the cubic order finite elements which havee 12 boundary nodes at the nodal coordinates {(-1,-1),(1,-1),(1,1),(-1,1),(-1/3,-1), (1/3,-1),(1,-1/3),(1,1/3),(1/3,1),(-1/3,1),(-1,1/3),(-1,-1/3)} and the four interoior nodal coordinates at the points (-1/3,-1/3),(1/3,-1/3),(1/3,1/3),(-1/3,1/3)} in the local parametric space ( In this paper, we have computed the integrals of local derivative products with linear denominator (4+) in exact forms using the symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme can be then applied to solve boundary value problems in continuum mechanics over convex polygonal domains. We have also developed a novel auto mesh generation technique of all 12-node and 16-node linear(straight edge) convex quadrilaterals for a polygonal domain which provides the nodal coordinates and the element connectivity. We have used the explicit integration scheme and this novel auto mesh generation technique to solve the Poisson equation u ,where u is an unknown physical variable and in with Dirichlet boundary conditions over the convex polygonal domain.

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Fu, Xiang-Rong, Li-Na Ge, Ge Tian, and Ming-Wu Yuan. "Study on the Explicit Formula of the Triangular Flat Shell Element Based on the Analytical Trial Functions for Anisotropy Material." Mathematical Problems in Engineering 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/486453.

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This paper presents a novel way to formulate the triangular flat shell element. The basic analytical solutions of membrane and bending plate problem for anisotropy material are studied separately. Combining with the conforming displacement along the sides and hybrid element strategy, the triangular flat shell elements based on the analytical trial functions (ATF) for anisotropy material are formulated. By using the explicit integral formulae of the triangular element, the matrices used in proposed shell element are calculated efficiently. The benchmark examples showed the high accuracy and high efficiency.

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Fergusson, N. J., and W. D. Pilkey. "Frequency-Dependent Element Mass Matrices." Journal of Applied Mechanics 59, no. 1 (March 1, 1992): 136–39. http://dx.doi.org/10.1115/1.2899418.

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This paper considers some of the theoretical aspects of the formulation of frequency-dependent structural matrices. Two types of mass matrices are examined, the consistent mass matrix found by integrating frequency-dependent shape functions, and the mixed mass matrix found by integrating a frequency-dependent shape function against a static shape function. The coefficients in the power series expansion for the consistent mass matrix are found to be determinable from those in the expansion for the mixed mass matrix by multiplication by the appropriate constant. Both of the mass matrices are related in a similar manner to the coefficients in the frequency-dependent stiffness matrix expansion. A formulation is derived which allows one to calculate, using a shape function truncated at a given order, the mass matrix expansion truncated at twice that order. That is the terms for either of the two mass matrix expansions of order 2n are shown to be expressible using shape functions terms of order n. Finally, the terms in the matrix expansions are given by formulas which depend only on the values of the shape function terms at the boundary.

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Pałasińska, Katarzyna. "Three-element nonfinitely axiomatizable matrices." Studia Logica 53, no. 3 (September 1994): 361–72. http://dx.doi.org/10.1007/bf01057933.

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41

Bebendorf, Mario. "Approximation of boundary element matrices." Numerische Mathematik 86, no. 4 (October 2000): 565–89. http://dx.doi.org/10.1007/pl00005410.

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Tang, Y. Q., Y. P. Liu, and S. L. Chan. "Element-Independent Pure Deformational and Co-Rotational Methods for Triangular Shell Elements in Geometrically Nonlinear Analysis." International Journal of Structural Stability and Dynamics 18, no. 05 (May 2018): 1850065. http://dx.doi.org/10.1142/s0219455418500657.

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Proposed herein is a novel pure deformational method for triangular shell elements that can decrease the element quantities and simplify the element formulation. This approach has computational advantages over the conventional finite element method for linear and nonlinear problems. In the element level, this method saves time for computing stresses, internal forces and stiffness matrices. A flat shell element is formed by a membrane element and a plate element, so that the pure deformational membrane and plate elements are derived and discussed separately in this paper. Also, it is very convenient to incorporate the proposed pure deformational method into the element-independent co-rotational (EICR) framework for geometrically nonlinear analysis. Thus, on the basis of the pure deformational method, a novel EICR formulation is proposed which is simpler and has more clear physical characteristics than the traditional formulation. In addition, a triangular membrane element with drilling rotations and the discrete Kirchhoff triangular plate element are used to verify the proposed pure deformational method, although several benchmark problems are employed to verify the robustness and accuracy of the proposed EICR formulations.

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Pham, Quoc-Hoa, The-Van Tran, Tien-Dat Pham, and Duc-Huynh Phan. "An Edge-Based Smoothed MITC3 (ES-MITC3) Shell Finite Element in Laminated Composite Shell Structures Analysis." International Journal of Computational Methods 15, no. 07 (October 12, 2018): 1850060. http://dx.doi.org/10.1142/s0219876218500603.

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This paper proposes an improvement of the MITC3 shell finite element to analyze of laminated composite shell structures. In order to enhance the accuracy and convergence of MITC3 element, an edge-based smoothed finite element method (ES-FEM) is applied to the derivation of the membrane, bending and shear stiffness terms of the MITC3 element, named ES-MICT3. In the ES-FEM, the smoothed strain is calculated in the domain that constructed by two adjacent MITC3 triangular elements sharing an edge. On a curved geometry of shell models, two adjacent MITC3 triangular elements may not be placed on the same plane. In this case, the edge-based smoothed strain can be performed on the virtual plane based on strain transformation matrices between the global coordinate and this virtual coordinate. Furthermore, a simple modification coefficient is chosen to be [Formula: see text] times the maximum diagonal value of the element stiffness matrix at the zero drilling degree of freedom to avoid the drill rotation locking when all elements meeting at a node are coplanar. The numerical examples demonstrated that the proposed method achieves the high accuracy in comparison to others existing elements in the literature.

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Perumal, Logah. "A Novel Virtual Node Hexahedral Element with Exact Integration and Octree Meshing." Mathematical Problems in Engineering 2016 (2016): 1–19. http://dx.doi.org/10.1155/2016/3261391.

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The method presented in this work is a 3-dimensional polyhedral finite element (3D PFEM) based on virtual node method. Novel virtual node polyhedral elements (termed as VPHE) are developed here, particularly virtual node hexahedral element (termed as VHE). Stiffness matrices of these polyhedral elements consist of simple polynomials. Thus, a new algorithm is introduced in this paper, which enables exact integration of monomials without a need for high number of integration points and weights. The number of nodes for VHE elements is not restricted, as opposed to the conventional hexahedral elements. This feature enables formulation of transition elements (termed as T-VHE) which are useful to adaptive computation. Performances of the new VHE elements in solid mechanics and conductive heat transfer phenomena are examined through numerical simulations. The new T-VHE elements are utilized in octree mesh. The VHE elements are found to produce good results and T-VHE elements help to reduce number of global nodes for the analysis.

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Gladwell, G. M. L., and H. Ahmadian. "Generic element matrices suitable for finite element model updating." Mechanical Systems and Signal Processing 9, no. 6 (November 1995): 601–14. http://dx.doi.org/10.1006/mssp.1995.0045.

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Akin, J. E., T. Tezduyar, M. Ungor, and S. Mittal. "Stabilization Parameters and Smagorinsky Turbulence Model." Journal of Applied Mechanics 70, no. 1 (January 1, 2003): 2–9. http://dx.doi.org/10.1115/1.1526569.

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For the streamline-upwind/Petrov-Galerkin and pressure-stabilizing/Petrov-Galerkin formulations for flow problems, we present in this paper a comparative study of the stabilization parameters defined in different ways. The stabilization parameters are closely related to the local length scales (“element length”), and our comparisons include parameters defined based on the element-level matrices and vectors, some earlier definitions of element lengths, and extensions of these to higher-order elements. We also compare the numerical viscosities generated by these stabilized formulations with the eddy viscosity associated with a Smagorinsky turbulence model that is based on element length scales.

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He, Peng, Zhansheng Liu, and Chun Li. "An Improved Beam Element for Beams with Variable Axial Parameters." Shock and Vibration 20, no. 4 (2013): 601–17. http://dx.doi.org/10.1155/2013/708910.

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The traditional beam element was improved to consider the variable axial parameters. The variable axial parameters were formulated in terms of a power series, and the general forms of elementary mass and stiffness matrices which depend on the power order were derived. The mass and stiffness matrices of the improved beam element were obtained in terms of an elementary matrix series. The beam elements for various tapered beams and a beam under linearly axial temperature distribution were derived. The vibrations of the beams with various taper shapes were studied and the variations of natural frequencies and modal shapes were investigated. A uniform beam under linearly axial temperature distribution was modeled and studied. The influences of axial temperature difference on the natural frequencies and modal shapes were investigated. Results show that the improved beam element could consider the variable axial parameters of beam conveniently.

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Senjanović, Ivo, Marko Tomić, Smiljko Rudan, and Neven Hadžić. "Conforming shear-locking-free four-node rectangular finite element of moderately thick plate." Journal of the Mechanical Behavior of Materials 25, no. 5-6 (December 20, 2016): 141–52. http://dx.doi.org/10.1515/jmbm-2017-0001.

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AbstractAn outline of the modified Mindlin plate theory, which deals with bending deflection as a single variable, is presented. Shear deflection and cross-section rotation angles are functions of bending deflection. A new four-node rectangular finite element of moderately thick plate is formulated by utilizing the modified Mindlin theory. Shape functions of total (bending+shear) deflections are defined as a product of the Timshenko beam shape functions in the plate longitudinal and transversal direction. The bending and shear stiffness matrices, and translational and rotary mass matrices are specified. In this way conforming and shear-locking-free finite element is obtained. Numerical examples of plate vibration analysis, performed for various combinations of boundary conditions, show high level of accuracy and monotonic convergence of natural frequencies to analytical values. The new finite element is superior to some sophisticated finite elements incorporated in commercial software.

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Vrebos, Bruno A. R., and Gjalt T. J. Kuipéres. "Areas for Improvement in XRF Analysis of Low Atomic Number Elements." Advances in X-ray Analysis 36 (1992): 73–80. http://dx.doi.org/10.1154/s0376030800018668.

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Accurate analysis of the light elements has been, from the early applications of X-ray fluorescence spectrometry a struggle compared to the determination of heavy elements in the same matrices. In contrast, there has been virtually no upper limit to the atomic number of the element that could be determined. The lower limit, however, has been continuously adjusted downward through the years. Clearly, the sensitivity as well as the lower limit of detection for the heavy elements have also been improved, but the effect is Jess striking than the advances made in the region of tight element performance. This paper deals specifically with wavelength dispersive sequential x-ray fluorescence spectrometry, although some of the observations made are equally applicable to energy dispersive spectrometry.

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Díaz Díaz, Alberto, Rubén Castañeda Balderas, Axel Fernando Domínguez Alvarado, and Claudio Iván Martínez Morfín. "Soluciones de Ecuaciones Diferenciales por Elemento Finito (SEDEF)." Ingeniería Investigación y Tecnología 21, no. 1 (January 1, 2020): 1–11. http://dx.doi.org/10.22201/fi.25940732e.2020.21n1.002.

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En este trabajo se expone el desarrollo de la primera versión de un software amigable de elemento finito gratuito y de código abierto llamado SEDEF ©, el cual propone y resuelve un sistema de ecuaciones diferenciales generales dependientes de dos dimensiones, una espacial y una temporal. Estas ecuaciones son propuestas de tal manera que se puedan modelar diferentes fenómenos físicos. Cabe mencionar que se da solución a la parte espacial de las ecuaciones generales mediante el método del elemento finto y a la temporal por medio del método BDF (“Backward Differentiation Formula”), lo que resulta en sistemas de ecuaciones lineales con matrices dispersas. Estos sistemas son programados y resueltos en el lenguaje C++; el álgebra de matrices dispersas se soluciona a través de la librería CSparse creada por Timothy A. Davis (2006). El software está dotado de una interfaz gráfica intuitiva y fácil de manejar programada en RAD Studio XE, la cual permite tener un módulo exclusivo para problemas de transferencia de calor, un módulo para problemas de vigas de Euler en un plano y un módulo general donde el usuario puede adaptar sus propias ecuaciones constitutivas para problemas particulares unidimensionales. La validación del software se hace a través del planteamiento de tres problemas: uno matemático, otro de transferencia de calor y un último de vigas en el plano. En estos, la solución de SEDEF se compara con soluciones analíticas y los resultados del software comercial de elementos finitos COMSOL Multiphysics. El fin que persigue SEDEF es el de dotar a las instituciones educativas de una herramienta que ayude en el entendimiento de matemáticas, física y varias ramas de ingeniería gracias a la resolución de ecuaciones y la interfaz gráfica amigable para las etapas de preprocesamiento y postprocesamiento de resultados.

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Journal articles on the topic 'Element matriciel'

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