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Mark Ainsworth - One of the best experts on this subject based on the ideXlab platform.
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High-order composite finite element exact sequences based on tetrahedral-hexahedral-prismatic-pyramidal partitions
2019Co-Authors: Philippe R.b. Devloo, Omar Duran, Sonia Gomes, Mark AinsworthAbstract:The combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and Quadrilateral Faces. This paper presents high order exact sequences of finite element approximations in H^1 (Ω), H(curl, Ω), H(div, Ω), and L^2(Ω) based on such kind of three dimensional mesh configurations. The approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra. The traces associated with triangular Faces of these tetrahedral elements are constrained to match the Quadrilateral shape functions on the Quadrilateral Face of the pyramid, in order to maintain conformity with shared neighboring hexahedron, or prism. Two classes of composite exact sequences are constructed, one using classic Nédélec spaces of first kind, and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces. Projection-based interpolants satisfying the commuting diagram property are presented in a general form for each type of element. The interpolants are expressed as the sum of linearly independent contributions associated with vertices, edges, Faces, and volume, according to the kind of traces appropriate to the space under consideration. Furthermore, we study applications to the mixed formulation of Darcy's problems based on compatible pairs of approximations in {H(div, Ω), L^2 (Ω)} for such tetrahedral-hexahedral-prismatic-pyramidal meshes. An error analysis is outlined, showing same (optimal) orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions. Enhanced accuracy for potential and flux divergence variables are obtained when enriched space configurations are applied. The predicted convergence orders are verified for some test problems.
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High-order composite finite element exact sequences based on tetrahedral─hexahedral─prismatic─pyramidal partitions
Computer Methods in Applied Mechanics and Engineering, 2019Co-Authors: Philippe R.b. Devloo, Omar Duran, Sônia M. Gomes, Mark AinsworthAbstract:Abstract The combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and Quadrilateral Faces. This paper presents high order exact sequences of finite element approximations in H 1 ( Ω ) , H ( curl , Ω ) , H ( div , Ω ) , and L 2 ( Ω ) based on such kind of three dimensional mesh configurations. The approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra. The traces associated with triangular Faces of these tetrahedral elements are constrained to match the Quadrilateral shape functions on the Quadrilateral Face of the pyramid, in order to maintain conformity with shared neighboring hexahedron, or prism. Two classes of composite exact sequences are constructed, one using classic Nedelec spaces of first kind, and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces. Projection-based interpolants satisfying the commuting diagram property are presented in a general form for each type of element. The interpolants are expressed as the sum of linearly independent contributions associated with vertices, edges, Faces, and volume, according to the kind of traces appropriate to the space under consideration. Furthermore, we study applications to the mixed formulation of Darcy’s problems based on compatible pairs of approximations in { H ( div , Ω ) , L 2 ( Ω ) } for such tetrahedral–hexahedral–prismatic–pyramidal meshes. An error analysis is outlined, showing same (optimal) orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions. Enhanced accuracy for potential and flux divergence variables is obtained when enriched space configurations are applied. The predicted convergence orders are verified for some test problems.
R. Bruce King - One of the best experts on this subject based on the ideXlab platform.
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Group 9 metallatelluraboranes: Comparison with their sulfur analogues
Journal of Organometallic Chemistry, 2018Co-Authors: Amr A. A. Attia, Alexandru Lupan, R. Bruce KingAbstract:Abstract The lowest energy structures for the Group 9 metallachalcaboranes CpMEB n -2 H n -2 (M = Co, Rh, Ir; E = S, Te) are found by density functional theory to have a central closo MEB n –2 deltahedron consistent with their 2 n + 2 skeletal electrons. Structures having metal atoms at degree 5 vertices and sulfur atoms at degree 4 vertices are clearly energetically preferred. However, for the 9- and 10-vertex metallatelluraboranes CpMTeB n –2 H n –2 ( n = 9, 10; M = Co, Rh, Ir) otherwise related structures having tellurium atoms at degree 4 and degree 5 vertices are energetically comparable. The low-energy 11-vertex CpMEB 9 H 9 structures are all based on the 11-vertex closo deltahedron with a unique degree 6 vertex. However, in many of the low-energy 11-vertex structures one of the edges connecting the degree 6 vertex with an adjacent degree 5 vertex is stretched to a non-bonding distance leading to an isonido structure having a Quadrilateral Face. The three lowest energy structures for the 12-vertex CpMEB 10 H 10 (M = Co, Rh, Ir; E = S, Te) systems are the three possible icosahedral structures with the energy ordering ortho meta para .
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Hypoelectronicity and Chirality in Dimetallaboranes of Group 9 Metals.
Inorganic chemistry, 2016Co-Authors: Szabolcs Jákó, Alexandru Lupan, Attila-zsolt Kun, R. Bruce KingAbstract:The structures and energetics of the dimetallaboranes Cp2M2Bn–2Hn–2 (n = 8, 9, 10, 11, 12; M = Co, Rh, Ir; Cp = η5-C5H5) were studied using density functional theory. The lowest energy Cp2M2B6H6 and Cp2M2B7H7 structures are chiral C2 structures based on the corresponding closo deltahedra, namely the bisdisphenoid and the tricapped trigonal prism. The permethylated iridaboranes Cp*2Ir2B6H6 and Cp*2Ir2B7H7 (Cp* = η5-Me5C5) were synthesized by Ghosh and co-workers. However, they were found by X-ray crystallography to have nondeltahedral structures containing a Quadrilateral Face, namely a bicapped trigonal prism and a capped square antiprism for the 8- and 9-vertex systems, respectively. These structures correspond to a mean of the two opposite enantiomers and can also represent the “square” intermediate in the interconversion of the two enantiomers. The lowest energy structures for the 10-vertex Cp2M2B8H8 systems are two isocloso deltahedra with one metal atom at a degree 6 vertex and the other metal atom a...
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Metal–metal interactions in deltahedral dirhoda- and diiridadicarbaboranes
Inorganica Chimica Acta, 2013Co-Authors: Alexandru Lupan, R. Bruce KingAbstract:Abstract The lowest energy Cp 2 M 2 C 2 B n −4 H n −2 (M = Rh, Ir; n = 9, 10, and 12) structures are found by density functional theory to be the most spherical deltahedra having the metal atoms at degree 5 vertices and the carbon atoms preferably at degree 4 vertices. Structures with direct surFace M–M bonds appear to be energetically more favorable than for the corresponding cobalt derivatives. For the 11-vertex systems Cp 2 M 2 C 2 B 7 H 9 one of the three lowest energy structures is based on the 11-vertex most spherical deltahedron with one of the metal atoms at the unique degree 6 vertex. The 11-vertex polyhedron in one of the other low-energy Cp 2 M 2 C 2 B 7 H 9 structures has a Quadrilateral Face thereby avoiding the presence of a degree 6 vertex. The Wiberg bond indices (WBIs) for the M–M interactions along deltahedral edges range from 0.36 to 0.40 and can be considered to be formal single bonds. The WBIs for non-adjacent M⋯M interactions are much lower at 0.07–0.12 suggesting weak metal–metal interactions through the center of the deltahedron.
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Supraicosahedral polyhedra in carboranes and metallacarboranes: The role of local vertex environments in determining polyhedral topology and the anomaly of 13-vertex closo polyhedra☆
Journal of Organometallic Chemistry, 2007Co-Authors: R. Bruce KingAbstract:Metal-free carboranes having 13 vertices are anomalous since their closo polyhedra having the expected 28 skeletal electrons are not the usual deltahedra with exclusively triangular Faces but instead polyhedra with one or two trapezoidal Faces obtained by removal of one or more edges from the corresponding 13-vertex deltahedron. Removal of such edges converts degree 6 boron vertices in the 13-vertex deltahedron into more favorable degree 5 boron vertices while lowering the degree of nearby carbon vertices. Thus the anomaly of the 13-vertex carborane closo polyhedron can be rationalized by the preference of boron for degree 5 vertices. The 12-vertex tetracarbon carborane (CH3)4C4B8H8 with a nido electron count of 28 skeletal electrons but with two Quadrilateral Faces has a solid state structure derived from a 13-vertex “closo” polyhedron with one Quadrilateral Face by removal of a degree 4 vertex to give the second Quadrilateral Face. However, the corresponding tetraethyl derivative (C2H5)4C4B8H8 has a different solid state structure derived from removal of a degree 6 vertex from an unusual 13-vertex deltahedron with three degree 6 vertices to give an open hexagonal Face rather than two Quadrilateral Faces. In contrast to the 13-vertex closo polyhedra, the 14-vertex closo polyhedron is a true deltahedron, namely the D6d bicapped hexagonal antiprism, which is found in a carborane derivative as well as in several dimetallacarboranes with the metal atoms always at the degree 6 vertices. However, the 15-vertex closo polyhedron, so far found only in the metallaborane 1,2-μ-(CH2)3C2B12H12Ru(η6-p-cymene), is a non-deltahedron with one Quadrilateral Face.
Philippe R.b. Devloo - One of the best experts on this subject based on the ideXlab platform.
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High-order composite finite element exact sequences based on tetrahedral-hexahedral-prismatic-pyramidal partitions
2019Co-Authors: Philippe R.b. Devloo, Omar Duran, Sonia Gomes, Mark AinsworthAbstract:The combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and Quadrilateral Faces. This paper presents high order exact sequences of finite element approximations in H^1 (Ω), H(curl, Ω), H(div, Ω), and L^2(Ω) based on such kind of three dimensional mesh configurations. The approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra. The traces associated with triangular Faces of these tetrahedral elements are constrained to match the Quadrilateral shape functions on the Quadrilateral Face of the pyramid, in order to maintain conformity with shared neighboring hexahedron, or prism. Two classes of composite exact sequences are constructed, one using classic Nédélec spaces of first kind, and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces. Projection-based interpolants satisfying the commuting diagram property are presented in a general form for each type of element. The interpolants are expressed as the sum of linearly independent contributions associated with vertices, edges, Faces, and volume, according to the kind of traces appropriate to the space under consideration. Furthermore, we study applications to the mixed formulation of Darcy's problems based on compatible pairs of approximations in {H(div, Ω), L^2 (Ω)} for such tetrahedral-hexahedral-prismatic-pyramidal meshes. An error analysis is outlined, showing same (optimal) orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions. Enhanced accuracy for potential and flux divergence variables are obtained when enriched space configurations are applied. The predicted convergence orders are verified for some test problems.
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High-order composite finite element exact sequences based on tetrahedral─hexahedral─prismatic─pyramidal partitions
Computer Methods in Applied Mechanics and Engineering, 2019Co-Authors: Philippe R.b. Devloo, Omar Duran, Sônia M. Gomes, Mark AinsworthAbstract:Abstract The combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and Quadrilateral Faces. This paper presents high order exact sequences of finite element approximations in H 1 ( Ω ) , H ( curl , Ω ) , H ( div , Ω ) , and L 2 ( Ω ) based on such kind of three dimensional mesh configurations. The approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra. The traces associated with triangular Faces of these tetrahedral elements are constrained to match the Quadrilateral shape functions on the Quadrilateral Face of the pyramid, in order to maintain conformity with shared neighboring hexahedron, or prism. Two classes of composite exact sequences are constructed, one using classic Nedelec spaces of first kind, and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces. Projection-based interpolants satisfying the commuting diagram property are presented in a general form for each type of element. The interpolants are expressed as the sum of linearly independent contributions associated with vertices, edges, Faces, and volume, according to the kind of traces appropriate to the space under consideration. Furthermore, we study applications to the mixed formulation of Darcy’s problems based on compatible pairs of approximations in { H ( div , Ω ) , L 2 ( Ω ) } for such tetrahedral–hexahedral–prismatic–pyramidal meshes. An error analysis is outlined, showing same (optimal) orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions. Enhanced accuracy for potential and flux divergence variables is obtained when enriched space configurations are applied. The predicted convergence orders are verified for some test problems.
Omar Duran - One of the best experts on this subject based on the ideXlab platform.
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High-order composite finite element exact sequences based on tetrahedral-hexahedral-prismatic-pyramidal partitions
2019Co-Authors: Philippe R.b. Devloo, Omar Duran, Sonia Gomes, Mark AinsworthAbstract:The combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and Quadrilateral Faces. This paper presents high order exact sequences of finite element approximations in H^1 (Ω), H(curl, Ω), H(div, Ω), and L^2(Ω) based on such kind of three dimensional mesh configurations. The approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra. The traces associated with triangular Faces of these tetrahedral elements are constrained to match the Quadrilateral shape functions on the Quadrilateral Face of the pyramid, in order to maintain conformity with shared neighboring hexahedron, or prism. Two classes of composite exact sequences are constructed, one using classic Nédélec spaces of first kind, and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces. Projection-based interpolants satisfying the commuting diagram property are presented in a general form for each type of element. The interpolants are expressed as the sum of linearly independent contributions associated with vertices, edges, Faces, and volume, according to the kind of traces appropriate to the space under consideration. Furthermore, we study applications to the mixed formulation of Darcy's problems based on compatible pairs of approximations in {H(div, Ω), L^2 (Ω)} for such tetrahedral-hexahedral-prismatic-pyramidal meshes. An error analysis is outlined, showing same (optimal) orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions. Enhanced accuracy for potential and flux divergence variables are obtained when enriched space configurations are applied. The predicted convergence orders are verified for some test problems.
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High-order composite finite element exact sequences based on tetrahedral─hexahedral─prismatic─pyramidal partitions
Computer Methods in Applied Mechanics and Engineering, 2019Co-Authors: Philippe R.b. Devloo, Omar Duran, Sônia M. Gomes, Mark AinsworthAbstract:Abstract The combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and Quadrilateral Faces. This paper presents high order exact sequences of finite element approximations in H 1 ( Ω ) , H ( curl , Ω ) , H ( div , Ω ) , and L 2 ( Ω ) based on such kind of three dimensional mesh configurations. The approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra. The traces associated with triangular Faces of these tetrahedral elements are constrained to match the Quadrilateral shape functions on the Quadrilateral Face of the pyramid, in order to maintain conformity with shared neighboring hexahedron, or prism. Two classes of composite exact sequences are constructed, one using classic Nedelec spaces of first kind, and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces. Projection-based interpolants satisfying the commuting diagram property are presented in a general form for each type of element. The interpolants are expressed as the sum of linearly independent contributions associated with vertices, edges, Faces, and volume, according to the kind of traces appropriate to the space under consideration. Furthermore, we study applications to the mixed formulation of Darcy’s problems based on compatible pairs of approximations in { H ( div , Ω ) , L 2 ( Ω ) } for such tetrahedral–hexahedral–prismatic–pyramidal meshes. An error analysis is outlined, showing same (optimal) orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions. Enhanced accuracy for potential and flux divergence variables is obtained when enriched space configurations are applied. The predicted convergence orders are verified for some test problems.
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Effects of mesh deformation on the accuracy of mixed finite element approximations for 3D Darcy's flows
2018Co-Authors: Philippe Devloo, Omar Duran, Agnaldo Farias, Sonia GomesAbstract:H(div)-conforming finite element approximation spaces are usually formed by locally backtracking vector polynomial spaces defined on the master element by the Piola transformation. The main focus of the present paper is to study the effect of using non-affine elements on the accuracy of three dimensional flux approximations based on such spaces. For instance , instead of order k + 1 for flux and flux divergence obtained with Raviart-Thomas or Nédélec spaces with normal fluxes of degree k, based on affine hexahedra or triangular prisms, reduced orders k for flux and k − 1 for flux divergence may occur for distorted elements. To improve this scenario, a hierarchy of enriched flux approximations is considered, with increasing orders of divergence accuracy, holding for general space stable configurations. The original vector polynomial space is required to be expressed by a decomposition in terms of edge and internal shape functions. The enriched versions are defined by adding internal shape functions of the original family of spaces up to higher degree level k + n, n > 0, while keeping fixed the original border fluxes of degree k. This procedure gives approximations with the same original flux accuracy, but with enhanced divergence order k + n + 1, in the affine case, or k + n − 1 for elements mapped by general multi-linear mappings. The loss of convergence in the flux variable due to Quadrilateral Face distortions cannot be corrected by including higher order internal functions. Application of these enriched flux spaces to the mixed finite element formulation of a Darcy's model problem is discussed. The computational cost of matrix assembly increases with n, but the global condensed systems to be solved have same dimension and structure of the original scheme.
Tsutomu Mizuta - One of the best experts on this subject based on the ideXlab platform.
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An X-ray crystallographic study on the molecular structures of seven-coordinate (ethylenediamine-N,N,N′-triacetato-N′-acetic acid) (aqua) -titanium(III) and -vanadium(III), [TiIII(H-edta)(H2O)]·H2O and [VIII(Hedta)(H2O)]·H2O
Inorganica Chimica Acta, 1995Co-Authors: Katsuhiko Miyoshi, Jun Wang, Tsutomu MizutaAbstract:Abstract The molecular structures of the title complexes,[Ti III (Hedta)(H 2 O)]·H 2 O ( I ) and [V III (Hedta)(H 2 O)]·H2O( II ) (Heda 3− = mono-protonated ethylenediamine- N,N,N′,N′ -tetraacetate) have been determined by single-crystal X-ray analyses. The crystal data are as follows: I : monoclinic, Aa, a = 13.401(1), b = 12.311(1), c = 8.552(1) A , β = 97.35(1)°, V = 1399.3(1) A 3 , Z = 4, R = 0.018 and R w =0.022; II : monoclinic, Aa, a = 13.891(1), b = 8.558(1), c = 12.135(1) A , β = 95.77(1)°, V = 1435.4(1) A 3 , Z = 4, R = 0.020 and R w = 0.028. The former complex has a seven-coordinate and approximately pentagonal-bipyramidal structure in which Hedta 3− acts as a hexadentate ligand, a proton is attached to the carbonyl oxygen atom on one of the equatorial glycine rings (G-rings), and a water molecule occupies one of the five basal coordination sites. The latter is also sevencoordinate but has a structure close to a mono-capped trigonal-prism in which Hedta 3− is also hexadentate and a water molecule caps a Quadrilateral Face as a seventh ligand. A structural comparison of these and other Hedta complexes with the corresponding edta complexes revealed that Hedta 3− serves well as a hexadentate ligand, for those metal ions which have a propensity to form seven-coordinate edta complexes, and that protonation takes place in most edta complexes on the carboxylate group of the more constrained equatorial glycine arm (G-ring).
