Finite Element Approximations
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In Chapter 5 we used some important results from functional analysis to prove some convergence theorems for finite element methods. Careful readers of that chapter will notice at least two very important unresolved issues. First, we did not prove that the approximation assumption in (5.58) is satisfied for any finite dimensional set of functions. To resolve this issue in this chapter, we will discuss the approximation assumption in Section 6.3. Second, in finite element methods, we approximate the weak solution of partial differential equations by mapped shape functions. For unit shapes other than triangles or tetrahedra and for nonlinear coordinate mappings, the mapped shape functions are generally not polynomials, even though the shape functions are polynomials before applying the coordinate maps. We will analyze the error in finite element approximations employing coordinate transformations in Sections 6.4.2 and 6.4.3.Keywords:
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This paper improves tetrahedral linear finite element with nodal integration points for transient dynamics analysis. Although tetrahedral finite element is effective for practical auto mesh generation, the simplest tetrahedral linear finite element brings numerical locking. In addition to this, tetrahedral high order finite elements are unsuitable for transient dynamics analysis because of inconsistent high-frequency vibrations in finite elements. To avoid numerical locking of tetrahedron linear finite element, the present study approximates constitutive variables on nodes instead of general integration points and extends to conventional explicit dynamic method for transient dynamics analysis. We test the present approach in representative computational examples comparing to results by standard tetrahedron linear finite element and reasonable hexahedron linear finite element with numerical selective reduced integration.
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Abstract The volume of a tetrahedron is represented in terms of the twelve face angles, inradii of the faces of tetrahedron, circumradii of the faces and the radius of the sphere circumscribing the tetrahedron
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1. In a tetrahedron ABCD with its opposite edges perpendicular there are two tetrahedra which can be described as pedal tetrahedra. (1) the tetrahedron A 0 B 0 C 0 D 0 where these points are the feet of the perpendiculars from ABCD on to the faces BCD .
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We systematically investigate properties of various triangle centers (such as orthocenter or incenter) located on the four faces of a tetrahedron. For each of six types of tetrahedra, we examine over 100 centers located on the four faces of the tetrahedron. Using a computer, we determine when any of 16 conditions occur (such as the four centers being coplanar). A typical result is: The lines from each vertex of a circumscriptible tetrahedron to the Gergonne points of the opposite face are concurrent.
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A fully encapsulated Pt4 tetrahedron in an incomplete tetrahedron of 36 nickel atoms is present in [Ni36 Pt4 (CO)45 ]6- (1; see picture for the metal framework), which is obtained as an inseparable mixture with [Ni37 Pt4 (CO)46 ]6- (2) by reaction of [Ni6 (CO)12 ]2- with K2 [PtCl4 ]. The trimethylbenzylammonium salts of 1 and 2 cocrystallize in a 1:1 ratio. The additional Ni atom of 2 caps the truncated vertex of 1.
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The common question of why the tetrahedral angle is 109.471° can be answered using a tetrahedron-in-a-cube, along with some Year 10 level mathematics. The tetrahedron-in-a-cube can also be used to demonstrate the non-polarity of tetrahedral molecules, the relationship between different types of lattice structures, and to demonstrate that inductive reasoning does not always provide the correct answer.
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A uniform refinement strategy for a tetrahedron is presented. Most finite element theories are based on the assumption that the tetrahedral elements in the refinement do not degenerate. In this paper, the author presents a refinement strategy that is nondegenerate and uniform for the model tetrahedra considered and quasi uniform for arbitrary tetrahedra. It can be used to construct nested, multilevel triangulations. At level j of refinement, an arbitrary nondegenerate tetrahedron in the initial triangulation is partitioned into $2^{3j} $ tetrahedra of equal volume. This refinement strategy can be implemented easily by partitioning block elements instead of the more complicated tetrahedral elements. This feature makes the use of tetrahedral elements attractive in a computer code.
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In this paper we have tried to study the geometry of tetrahedron as a member of polyhedron family and hence calculate the face angles on each face of a tetrahedron with help of “C” Programming. We will also be looking into the applications of tetrahedron structure in chemistry in the structure of various molecules, in aviation it is used as a Wind Sock which serves as a reference to pilots indicating the direction of the wind as well as Geology where the “Tetrahedral Hypothesis” was used to explain the formation of earth.
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Simulations of spherical tetrahedron shaped body for ball mill are made. The impact of spherical tetrahedron shaped body against a tip of fixed spherical tetrahedron was modeled via explicit Finite Element Method. Distributions of Strain and Stress and Force to time diagrams are obtained for three cases of impact. 1. Impact of spherical tetrahedron spherical surface against a tip of fixed spherical tetrahedron. 2. Impact of spherical tetrahedron tip against a tip of fixed spherical tetrahedron. 3. Impact of spherical tetrahedron edge against a tip of fixed spherical tetrahedron. Conclusions about the workability of the spherical tetrahedron shaped body are made. PACS: 45.50.Tn, 83.50.-v, 89.20.Bb
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Spherical Geometry
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A regular tetrahedron has six spatial angles. The sum of these spatial angles for a tetrahedron with all four groups the same (CA4) is 6 × 109.4712206°. To the nearest degree that value is 657°. For a tetrahedron with one group different (CA3B), there are two different bond angles. The sum of these two angles can be approximated to be 219°, one third of 657°. For a tetrahedron with disubstitution by the same group (CA2B2), the approximate sum involves three different angles: ∠ACA + ∠BCB + 4 ∠ACB = 657°. For a tetrahedron with disubstitution by two different groups (CA2BD), the approximate sum uses four different angles: ∠ACA + ∠BCD + 2 ∠ACB + 2 ∠ACD = 657°. For a tetrahedron with all groups different (CABDE), the approximate sum comes from six angles. Examples of each type are given along with the limitations of the method.
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