Numerical integration of some functions over an arbitrary linear tetrahedra in Euclidean three-dimensional space

Rathod, H.T.; Nagaraja, K.V.; Venkatesudu, B.

doi:10.1016/j.amc.2007.02.104

Cited by 8 publications

(4 citation statements)

References 12 publications

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“…However, as the density attributes at each vertex can be individually specified, expressing a linear density field inside the tetrahedron, the computation of mass and mass center changes. Instead, as in FEM [23,24], we utilize a simple basis case in a linear density field, for which the integration is solved analytically. A linear combination of four base cases (one per vertex) already gives the desired properties for a general tetrahedron.…”

Section: Geometry Integrationmentioning

confidence: 99%

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Tetrahedra of varying density and their applications

Bukenberger

Lensch

2021

Vis Comput

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We propose concepts to utilize basic mathematical principles for computing the exact mass properties of objects with varying densities. For objects given as 3D triangle meshes, the method is analytically accurate and at the same time faster than any established approximation method. Our concept is based on tetrahedra as underlying primitives, which allows for the object’s actual mesh surface to be incorporated in the computation. The density within a tetrahedron is allowed to vary linearly, i.e., arbitrary density fields can be approximated by specifying the density at all vertices of a tetrahedral mesh. Involved integrals are formulated in closed form and can be evaluated by simple, easily parallelized, vector-matrix multiplications. The ability to compute exact masses and centroids for objects of varying density enables novel or more exact solutions to several interesting problems: besides the accurate analysis of objects under given density fields, this includes the synthesis of parameterized density functions for the make-it-stand challenge or manufacturing of objects with controlled rotational inertia. In addition, based on the tetrahedralization of Voronoi cells we introduce a precise method to solve $$L_{2|\infty }$$ L 2 | ∞ Lloyd relaxations by exact integration of the Chebyshev norm. In the context of additive manufacturing research, objects of varying density are a prominent topic. However, current state-of-the-art algorithms are still based on voxelizations, which produce rather crude approximations of masses and mass centers of 3D objects. Many existing frameworks will benefit by replacing approximations with fast and exact calculations. Graphic abstract

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Section: Geometry Integrationmentioning

confidence: 99%

“…Surprisingly, it is neither a very complex problem nor bound to numerical tradeoffs between computation cost and accuracy. As used in finite element methods (FEM) [23,24], our techniques are based on tetrahedra as the underlying geometry primitive. Density fields are accurately represented by specifying the density at the four vertices.…”

Section: Introductionmentioning

confidence: 99%

Tetrahedra of varying density and their applications

Bukenberger

Lensch

2021

Vis Comput

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“…When polyhedral cells are used as an alternative cell/unit in large-scale or global-scale problems [28], significant polyhedral cells/units are required to fit the spherical curvature [21]. Furthermore, at present, gravity and magnetic forward modeling methods based on polyhedral cells generally use the Gauss/divergence theorem [29] to transform the volume integral of a polyhedral into an area/line integral of polygonal surfaces/lines of the polyhedral through various methods [28,30,31], and subsequently obtain the corresponding analytical solution/numerical solution [32]. How to effectively avoid the numerical ambiguity inherent in the divergence theorem [31] while exploiting the efficient computational properties of area/line integrals [33] is a core issue in the gravitational field forward modeling based on the divergence theorem.…”

Section: Introductionmentioning

confidence: 99%

Spherical Gravity Forwarding of Global Discrete Grid Cells by Isoparametric Transformation

Cao,

Chen,

et al. 2024

Mathematics

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For regional or even global geophysical problems, the curvature of the geophysical model cannot be approximated as a plane, and its curvature must be considered. Tesseroids can fit the curvature, but their shapes vary from almost rectangular at the equator to almost triangular at the poles, i.e., degradation phenomena. Unlike other spherical discrete grids (e.g., square, triangular, and rhombic grids) that can fit the curvature, the Discrete Global Grid System (DGGS) grid can not only fit the curvature but also effectively avoid degradation phenomena at the poles. In addition, since it has only edge-adjacent grids, DGGS grids have consistent adjacency and excellent angular resolution. Hence, DGGS grids are the best choice for discretizing the sphere into cells with an approximate shape and continuous scale. Compared with the tesseroid, which has no analytical solution but has a well-defined integral limit, the DGGS cell (prisms obtained from DGGS grids) has neither an analytical solution nor a fixed integral limit. Therefore, based on the isoparametric transformation, the non-regular DGGS cell in the system coordinate system is transformed into the regular hexagonal prism in the local coordinate system, and the DGGS-based forwarding algorithm of the gravitational field is realized in the spherical coordinate system. Different coordinate systems have differences in the integral kernels of gravity fields. In the current literature, the forward modeling research of polyhedrons (the DGGS cell, which is a polyhedral cell) is mostly concentrated in the Cartesian coordinate system. Therefore, the reliability of the DGGS-based forwarding algorithm is verified using the tetrahedron-based forwarding algorithm and the tesseroid-based forwarding algorithm with tiny tesseroids. From the numerical results, it can be concluded that if the distance from observations to sources is too small, the corresponding gravity field forwarding results may also have ambiguous values. Therefore, the minimum distance is not recommended for practical applications.

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“…With respect to the order N up to which the moments are computed, all the proposed exact algorithms have computational complexity N 9 . Some approximate integration formulas using quadratures have been also proposed for the moments of a single tetrahedron [20], [21], [22], [23], [24]. However an evaluation of their accuracy for high orders in general polyhedra is not available.…”

Section: Introductionmentioning

confidence: 99%

Efficient 3D Geometric and Zernike Moments Computation from Unstructured Surface Meshes

Pozo

Villa‐Uriol

Frangi

2011

IEEE Trans. Pattern Anal. Mach. Intell.

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This paper introduces and evaluates a fast exact algorithm and a series of faster approximate algorithms for the computation of 3D geometric moments from an unstructured surface mesh of triangles. Being based on the object surface reduces the computational complexity of these algorithms with respect to volumetric grid-based algorithms. In contrast, it can only be applied for the computation of geometric moments of homogeneous objects. This advantage and restriction is shared with other proposed algorithms based on the object boundary. The proposed exact algorithm reduces the computational complexity for computing geometric moments up to order N with respect to previously proposed exact algorithms, from N(9) to N(6). The approximate series algorithm appears as a power series on the rate between triangle size and object size, which can be truncated at any desired degree. The higher the number and quality of the triangles, the better the approximation. This approximate algorithm reduces the computational complexity to N(3). In addition, the paper introduces a fast algorithm for the computation of 3D Zernike moments from the computed geometric moments, with a computational complexity N(4), while the previously proposed algorithm is of order N(6). The error introduced by the proposed approximate algorithms is evaluated in different shapes and the cost-benefit ratio in terms of error, and computational time is analyzed for different moment orders.

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Numerical integration of some functions over an arbitrary linear tetrahedra in Euclidean three-dimensional space

Cited by 8 publications

References 12 publications

Tetrahedra of varying density and their applications

Tetrahedra of varying density and their applications

Spherical Gravity Forwarding of Global Discrete Grid Cells by Isoparametric Transformation

Efficient 3D Geometric and Zernike Moments Computation from Unstructured Surface Meshes

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