On finite element integration in natural co‐ordinates

Eisenberg, Martin A.; Malvern, L. E.

doi:10.1002/nme.1620070421

Cited by 91 publications

(25 citation statements)

References 2 publications

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“…The integration scheme used to integrate the left hand side of the Eq. 3.15 is analytical using the formula derived by Eisenberg and Malvern (1973):…”

Section: Weak Formulation Of the Governing Equationsmentioning

confidence: 99%

Numerical analysis of plane cracks in strain-gradient elastic materials

Akarapu

Zbib

2006

Int J Fract

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The classical linear elastic fracture mechanics is not valid near the crack tip because of the unrealistic singular stress at the tip. The study of the physical nature of the deformation around the crack tip reveals the dominance of long-range atomic interactive forces. Unlike the classical theory which incorporates only short range forces, a higher-order continuum theory which could predict the effect of long range interactions at a macro scale would be appropriate to understand the deformation around the crack tip. A simplified theory of gradient elasticity proposed by Aifantis is one such grade-2 theory. This theory is used in the present work to numerically analyze plane cracks in strain-gradient elastic materials. Towards this end, a 36 DOF C 1 finite element is used to discretize the displacement field. The results show that the crack tip singularity still persists but with a different nature which is physically more reasonable. A smooth closure of the structure of the crack tip is also achieved.

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“…The integration scheme used to integrate the left hand side of the Eq. 3.15 is analytical using the formula derived by Eisenberg and Malvern (1973):…”

Section: Weak Formulation Of the Governing Equationsmentioning

confidence: 99%

Numerical analysis of plane cracks in strain-gradient elastic materials

Akarapu

Zbib

2006

Int J Fract

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“…The success of the use of barycentric coordinates for the exact integration of polynomials in the classical finite element method lies partly in the availability of exact integration formulas for integrals along the edges L and the area A of a triangular element, and over the volume V of a tetrahedron; see [12]:…”

Section: Discussionmentioning

confidence: 99%

“…In finite element methods on simplicial grids, the polynomial basis functions are usually expressed in terms of barycentric simplex coordinates. Then, formulas for the exact integration of the generic monomials in barycentric coordinates along an edge of a triangle, over a triangular element, and over a tetrahedron, as provided, e.g., by Eisenberg and Malvern [12], are used to complete the approximation. For finite volume discretization, a general approach for the integration of polynomial basis functions over arbitrary polygonal and polyhedral grids was given by Liu and Vinokur [13].…”

Section: Introductionmentioning

confidence: 99%

Exact integration formulas for the finite volume element method on simplicial meshes

Voitovich

Vandewalle

2007

Numerical Methods Partial

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This article considers the technological aspects of the finite volume element method for the numerical solution of partial differential equations on simplicial grids in two and three dimensions. We derive new classes of integration formulas for the exact integration of generic monomials of barycentric coordinates over different types of fundamental shapes corresponding to a barycentric dual mesh. These integration formulas constitute an essential component for the development of high-order accurate finite volume element schemes. Numerical examples are presented that illustrate the validity of the technology.

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“…These coordinates give the ratio of the area of a sub triangular 170 The advantage of using the area coordinate system is the existence of an integration formula that simplifies the evaluation of area integrals (Eisenberg and Malvern, 1973). This integral equation is (0.18) where i, j, and k are exponential powers.…”

Section: Discussionmentioning

confidence: 99%

An Efficient, Semi-implicit Pressure-based Scheme Employing a High-resolution Finitie Element Method for Simulating Transient and Steady, Inviscid and Viscous, Compressible Flows on Unstructured Grids

Martineau¹,

Berry²

2003

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A new semi-implicit pressure-based Computational Fluid Dynamics (CFD) scheme for simulating a wide range of transient and steady, inviscid and viscous compressible flow on unstructured finite elements is presented here. This new CFD scheme, termed the PCICE-FEM (Pressure-Corrected ICE-Finite Element Method) scheme, is composed of three computational phases, an explicit predictor, an elliptic pressure Poisson solution, and a semiimplicit pressure-correction of the flow variables. The PCICE-FEM scheme is capable of second-order temporal accuracy by incorporating a combination of a time-weighted form of the two-step Taylor-Galerkin Finite Element Method scheme as an explicit predictor for the balance of momentum equations and the finite element form of a time-weighted trapezoid rule method for the semi-implicit form of the governing hydrodynamic equations. Second-order spatial accuracy is accomplished by linear unstructured finite element discretization. The PCICE-FEM scheme employs Flux-Corrected Transport as a high-resolution filter for shock capturing. The scheme is capable of simulating flows from the nearly incompressible to the high supersonic flow regimes.The PCICE-FEM scheme represents an advancement in mass-momentum coupled, pressurebased schemes. The governing hydrodynamic equations for this scheme are the conservative form of the balance of momentum equations (Navier-Stokes), mass conservation equation, and total energy equation. An operator splitting process is performed along explicit and implicit operators of the semi-implicit governing equations to render the PCICE-FEM scheme in the class of predictor-corrector schemes. The complete set of semi-implicit governing equations in the PCICE-FEM scheme are cast in this form, an explicit predictor phase and a semi-implicit pressure-correction phase with the elliptic pressure Poisson solution coupling the predictor-corrector phases. The result of this predictor-corrector formulation is that the pressure Poisson equation in the PCICE-FEM scheme is provided with sufficient internal energy information to avoid iteration. The ability of the PCICE-FEM scheme to accurately and efficiently simulate a wide variety of inviscid and viscous compressible flows is demonstrated here.

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On finite element integration in natural co‐ordinates

Cited by 91 publications

References 2 publications

Numerical analysis of plane cracks in strain-gradient elastic materials

Numerical analysis of plane cracks in strain-gradient elastic materials

Exact integration formulas for the finite volume element method on simplicial meshes

An Efficient, Semi-implicit Pressure-based Scheme Employing a High-resolution Finitie Element Method for Simulating Transient and Steady, Inviscid and Viscous, Compressible Flows on Unstructured Grids

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