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Gautam Dasgupta - One of the best experts on this subject based on the ideXlab platform.
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incompressible and locking free finite elements from rayleigh mode vectors quadratic polynomial displacement fields
Acta Mechanica, 2012Co-Authors: Gautam DasguptaAbstract:Under pure bending, with an arbitrary patch of plane four-Node finite elements, the exact analytical algebraic expressions of deformation, strain and stress fields are numerically captured by a computer algebra program for both compressible and incompressible continua. Linear combinations of Rayleigh displacement vectors yield the Ritz test functions. These coupled fields model pure bending of an Euler-Bernoulli beam with appropriate linearly varying axial strains devoid of shear. Such Courant admissible functions allow an undeformed straight Side to curve in flexure. Since these displacement vectors satisfy equilibrium conditions, they are necessarily functions of the Poisson’s ratio. Applications in bio-, micro- and nano-mechanics motivated this formulation that blurs the frontier between the finite and the boundary element methods. Exact integration yields the element stiffness matrix of a compressible convex or concave quadrilateral, or a triangular element with a Side Node. For the generic energy density integral, the paper furnishes an analytical expression that can be incorporated in Fortran or C ++. In isochoric plane strain problems, the Rayleigh kinematic mode of dilatation is replaced by a constant element pressure. The equivalent nodal loadings are calculated according to the Ritz variational statement. Subsequently, without assembling the global stiffness matrix, nodal compatibility and equilibrium equations are solved in terms of Rayleigh modal participation factors.
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stiffness matrices of isoparametric four Node finite elements by exact analytical integration
Journal of Aerospace Engineering, 2008Co-Authors: Gautam DasguptaAbstract:Explicit algebraic expressions needed to compute element stiffness matrices using procedural (FORTRAN) and object oriented ( C++ ) computer programs are presented. Numerical illustrations for a convex quadrilateral and a triangle with a Side Node are included. The wide controversy due to conventional element level approximate numerical quadrature within the computational square domain in η and ξ coordinates is completely resolved here by the closed form analytical integration within the physical element, in x and y coordinates.
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Closed Form Isoparametric Shape Functions of Four-Node Convex Finite Elements
2006Co-Authors: Gautam DasguptaAbstract:On arbitrary plane quadrilaterals, difficulties in integrating energy densities prevented analysts from directly using shape function expressions in terms of the physical coordinate variables (x and y). With the availability of an exact integration procedure, shape functions are sought here as explicit expressions (in x and y). Conventional isoparametric (indirect) representation, via canonical coordinate variables (η and ξ) on a unit square in the computational domain, do not reveal the presence of irrational algebraic expressions that were first elaborated by Wachspress. Computer algebra systems demonstrate that the isoparametric shape functions are: for a general quadrilateral — linear and a square root of a quadratic (in x and y); for a trapezoid — identical to the Wachspress rational polynomials; for a parallelogram — bilinear functions; and, moreover, even valid for a triangle with a Side Node. The failure of the isoparametric formulation for concave domains is traced to the negative argument of irrational parts. Shape functions in the physical domain (x − and y − ) facilitate contour plotting of responses (e.g., temperature distributions) within quadrilateral elements. A subsequent paper details exact calculation of stiffness matrices where the presented shape functions (in x and y) are indispensable
A Tessler - One of the best experts on this subject based on the ideXlab platform.
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c0 continuous triangular plate element for laminated composite and sandwich plates using the 2 2 refined zigzag theory
Composite Structures, 2013Co-Authors: Atila Barut, Erdogan Madenci, A TesslerAbstract:Abstract Most of the existing plate elements assume constant transverse displacement across the thickness resulting in zero transverse stretch deformation. This study presents a new triangular finite element for modeling thick laminates and sandwich panels based on the {2, 2}-order refined zigzag plate theory. It adopts quadratic through-thickness variation of the in-plane and transverse displacement components. The transverse normal strain is calculated based on the assumption of cubic representation of the transverse normal stress. The zigzag functions are piecewise linear through the thickness. The element consists of 3 corner Nodes and 3 mid-Side Nodes along the edges. Each corner and mid-Side Node has 11 and 3 degrees of freedom (DOF), respectively. This C 0 continuous element is free of geometric locking, and does not require shear correction factors. It provides robust and accurate prediction of all six stress components (in-plane and transverse normal and shear stresses) in the analysis of highly heterogeneous laminates and sandwich plates.
Ted Belytschko - One of the best experts on this subject based on the ideXlab platform.
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a finite strain quadrilateral shell element based on discrete kirchhoff love constraints
International Journal for Numerical Methods in Engineering, 2005Co-Authors: P Areias, Jeonghoon Song, Ted BelytschkoAbstract:This paper improves the 16 degrees-of-freedom quadrilateral shell element based on pointwise Kirchhoff–Love constraints and introduces a consistent large strain formulation for this element. The model is based on classical shell kinematics combined with continuum constitutive laws. The resulting element is valid for large rotations and displacements. The degrees-of-freedom are the displacements at the corner Nodes and one rotation at each mid-Side Node. The formulation is free of enhancements, it is almost fully integrated and is found to be immune to locking or unstable modes. The patch test is satisfied. In addition, the formulation is simple and amenable to efficient incorporation in large-scale codes as no internal degrees-of-freedom are employed, and the overall calculations are very efficient. Results are presented for linear and non-linear problems. Copyright © 2005 John Wiley & Sons, Ltd.
Atila Barut - One of the best experts on this subject based on the ideXlab platform.
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c0 continuous triangular plate element for laminated composite and sandwich plates using the 2 2 refined zigzag theory
Composite Structures, 2013Co-Authors: Atila Barut, Erdogan Madenci, A TesslerAbstract:Abstract Most of the existing plate elements assume constant transverse displacement across the thickness resulting in zero transverse stretch deformation. This study presents a new triangular finite element for modeling thick laminates and sandwich panels based on the {2, 2}-order refined zigzag plate theory. It adopts quadratic through-thickness variation of the in-plane and transverse displacement components. The transverse normal strain is calculated based on the assumption of cubic representation of the transverse normal stress. The zigzag functions are piecewise linear through the thickness. The element consists of 3 corner Nodes and 3 mid-Side Nodes along the edges. Each corner and mid-Side Node has 11 and 3 degrees of freedom (DOF), respectively. This C 0 continuous element is free of geometric locking, and does not require shear correction factors. It provides robust and accurate prediction of all six stress components (in-plane and transverse normal and shear stresses) in the analysis of highly heterogeneous laminates and sandwich plates.
P Areias - One of the best experts on this subject based on the ideXlab platform.
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a finite strain quadrilateral shell element based on discrete kirchhoff love constraints
International Journal for Numerical Methods in Engineering, 2005Co-Authors: P Areias, Jeonghoon Song, Ted BelytschkoAbstract:This paper improves the 16 degrees-of-freedom quadrilateral shell element based on pointwise Kirchhoff–Love constraints and introduces a consistent large strain formulation for this element. The model is based on classical shell kinematics combined with continuum constitutive laws. The resulting element is valid for large rotations and displacements. The degrees-of-freedom are the displacements at the corner Nodes and one rotation at each mid-Side Node. The formulation is free of enhancements, it is almost fully integrated and is found to be immune to locking or unstable modes. The patch test is satisfied. In addition, the formulation is simple and amenable to efficient incorporation in large-scale codes as no internal degrees-of-freedom are employed, and the overall calculations are very efficient. Results are presented for linear and non-linear problems. Copyright © 2005 John Wiley & Sons, Ltd.