Construction of high-order complete scaled boundary shape functions over arbitrary polygons with bubble functions

2016 ◽  
Vol 108 (9) ◽  
pp. 1086-1120 ◽  
Author(s):  
Ean Tat Ooi ◽  
Chongmin Song ◽  
Sundararajan Natarajan
Keyword(s):  
High Order ◽  
Shape Functions ◽  
Boundary Shape ◽  
Bubble Functions

scholarly journals Detecting Inverse Boundaries by Weighted High-Order Gradient Collocation Method

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1297 ◽  
Author(s):  
Judy P. Yang ◽  
Hon Fung Samuel Lam
Keyword(s):  
Collocation Method ◽  
Reproducing Kernel ◽  
Outer Boundary ◽  
Computational Cost ◽  
Weak Form ◽  
High Order ◽  
Unknown Boundary ◽  
Shape Functions ◽  
High Order Derivative ◽  
Special Cases

The weighted reproducing kernel collocation method exhibits high accuracy and efficiency in solving inverse problems as compared with traditional mesh-based methods. Nevertheless, it is known that computing higher order reproducing kernel (RK) shape functions is generally an expensive process. Computational cost may dramatically increase, especially when dealing with strong-from equations where high-order derivative operators are required as compared to weak-form approaches for obtaining results with promising levels of accuracy. Under the framework of gradient approximation, the derivatives of reproducing kernel shape functions can be constructed synchronically, thereby alleviating the complexity in computation. In view of this, the present work first introduces the weighted high-order gradient reproducing kernel collocation method in the inverse analysis. The convergence of the method is examined through the weights imposed on the boundary conditions. Then, several configurations of multiply connected domains are provided to numerically investigate the stability and efficiency of the method. To reach the desired accuracy in detecting the outer boundary for two special cases, special treatments including allocation of points and use of ghost points are adopted as the solution strategy. From four benchmark examples, the efficacy of the method in detecting the unknown boundary is demonstrated.

A high-order control volume finite element method for thermoelastic analysis of functionally graded solids with mixed grids

2018 ◽  
Vol 233 (11) ◽  
pp. 3994-4013
Author(s):  
Qi Liu ◽  
Yan Yu ◽  
Pingjian Ming
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Material Properties ◽  
Control Volume ◽  
Functionally Graded ◽  
High Order ◽  
Shape Functions ◽  
Quadrilateral Element ◽  
Thermoelastic Analysis ◽  
Control Volume Finite Element

In this article, a new two-dimensional control volume finite element method has been developed for thermoelastic analysis in functionally graded materials. A nine-node quadrilateral element and a six-node triangular element are employed to deal with the mixed-grid problem. The unknown variables and material properties are defined at the node. The high-order shape functions of six-node triangular and nine-node quadrilateral element are employed to obtain the unknown variables and their derivatives. In addition, the material properties in functionally graded structure are also modeled by applying the high-order shape functions. The capabilities of the presented method to heat conduction problem, elastic problem, and thermoelastic problem have been validated. First, the defined location of material properties is found to be important for the accuracy of the numerical results. Second, the presented method is proven to be efficient and reliable for the elastic analysis in multi-phase materials. Third, the presented method is capable of high-order mixed grids. The memory and computational costs of the presented method are also compared with other numerical methods.

scholarly journals FINITE ELEMENT SOLUTION OF MULTI-SCALE TRANSPORT PROBLEMS USING THE LEAST SQUARES-BASED BUBBLE FUNCTION ENRICHMENT

2012 ◽  
Vol 03 (04) ◽  
pp. 1250019
Author(s):  
A. YAZDANI ◽  
V. NASSEHI
Keyword(s):  
Finite Elements ◽  
Least Squares ◽  
Computational Cost ◽  
Shape Functions ◽  
Element Level ◽  
Polynomial Coefficients ◽  
Stiffness Matrices ◽  
Standard Linear ◽  
Bubble Functions ◽  
Transport Problems

This paper presents a technique for deriving least-squares-based polynomial bubble functions to enrich the standard linear finite elements, employed in the formulation of Galerkin weighted-residual statements. The element-level linear shape functions are enhanced using supplementary polynomial bubble functions with undetermined coefficients. The enhanced shape functions are inserted into the model equation and the residual functional is constructed and minimized by using the method of the least squares, resulting in an algebraic system of equations which can be solved to determine the unknown polynomial coefficients in terms of element-level nodal values. The stiffness matrices are subsequently formed with the standard finite elements assembly procedures followed by using these enriched elements which require no additional nodes to be introduced and no extra degree of freedom incurred. Furthermore, the proposed technique is tested on a number of benchmark linear transport equations where the quadratic and cubic bubble functions are derived and the numerical results are compared against the exact and standard linear element solutions. It is demonstrated that low order bubble enriched elements provide more accurate approximations for the exact analytical solutions than the standard linear elements at no extra computational cost in spite of using relatively crude meshes. On the other hand, it is observed that a satisfactory solution of the strongly convection-dominated transport problems may require element enrichment by using significantly higher order polynomial bubble functions in addition to the use of extremely fine computational meshes.

scholarly journals Shape functions for high-order shear deformation beam element

2021 ◽  
Author(s):  
Himanshu Gaur ◽  
Mahmoud Dawood ◽  
Ram Kishore Manchiryal
Keyword(s):  
Boundary Conditions ◽  
Shear Deformation ◽  
Beam Theory ◽  
Order Theory ◽  
Beam Element ◽  
High Order ◽  
Shape Functions ◽  
High Order Theory ◽  
Cubic Polynomial ◽  
Transverse Deflection

In this article, shape functions for higher-order shear deformation beam theory are derived. For the two nodded beam element, transverse deflection is assumed as cubic polynomial. By using equations of equilibrium of high-order theory that are already derived by J. N. Reddy in 1997, equation for slope of high- order theory is found. Finally with the boundary conditions of beam element and assumed kinematics of high-order theory, shape functions are derived.

scholarly journals High‐order shape functions for interior acoustics

PAMM ◽  
2019 ◽  
Vol 19 (1) ◽  
Author(s):  
Fabian Duvigneau ◽  
Sascha Duczek
Keyword(s):  
High Order ◽  
Shape Functions

A Well-Conditioned Hierarchical Basis for Triangular H(curl)-Conforming Elements

2011 ◽  
Vol 9 (3) ◽  
pp. 780-806 ◽  
Author(s):  
Jianguo Xin ◽  
Wei Cai
Keyword(s):  
Orthogonal Polynomials ◽  
Stiffness Matrix ◽  
Jacobi Polynomials ◽  
Mass Matrix ◽  
Basis Functions ◽  
Shape Functions ◽  
Reference Element ◽  
Hierarchical Basis ◽  
Normal Functions ◽  
Bubble Functions

AbstractWe construct a well-conditioned hierarchical basis for triangular H(curl)-conforming elements with selected orthogonality. The basis functions are grouped into edge and interior functions, and the later is further grouped into normal and bubble functions. In our construction, the trace of the edge shape functions are orthonormal on the associated edge. The interior normal functions, which are perpendicular to an edge, and the bubble functions are both orthonormal among themselves over the reference element. The construction is made possible with classic orthogonal polynomials, viz., Legendre and Jacobi polynomials. For both the mass matrix and the quasi-stiffness matrix, better conditioning of the new basis is shown by a comparison with the basis previously proposed by Ainsworth and Coyle [Comput. Methods. Appl. Mech. Engrg., 190 (2001), 6709-6733].

Modeling Slope Anchor Reinforcement Using Enhanced Discontinuous Deformation Analysis Method (EDDA) with Natural Neighbor Interpolation

2014 ◽  
Vol 638-640 ◽  
pp. 680-683 ◽  
Author(s):  
Yong Zheng Ma ◽  
Ke Jian Cai ◽  
Zhan Tao Li ◽  
Chun Xia Song
Keyword(s):  
High Order ◽  
Numerical Results ◽  
Discontinuous Deformation Analysis ◽  
Deformation Analysis ◽  
Analysis Method ◽  
Shape Functions ◽  
Deformable Solids ◽  
Slope Reinforcement ◽  
Natural Neighbor ◽  
Discontinuous Deformation

A new enhanced Discontinuous Deformation Analysis method (EDDA) in conjunction with Natural Neighbor Interpolative (NNI) bases for modeling the system composed of high order deformable solids is developed. The advantages of NNI lie in its efficiency and the interpolative property when employed as the shape functions. The anchor reinforcement algorithm is also implemented in the EDDA for modeling high order deformable solids. The numerical results of simple problems by using the proposed method agree well with the corresponding analytical results, and certain slope reinforcement problems are also simulated with rational numerical results, which verify efficiency and accuracy of the EDDA.

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