scholarly journals A Finite Circular Arch Element Based on Trigonometric Shape Functions

2007 ◽  
Vol 2007 ◽  
pp. 1-19 ◽  
Author(s):  
H. Saffari ◽  
R. Tabatabaei
Keyword(s):  
Equilibrium Equation ◽  
Displacement Function ◽  
Trigonometric Function ◽  
Polar Coordinates ◽  
Element Formulation ◽  
Shape Functions ◽  
Total Potential ◽  
Matrix Operations ◽  
Displacement Vectors ◽  
Locking Phenomena

The curved-beam finite element formulation by trigonometric function for curvature is presented. Instead of displacement function, trigonometric function is introduced for curvature to avoid the shear and membrane locking phenomena. Element formulation is carried out in polar coordinates. The element with three nodal parameters is chosen on curvature. Then, curvature field in the element is interpolated as the conventional trigonometric functions. Shape functions are obtained as usual by matrix operations. To consider the boundary conditions, a transformation matrix between nodal curvature and nodal displacement vectors is introduced. The equilibrium equation is written by minimizing the total potential energy in terms of the displacement components. In such equilibrium equation, the locking phenomenon is eliminated. The interesting point in this method is that for most problems, it is sufficient to use only one element to obtain the solution. Four examples are presented in order to verify the element formulation and to show the accuracy and efficiency of the method. The results are compared with those of other concepts.

A new formulation based upon trigonometric function for finite circular arch elements

2008 ◽  
Vol 222 (8) ◽  
pp. 1371-1380 ◽  
Author(s):  
H Saffari ◽  
M J Fadaee ◽  
R Tabatabaei
Keyword(s):  
Arbitrary Point ◽  
Displacement Function ◽  
Trigonometric Function ◽  
Polar Coordinate System ◽  
Curved Beam ◽  
Stationary Condition ◽  
Total Potential ◽  
Curved Beam Element ◽  
Polar Coordinate ◽  
New Formulation

In this paper, the formulation of the curved beam element by trigonometric functions for the curvature, which is an alternative to the displacement function is presented. Trigonometric function is chosen for the curvature to avoid the shear and membrane locking phenomena. In the developed formulation, the force—curvature relationships in polar coordinate system have been obtained first; then the curvature of the element has been assumed to have a trigonometric function form; and the radial and tangential displacements, and rotation of the cross-section have been found to be a function of the curvature. Moreover, the relationship between the nodal curvatures and the nodal deformations has been calculated and used for determining the deformations in terms of curvature at an arbitrary point. The total potential energy has been calculated accounting for bending, shear, and tangential deformations. Invoking the stationary condition of the system, the force—deformation relationship for the element has been obtained. Using this relationship, the stiffness matrix and the equivalent fixed loads applying at the nodes have been computed. In such an equilibrium equation, the locking phenomenon is eliminated. The formulation is applied to six examples to verify its capabilities. The results demonstrate that the presented element is capable of representing the behaviour of the curved beam with adequate accuracy and efficiency as compared with the previous methods.

scholarly journals Plate Finite Element with Physical Shape Functions: Correctness of the Formulation

2017 ◽  
Vol 63 (3) ◽  
pp. 19-37
Author(s):  
W. Gilewski ◽  
M. Sitek
Keyword(s):  
Finite Element ◽  
Good Choice ◽  
Test Case ◽  
Element Formulation ◽  
Structural Elements ◽  
Shape Functions ◽  
Solution Convergence ◽  
Wide Range ◽  
Physical Shape ◽  
Moderately Thick Plates

Abstract The formulation of a plate finite element with so called ‘physical’ shape functions is revisited. The derivation of the ‘physical’ shape functions is based on Hencky-Bolle theory of moderately thick plates. The considered finite element was assessed in the past, and the tests showed that the solution convergence was achieved in a wide range of thickness to in-plane dimensions ratios. In this paper a holistic correctness assessment is presented, which covers three criteria: the ellipticity, the consistency and the inf-sup conditions. Fulfilment of these criteria assures the existence of a unique solution, and a stable and optimal convergence to the correct solution. The algorithms of the numerical tests for each test case are presented and the tests are performed for the considered formulation. In result it is concluded that the finite element formulation passes every test and therefore is a good choice for modeling plate structural elements regardless of their thickness.

Automatic Refinement of FE Shell Models Based on a Local Energy Function

2013 ◽  
Author(s):  
Antonio Carminelli ◽  
Giuseppe Catania
Keyword(s):  
Potential Energy ◽  
Local Density ◽  
Free Vibration Analysis ◽  
Element Model ◽  
Companion Paper ◽  
Total Potential Energy ◽  
Shape Functions ◽  
Total Potential ◽  
Error Functional ◽  
Local Patch

This paper deals with an adaptive refinement technique of a B-spline degenerate shell finite element model, for the free vibration analysis of curved thin and moderately thick-walled structures. The automatic refinement of the solution is based on an error functional related to the density of the total potential energy. The model refinement is generated by locally increasing, in a sub-domain R of a local patch domain, the number of shape functions while maintaining constant the functions polynomial order. The local refinement strategy is described in a companion paper, written by the same authors of this paper and presented in this Conference. A two-step iterative procedure is proposed. In the first step, one or more sub domains to be refined are identified by means of a point-wise error functional based on the system total potential energy local density. In the second step, the number of shape functions to be added is iteratively increased until the difference of the total potential energy, calculated on the sub domain between two iteration, is below a user defined tolerance. A numerical example is presented in order to test the proposed approach. Strengths and limits of the approach are critically discussed.

B-Spline Finite Element Formulation for Laminated Composite Shells

2008 ◽  
Author(s):  
Antonio Carminelli ◽  
Giuseppe Catania
Keyword(s):  
Finite Element ◽  
Tensor Product ◽  
Shear Deformation ◽  
Laminated Composite ◽  
Finite Element Formulation ◽  
Spline Functions ◽  
Element Formulation ◽  
Shape Functions ◽  
Composite Shells ◽  
B Spline

This paper presents a finite element formulation for the dynamical analysis of general double curvature laminated composite shell components, commonly used in many engineering applications. The Equivalent Single Layer theory (ESL) was successfully used to predict the dynamical response of composite laminate plates and shells. It is well known that the classic shell theory may not be effective to predict the deformational behavior with sufficient accuracy when dealing with composite shells. The effect of transverse shear deformation should be taken into account. In this paper a first order shear deformation ESL laminated shell model, adopting B-spline functions as approximation functions, is proposed and discussed. The geometry of the shell is described by means of the tensor product of B-spline functions. The displacement field is described by means of tensor product of B-spline shape functions with a different order and number of degrees of freedom with respect to the same formulation used in geometry description, resulting in a non-isoparametric formulation. A solution refinement method, making it possible to increase the order of the displacement shape functions without using the well known B-spline “degree elevation” algorithm, is also proposed. The locking effect was reduced by employing a low-order integration technique. To test the performance of the approach, the static solution of a single curvature shell and the eigensolutions of composite plates were obtained by numerical simulation and are then compared with known solutions. Discussion follows.

Nodal Collocation for the P-Convergent Scheme in Boundary Element Technique

2013 ◽  
Vol 405-408 ◽  
pp. 3139-3142
Author(s):  
Kwang Sung Woo ◽  
Won Seok Jang ◽  
Yoo Mi Kwon ◽  
Jun Hyung Jo
Keyword(s):  
Boundary Element ◽  
Higher Order ◽  
Kernel Functions ◽  
Element Formulation ◽  
Shape Functions ◽  
Collocation Points ◽  
Element Technique ◽  
Scheme Selection ◽  
Shaped Domain ◽  
Hierarchical Pattern

The concept of thep-convergent boundary element modeling has been presented to analyze the potential problem with L-shaped domain. The details of thep-convergent boundary element formulation are discussed. These include the equations of nodal collocation for thep-convergent scheme, selection of higher order hierarchical shape functions, techniques for integrating the product of the kernel functions and corresponding shape functions, strategies for selecting collocation points used in approximating the unknowns associated with the higher order shape functions, and program organizations. A numerical example that demonstrates the performance of thep-convergent boundary element formulation is shown with respect to different arrangement of collocation points including both symmetric non-hierarchical pattern and non-symmetric hierarchical pattern.

scholarly journals Accurate Element of Compressive Bar considering the Effect of Displacement

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Lifeng Tang ◽  
Jing Xu ◽  
Hongzhi Wang ◽  
Xinghua Chen
Keyword(s):  
Potential Energy ◽  
Shape Function ◽  
Variation Principle ◽  
Shape Functions ◽  
Governing Equations ◽  
Energy Principle ◽  
Potential Energy Principle ◽  
Total Potential ◽  
Stiffness Matrices ◽  
Function Element

By constructing the compressive bar element and developing the stiffness matrix, most issues about the compressive bar can be solved. In this paper, based on second derivative to the equilibrium differential governing equations, the displacement shape functions are got. And then the finite element formula of compressive bar element is developed by using the potential energy principle and analytical shape function. Based on the total potential energy variation principle, the static and geometrical stiffness matrices are proposed, in which the large deformation of compressive bar is considered. To verify the accurate and validity of the analytical trial function element proposed in this paper, a number of the numerical examples are presented. Comparisons show that the proposed element has high calculation efficiency and rapid speed of convergence.

Finite Element Analysis Over Tangled Meshes

2012 ◽  
Author(s):  
Josh Danczyk ◽  
Krishnan Suresh
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Functional Space ◽  
Mesh Optimization ◽  
Element Formulation ◽  
Shape Functions ◽  
Element Analysis ◽  
Patch Tests ◽  
Galerkin Formulation ◽  
Element Shape

In finite element analysis (FEA), tasks such as mesh optimization and mesh morphing can lead to overlapping elements, i.e., to a tangled mesh. Such meshes are considered ‘unacceptable’ today, and are therefore untangled using specialized procedures. Here it is shown that FEA can be easily extended to handle tangled meshes. Specifically, by defining the nodal functional space as an oriented linear combination of the element shape functions, it is shown that the classic Galerkin formulation leads to a valid finite element formulation over such meshes. Patch tests and numerical examples illustrate the correctness of the proposed methodology.

Nonlinear Finite Element Formulation for the Large Displacement Analysis of Plates

1990 ◽  
Vol 57 (3) ◽  
pp. 707-718 ◽  
Author(s):  
Bilin Chang ◽  
A. A. Shabana
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Coordinate System ◽  
Deformable Body ◽  
Finite Element Formulation ◽  
Deformation Analysis ◽  
Element Formulation ◽  
Rectangular Plates ◽  
Shape Functions ◽  
Intermediate Element

In this investigation a nonlinear total Lagrangian finite element formulation is developed for the dynamic analysis of plates that undergo large rigid body displacements. In this formulation shape functions are required to include rigid body modes that describe only large translational displacements. This does not represent any limitation on the technique presented in this study, since most of commonly used shape functions satisfy this requirement. For each finite plate element an intermediate element coordinate system, whose axes are initially parallel to the axes of the element coordinate system, is introduced. This intermediate element coordinate system, which has an origin which is rigidly attached to the origin of the deformable body, is used for the convenience of describing the configuration of the element with respect to the deformable body coordinate system in the undeformed state. The nonlinear dynamic equations developed in this investigation for the large rigid body displacement and small elastic deformation analysis of the rectangular plates are expressed in terms of a unique set of time invariant element matrices that depend on the assumed displacement field. The invariants of motion of the deformable body discretized using the plate elements are obtained by assembling the invariants of its elements using a standard finite element procedure.

Exploroing Polar Curves with GeoGebra

2012 ◽  
Vol 106 (3) ◽  
pp. 228-233
Author(s):  
Tuyetdong Phan-Yamada ◽  
Walter M. Yamada
Keyword(s):  
Trigonometric Function ◽  
Polar Coordinates ◽  
Cartesian Coordinates ◽  
Step Process ◽  
Cartesian Plane

Most trigonometry textbooks teach the graphing of polar equations as a two-step process: (1) plot the points corresponding to values of θ such as π, π/2, π/3, π/4, π/6, and so on; and then (2) connect these points with a curve that follows the behavior of the trigonometric function in the Cartesian plane. Many students have difficulty using this method to graph general polar curves. The difficulty seems to stem from an inability to convert changes in the value of the trigonometric equation as a function of angle (abscissa vs. ordinate in Cartesian coordinates) to changes of the radius as a function of angle (r[θ] in polar coordinates). GeoGebra provides a tool to help students visualize this relationship, thus significantly improving students' ability

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