High-order Triangular Elements Research Articles

Studying the Effects of Negative Skin Friction on Single Piles in Basrah Governorate

The finite element method capable of simulating the behavior of deep foundations subjected to negative skin friction in Basrah soil is investigated. Single piles under drag forces are analyzed using the PLAXIS program with an axisymmetric model. Linear elastic, Soft Soil and Mohr-Coulomb constitutive relations are adopted, where higher order triangular element is chosen for pile and soil clusters. Both pile and soil are modeled using (15)-node triangular elements. Three sites in Basrah province (Umm Qasr Port, Khor Al-Zubair, and Shatt AlArab Hotel) were selected to perform this study. The soil profile and layer characteristics are obtained from the soil investigation reports. Where the negative skin friction is evaluated due to filling loads. It is Conclusion thatSmall relative displacements are necessary to activate the negative skin friction. The elastic shorting for pile effect negative skin friction, due to increase relative displacement. The elastic shorting of the driven pile is more than that of the bored pile due to the less cross-sectional area of the driven pile. The results revealed proportional relation between the developed drag forces and pile section dimensions, interface friction factor, and fill height, with a maximum effect on the section dimension and minimum effect on the interface factor. The locations of neutral points are not sensitive to the above-mentioned factors.

L2-Norm Based a Posteriori Error Estimates of Compressible and Nearly-Incompressible Elastic Finite Element Solutions

The displacement and stress-based error estimates in a posteriori error recovery of compressible and nearly-incompressible elastic finite element solutions is investigated in the present study. The errors in the finite element solutions, i.e., in displacement and stress, at local and global levels are computed in L2-norm of quantity of interest, namely, displacements and gradients. The error estimation techniques are based on the least square fitting of higher order polynomials to stress and displacement in a patch comprising of node/elements surrounding and including the node/elements under consideration. The benchmark examples of compressible and incompressible elastic bodies, with known solutions employing triangular discretization schemes, are implemented to measure the finite element errors in displacements and gradients. The mixed formulation involving displacement and pressure is used for incompressible elastic analysis. The performance of error estimation is measured in terms of convergence properties, effectivity and mesh required for predefined precision. The error convergence rate, in FEM original solution, recovered solution using displacement recovery-based and stress-based error recovery technique for stresses, are obtained as (1.9714, 2.8999, and 2.5018) and (0.9818, 1.7805, and 1.4952) respectively for compressible and incompressible self-loaded elastic plate benchmark example using higher order triangular elements. It is concluded from the study that displacement fitting technique for extracting higher order derivatives shows a very effective technique for recovery of compressible and nearly-incompressible finite element analysis errors.

Linear smoothed finite element method for quasi-incompressible hyperelastic media

This work presents a linear smoothing scheme over high-order triangular elements within the framework of the cell-based strain smoothed finite element method for two-dimensional nonlinear problems. The main idea behind the proposed linear smoothing scheme is that it unlike the classical SFEM, it does not require the subdivision of the finite element cells into smoothing sub-cell. The other features of the classical SFEM are retained, such as: it does not require an explicit form of the derivatives of the basis functions, all the computations are done in the physical space, and the results are less sensitive to mesh distortion. A series of benchmark tests are done to demonstrate the validity and the stability of the proposed scheme. The validity and accuracy are confirmed by comparing the obtained numerical results with the standard FEM using quadratic triangular element and the exact solutions.

An automated higher order meshing for NACA0018 airfoil design using subparametric transformation

A novel higher order meshing around the airfoil design by using finite simple triangle elements has been presented in this paper. An automated meshing has been performed by linear to sextic order (n = 1, 2, 3, 4, 5 and 6) simple triangle elements using MATLAB code. The subparametric transformation approach has been implemented in the present accurate and efficient meshing. Linear, quadratic, cubic, quartic, quintic and sextic order triangle shaped elements are been utilized to discretize the area near the airfoil shape. An itemized extraction of the edge information, nodal coordinates, the total number of discretized elements and the triangle indices has been carried out from the present meshing. A NACA0018 airfoil design of 18% thickness of max camber is under consideration for the present work. Meshing is executed with the MATLAB code AirfoilHOmesh2d .m by implementing simple changes based on the present 4-digit NACA0018 airfoil profile. The accuracy of an eminent linear mesh generator scheme presented by Gilbert Strang and Persson has been proved and verified. Present work has extended these eminent linear meshing scheme by utilizing subparametric transformation on higher order triangular elements. The algorithm of meshing with higher order has been explained in detail and represented in flowchart form in the Appendix. The data extracted from the meshing can offer a simplified approach to optimize the airfoil shape, for the study of flow around the airfoil in fluid dynamics and the analysis of adverse effects on its external surface in aerospace applications. The advanced higher order meshing is beneficial for the optimization of any symmetric airfoil resulting a better economical fuel usage in the aircraft.

Using Higher-Order Strain Interpolation Function to Improve the Accuracy of Structural Responses

It is widely known that the accuracy of the finite element method has a direct relation with the type of elements and meshes. Another issue which has remained less treated is the impact of loading type on the accuracy of responses. Changing the applied forces from concentrated to distributed loading has a great effect on the accuracy of certain types of elements and this action can greatly reduce their accuracy. Particularly in the coarse meshes, it creates a critical situation. Some elements do not have the ability to provide the exact answers in stated conditions. For example, the well-known plane element, LST, demonstrates promising performance under concentrated shear and bending loading as well as surface traction. In the case of distributed loads and coarse meshes, its accuracy diminishes considerably. To remedy this defect, in this paper, a new higher-order triangular element is proposed by using natural assumed strain approximation. Various numerical examples demonstrate high accuracy and efficiency of the element in comparison with common well-known finite elements in analysis of structures under distributed loading.

Low-Order Preconditioning of High-Order Triangular Finite Elements

We propose a new formulation of a low-order elliptic preconditioner for high-order triangular elements. In the preconditioner, the nodes of the low-order finite element problem do not necessarily coincide with the high-order nodes. Instead, the two spaces are connected using least squares projection operators. The effectiveness of the preconditioner is demonstrated to be highly sensitive to the location and number of vertices used to construct the low-order finite element mesh on each high-order element. We treat the number of low-order vertices and their locations as optimizable quantities and chose them to minimize the condition number of the preconditioned stiffness matrix on the reference element. We present computational results that demonstrate that the condition number of the preconditioned high-order stiffness matrix on the reference element can be improved. The best performing preconditioners are formed with low-order finite element meshes that have more vertices than the high-order element has degrees of freedom, and have nodes grouped close to the element edges.

Delaminating Quadrature Method for Physical Optics Integrals over High-Order Triangular Mesh for Complex Source Beam Applications

ABSTRACTA novel method for calculating physical optics (PO) integrals over an electrically large target illuminated by a complex source beam is presented. A high-order triangular element is used to mesh the surface of this target. The PO integrals over these elements are calculated with a delaminating quadrature method. This method is very stable, accurate, and efficient, and its computation cost remains constant as the frequency increased. The influence of critical points is taken into account by subdividing the integral intervals. Numerical results are shown in this article to demonstrate the efficiency of this approach.

Stress Analysis of Free-Edge Laminated Composite Plates by Two Bending Elements

Two higher order triangular elements will be formulated for inter-laminar stresses analysis in free edge of the laminated composite bending plates. The analysis is based on a displacement model and includes the higher order shear deformation. To demonstrate the validity and efficiency of the proposed method, the orthotropic laminated composite plates with arbitrary laminations and boundary conditions will be analyzed. In this paper, the effects of increasing the number of nodes and the degree of interpolation functions will be studied. It will be proven that by using particular higher order interpolation, more accurate stress resultants for plates can be predicted. Furthermore, a good rate of convergence will be found by utilizing only a very coarse mesh.

Seleção de topologias ótimas de estruturas elásticas 2D com restrição de tensão – via Smooth Evolutionary Structural Optimization

This paper approaches the topology optimization problems in plane linear elasticity considering the minimization of the volume with restriction of the stress employing an index of performance for monitoring the meeting of the optimum region. It is used for this purpose the classical evolutionary structural optimization, or ESO ‐ evolutionary structural optimization. This procedure is based on systematic and gradual removal of the elements with lower stress compared with the maximum stress of the structure. This procedure also known as a process “hard‐kill”. It is proposed a variant of the ESO method, called SESO ‐ Smoothing ESO, which is based on the philosophy that if an element is not really necessary for the structure, its contribution to the structural stiffness will gradually diminish until it has no longer influence in the structure, so its removal is performed smoothly. That is, their removal is done smoothly, reducing the values of the constitutive matrix of the element as if it were in the process of damage. A new performance index for the monitoring of this evolutionary process smoothed is proposed herein. The applications of ESO and SESO are made with the finite element method, but considering a high order triangular element based on the free formulation. Finally, it is implemented a spatial filter in terms of stress control, which was associated with SESO technique proved to be very stable and efficient in eliminating the formation of the checkerboard.

Advantages of cubic arcs for approximating curved boundaries by subparametric transformations for some higher order triangular elements

In the finite element method, the most popular technique for dealing with curved boundaries is that of isoparametric coordinate transformations. In this paper, the 10-node (cubic), 15-node (quartic) and 21-node (quintic) curved boundary triangular elements having one curved side and two straight sides are analyzed using the isoparametric coordinate transformations. By this method, these curved triangles in the global coordinate system are mapped into a isosceles right angled unit triangle in the local coordinate system and the curved boundary of these triangular elements are implicitly replaced by cubic, quartic, and quintic arcs. The equations of these arcs involve parameters, which are the coordinates of points on the curved side. Relations are deduced for choosing the parameters in quartic and quintic arcs in such a way that each arc is always a cubic arc which passes through four points of the original curve, thus ensuring a good approximation. The point transformations thus determined with the above choice of parameters on the curved boundary and also in turn the other parameters in the interior of curved triangles will serve as a powerful subparametric coordinate transformation for higher order curved triangular elements. Numerical examples are given to demonstrate the accuracy and efficiency of the method.

Free vibration of tapered isotropic rectangular plates

In this paper, free vibration analysis of isotropic rectangular plates with linearly varying thickness in one direction has been investigated using a high-order triangular element. The first order shear deformation theory is used to include the effect of transverse shear deformation. The element has 18 nodes on the sides and six internal nodes. The geometry of element is expressed by three linear shape functions of area coordinates. The formulation is displacement-based. The element has 51 degrees-of-freedom, which can be reduced to 39 degrees-of-freedom by Guyan reduction, mass condensation, or eigenvalue economization scheme for the degrees-of-freedom associated with the internal nodes. Rotary inertia has been included in the consistent mass matrix. Numerical examples are presented to show the accuracy and convergence characteristics of the element. Isotropic plates with different thickness ratios (varies from 0.01 to 0.2), tapered ratios, aspect ratios and boundary conditions are analyzed. The results obtained by present element show a close agreement with the available published results. Some numerical results have been given as new results.

Computation of stress resultants in plate bending problems using higher-order triangular elements

This paper is concerned with the development of a higher-order triangular plate element based on the first-order shear deformation plate theory. The present study brings out the shortcomings of conventional lower-order finite element interpolation when applied for the bending analysis of plates with free edges. The stress resultants, especially the transverse shear forces and twisting moments, obtained using the lower-order displacement finite elements fail to satisfy the natural (or force) boundary conditions accurately. On the other hand, the plate element with higher-order interpolation of the field variables enables the accurate prediction of stress resultants for plates. The superior performance of the higher-order plate bending element is demonstrated through bending analyses of plates of various shapes and free edges and examining the distribution of the stress resultants and the satisfaction of the natural boundary conditions.

The dynamics of three-dimensional underwater explosion bubble

The fluid is assumed to be inviscid and incompressible and the flow irrotational. A time-integration boundary-integral method is used to solve the Laplace equation for the velocity potential to calculate the shape and position of the bubble. To improve the accuracy of the solution, the high-order triangular elements with curved sides and surfaces defined by six nodes are used to discretize the boundary surface in this investigation. Meanwhile, the singularity of the double-layer potential is eliminated by recasting the principal-value integral of the double-layer potential when the influence coefficient matrix is calculated. The material velocity vector at any node can be obtained by the potential of adjacent nodes with an appropriate weighted method. Elastic mesh technique (EMT), which is a new mesh regulation technique, is further applied to maintain the regularity of the triangular-element mesh used to discretize the dynamic boundary surface during the evolution of explosion bubble(s). All these efforts make the present approach viable and robust, and which is validated by computations of several bubble dynamics problems. Numerical analyses are carried out for the evolution of a bubble near a free surface and the interaction of two bubbles with a floating structure near a free surface. The robustness of the algorithm is demonstrated through simulating bubble jets near a free surface producing sharp free surface spikes and bubble(s) collapsing nearby a floating structure close to a free surface.

A high-order implicit finite element method for integrating the two-fluid magnetohydrodynamic equations in two dimensions

We describe a new method for solving the time-dependent two-fluid magnetohydrodynamic (2F-MHD) equations in two dimensions that has significant advantages over other methods. The stream-function/potential representation of the velocity and magnetic field vectors, while fully general, allows accurate description of nearly incompressible fluid motions and manifestly satisfies the divergence condition on the magnetic field. Through analytic manipulation, the split semi-implicit method breaks the full matrix time advance into four sequential time advances, each involving smaller matrices. The use of a high-order triangular element with continuous first derivatives (C1 continuity) allows the Galerkin method to be applied without introduction of new auxiliary variables (such as the vorticity or the current density). These features, along with the manifestly compact nature of the fully node-based C1 finite elements, lead to minimum size matrices for an unconditionally stable method with order of accuracy h4. The resulting matrices are compatible with direct factorization using SuperLU_dist. We demonstrate the accuracy of the method by presenting examples of two-fluid linear wave propagation, two-fluid linear eigenmodes of a tilting cylinder, and of a challenging nonlinear problem in two-fluid magnetic reconnection.

A Quasicontinuum Method for Deformations of Carbon Nanotubes

We present a coarse-graining computation for deformations of CNTs (carbon nanotubes) via QC (quasicontinuum), particularly targeting analysis of multi-walled carbon nanotubes. Higher order triangular elements are utilized for proper interpolation of atom po- sitions of the CNT on the basis of QC approach. The computing scheme enables one to differentiate between the fully atomistic zone and the coarse-grained zone in the framework of the multiscale computing. Several nu- merical examples demonstrate the effectiveness and ac- curacy of the present methodology.

Free vibration analysis of isotropic rectangular plates using a high-order triangular finite element with shear

In this paper, free vibration analysis of isotropic rectangular plates with different thickness ratios, different boundary conditions and different aspect ratios has been investigated using a high-order triangular element. The first-order shear deformation theory (FOSDT) is used to include the effect of transverse shear deformation. The element has 18 nodes on the sides and six internal nodes. The geometry of the element is expressed by three linear shape functions of area coordinates. The formulation is displacement type and the use of area coordinates makes the shape functions for field variables to be expressed explicitly. No numerical integration is required to get the element stiffness and consistent mass matrices. The element has 51 degrees of freedom, which can be reduced to 39 degrees of freedom by Guyan reduction, mass condensation, or eigenvalue economization scheme for the degrees of freedom associated with the internal nodes. Rotary inertia has been included in the consistent mass matrix. Numerical examples are presented to show the accuracy and convergence characteristics of the element.

A family of 13-node plate bending triangular elements

This paper investigates the importance and accuracy of a family of high-order triangular elements. After a brief review of characteristics of high-order triangular elements, six new incompatible 13-node triangular elements are presented. The accuracy of the proposed elements is illustrated by comparing their numerical results with the other investigators' solutions, and the best element is introduced. © 1998 John Wiley & Sons, Ltd.

Buckled plate vibrations and large amplitude vibrations using high-order triangular elements

A high-order triangular membrane finite element is combined with a fully conforming triangular plate bending element to solve the geometrically nonlinear problems of plates where the membrane and flexural behaviors are coupled and the effect of the inplane boundary conditions is as significant as the flexural boundary conditions. Each of the three orthogonal displacement components is represented by a two-dimensional polynomial of the same quintic order with no bias against one another giving the element a total of 54 degrees of freedom. The nonlinear stiffness matrices are formulated and a Newton-Raphson iteration procedure is used. Examples include the analyses of plane stresses of a parabolically loaded square plate, large deflections of a square plate under lateral pressure, postbuckling of a square plate, linear free vibration of a buckled rectangular plate, large amplitude free vibration of a square plate with and without inplane stresses. Various flexural and inplane boundary conditions are considered. Results are compared with those obtained by alternative finite element methods, analytical approximate methods, and an experiment. Physical interpretations of the results and explanations of the discrepancies among various solutions are provided. The results indicate that the present development is capable of accurately solving a wide variety of geometrically nonlinear plate problems.

Dual-order parallelogram finite elements for the axisymmetric vector Poisson equation

A new family of parallelogram finite elements, in which the order of approximation can be separately specified for the two pairs of opposing sides, is described. These elements are uniquely useful in problems involving long, narrow subregions and successfully complement high-order triangular elements for a wide range of applications. In addition to their use for narrow subregions they effectively fill up large, open spaces (a traditional use of rectangular elements) and uniquely allow different orders of approximation to be achieved in different regions without sacrificing the conformity of the solution. The assembly of these elements at run-time is accomplished without costly numerical integration by the use of pre-calculated universal element matrices. The element matrices are calculated exactly for one-dimensional elements through sixth order and are less expensive to generate and smaller than the corresponding matrices for triangular elements. Element matrices are given for approximation orders up to three and a sample problem is solved.