Mid-side Nodes Research Articles

Toward the application of the layer-wise displacement theory in passenger ships—a quasi-static response

This paper introduces an application of the layer-wise displacement theory for response analysis of multilayer ship girders. Quadrilateral, 6-node finite membrane elements are used for which 8 displacement degrees of freedom at corners, and 4 shear strain degrees of freedom at mid-side nodes, are defined. This approach enables accurate estimates of the shear strain between adjacent layers with different stiffness. A simple cantilever beam, non-uniform stepped girder with constant layer stiffness, and uniform girder with varying stiffness is used in the validation analysis. An excellent agreement between the fine mesh finite element model and the proposed model is observed.

Triangular C0 continuous finite elements based on refined zigzag theory {2,2} for free and forced vibration analyses of laminated plates

This study presents a finite element formulation to perform vibration analysis of laminated composite plates based on the Refined Zigzag Theory of order {2,2}, namely RZT-{2,2}. This theory considers the transverse stretching by introducing quadratic through-thickness variations of both in-plane and transverse displacement components. Also, it eliminates the use of shear correction factors and is highly suitable for the analyses of thick and heterogeneous laminated plates. The governing equations of the RZT-{2,2} are derived based on Hamilton’s principle. The stiffness matrix, consistent mass matrix, and load vector of the governing equations are constructed by adopting anisoparametric interpolation functions associated with a triangular element that involves three corner nodes and three mid-side nodes along its edges. The corner nodes consist of eleven kinematic variables while each mid-side node possesses only three deflection components. An adaptive time-stepping algorithm is employed with an optimum time increment determined automatically at each time step, thus, eliminating the stability concerns. The capability of the present approach is demonstrated by considering several benchmark cases regarding free vibration analysis. The effect of transverse stretching is revealed through the dynamic characteristics of the laminated composite plates. In the case of forced vibration analysis, the accuracy of the present approach is established through comparison with three-dimensional finite element models.

Mixed finite element approximations of a singular elliptic problem based on some anisotropic and hp-adaptive curved quarter-point elements

Mixed finite element methods are applied to a Poisson problem with a singularity at a boundary point. The approximation spaces are based on quarter-point elements, the shape functions inheriting the singular behavior of their quadratic geometric maps. Two mesh scenarios are considered, by fixing some macro quarter-point elements at the coarse level, and subdividing them by mapping uniformly refined square meshes on the master element by their corresponding geometric transforms. For eight-noded coarse quadrilateral quarter-point elements, placing two mid-side nodes near the singular vertex, the radial singularity is exactly captured along element edges, and their refinements reveal shape regular curved meshes. For an improved version, using collapsed quadrilateral quarter-point elements obtained by reducing one of the quadrilateral element edges to the singular point, the radial singularity is captured inside the coarse macro elements as well. Their uniform refinement generates anisotropic meshes, grading towards the singular point. The assembly of the required H(div)-conforming approximation spaces based on these kinds of meshes are described. Results for a typical test problem demonstrate superior effectiveness of the proposed techniques for convergence acceleration, when confronted with usual affine finite elements, for h, p and hp-adaptive refinements. Especially, collapsed quarter-point elements applied to the singular problem reveal accuracy rates equivalent to standard regular contexts, of smooth solutions discretized on uniform affine meshes.

Curved domain discretization with boundary defined by polar equations

A domain discretisation procedure for a planar curved domain with boundary in polar equations is presented. The curved domain is split into curved triangles and then to a fine mesh of linear triangles in the interior and curved triangles or linear triangles near to the boundary. Later by inserting midside nodes to these triangles 6-node triangles obtained further each one into four triangles. The mesh conformity is preserved by applying similar procedure to every triangle of the domain. This procedure is applied to discretize the star shaped curved domain or cracked convex curved domains into all triangles and then into all quadrilaterals. Thus we generate a triangular and a quadrangular finite element mesh. The refinements to the mesh are obtained by increasing the number of divisions of the boundary curve. This discretization of curved domains will reduce the computational complications in the evaluation of integrals, which has lot of practical applications.

New elements with harmonic shape functions in adaptive mesh refinement

New elements are proposed in the present work based on the harmonic coordinates in which the shape functions satisfy the Laplace equation. The harmonic functions have some appealing characteristics that made it possible to define elements with arbitrary shape functions on the element boundaries and arbitrary node arrangement. In the present work, without loss of generality, we formulate a set of quadrilateral elements with midside nodes and piecewise linear shape functions along the element boundaries. The analytical solution of Laplace equation in rectangular domains are used here to represent the harmonic shape function as Fourier series. To derive the shape functions of different elements with different node arrangements a systematic approach is proposed to represent all of the shape functions using a small set of harmonic functions. Two model boundary value problems, Poisson equation and linear elasticity are used here to evaluate the proposed method. Patch test and convergence analysis are done for each model problems via some examples. The elements with midside nodes have great potential to be used in mesh adaptive solutions. Therefore, some adaptive mesh refinement examples are solved based on the ZZ error estimation and quadtree grid structure as the mesh refinement technique.

Adaptive mesh refinement schemes for the localizing gradient damage method based on biquadratic-bilinear coupled-field elements

In the present work, adaptive h-refinement schemes are developed for the localizing gradient damage method. In general, the existing adaptive refinement schemes have been developed using the bilinear quadrilateral (Q4) elements for the weakly coupled systems such as XFEM and Phase field method. In contrast to this, the presently developed refinement schemes are designed for simultaneous refinement of the strongly coupled biquadratic serendipity (Q8) and Q4 elements, which is a typical requirement of the localizing gradient damage method. This work addresses mainly two types of numerical complexities in the presently developed refinement schemes, first one is the strong coupling of primary variables in the localizing gradient damage method and other is the presence of mid-side nodes in the Q8 element. A detailed comparative study of three adaptive refinement schemes i.e. hanging nodes based, template based and multiscale FEM based, is carried out. The suitability and effectiveness of the refinement schemes are demonstrated through the study of various parameters like damage profile, load-displacement response, discretization error and computation time. Additionally, a comparison matrix of implementation complexity based on developed algorithm and coding is also presented to highlight the difficulties in their implementation. A novel oversampling data transfer scheme is proposed for the transfer of Gauss point’s state variables from unrefined to refined mesh.

Study on transient heat flux intensity factor with interaction integral

Aim at determining of the heat flux intensity factor (HFIF) near a crack tip in structures under transient heat flux load, the transient JT integral is defined firstly and proved to be path-independent. Based on the path-independence of the transient JT integral, a transient interaction integral utilized to extract the transient HFIF is established by introducing a known auxiliary field. A singular isoparametric quadrilateral element with eight nodes, which is used to discretize the structure around the crack tip, by moving the midside nodes to one-fourth of the sides is presented. With this singular element, the 1/r singularity of heat flux near the crack tip can be exactly depicted. Numerical examples are carried out to verify the path-independence of the transient JT integral and the interaction integral method to extract the transient HFIF. Moreover, the transient HFIF in a structure with multiple cracks are numerically investigated by means of the presented method. The method proposed in this paper can be extended to investigate the thermo-mechanical problems.

Division model‐based distortion correction method for head‐mounted displays

AbstractIn this paper, we present a simple and efficient method for correcting the asymmetric and nonlinear geometrical distortion of a head‐mounted display (HMD). The method divides the object space into a number of quadratic triangular elements and applies a quadratic predistortion for each image section so that it can handle the distortion with any complex shapes caused by the decentering and tilted optics and improve the calibration accuracy. The errors introduced in the process of fabricating and assembling can also be eliminated. We investigated the use of a quadratic division model and two types of linear division models on an off‐axis HMD to correct the optical distortion. Experimental results demonstrated that the quadratic division model converged at a faster rate and produced higher accuracy over the typical linear model. The average root of mean square error (RMSE) after distortion correction was one pixel, and the maximum standard deviations in all rows and columns were 0.67 pixel and 0.57 pixel. In addition, the deformation continuity is maintained by using the quadratic element with connected midside nodes.

A New Approach to Automesh Generation of all β graded triangular and quadrilateral Finite Elements over Analytical Surfaces by using the Parabolic Arcs passing through four points on the Boundary Curve

This paper presents a new mesh generation method for a simply connected curved domain of a planar region which has curved boundary described by one or more analytical equations. We first decompose this curved domain into simple sub regions in the shape of curved triangles. These simple regions are then triangulated to generate a fine mesh of linear triangles in the interior and curved triangles near to the boundary of curved domain. These simple regions are then triangulated to create 6-node triangles by inserting midside nodes to these triangles. Each isolated 6-node triangle is then split into four triangles according to the usual scheme, that is, by using straight lines to join the midside nodes. To preserve the mesh conformity a similar procedure is also applied to every triangle of the domain to fully discretize the given convex curved or cracked convex curved domains into all triangles, thus propagating refinements and .The quadrangulation of graded 3-node linear triangles is done by inserting three midside nodes and a centroidal node.Then each graded triangle is split into three quadrilaterals by using straight lines to join the centroid to the midside nodes. This simple method generates a high quality mesh whose elements confirm well to the requested shape by refining the problem domain. We have approximated the curved boundary arcs by equivalent parabolic arcs.To preserve the mesh conformity, a similar procedure is also applied to every triangle of the domain to fully discretize the given curved domain into all triangles and quadrilaterals, thus propagating a uniform refinement. This simple method generates a high quality mesh whose elements confirm well to the requested shape by refining the problem domain. Examples of a circular disk and a cracked circular disk are presented to illustrate the simplicity and efficiency of the new mesh generation method. We have appended the MATLAB programs which incorporate the mesh generation scheme developed in this paper. These programs provide valuable output on the nodal coordinates ,element connectivity and graphic display of the all triangular and quadrilateral mesh for application to finite element analysis.

A New Approach to Automatic Generation of All Graded Triangular and Quadrilateral Finite Element Mesh Over Polygonal Domains

This paper presents a new automesh generation method for a convex and cracked convex polygonal domains. We first decompose the polygon into simple sub regions in the shape of triangles. These simple regions are then triangulated to generate a mesh of graded 3-node triangular elements. We propose then an automatic 6-node triangluar conversion scheme by inserting midside nodes to these triangles. Each isolated 6-node triangle is split into four triangles according to the usual scheme,that is,by joining the midside nodes by straight lines. To preserve the mesh conformity a similar procedure is also applied to every triangle of the domain to fully discretize the given convex or cracked convex polygonal domain into all triangles, thus propagating a uniform or graded refinement. The quadrangulation of graded 3-node linear triangles is done by inserting three midside nodes and a centroidal node.Then each graded triangle is split into three quadrilaterals by joining the centroid to the midside nodes. This simple method generates a high quality mesh whose elements confirm well to the requested shape by refining the problem domain. Examples are presented to illustrate the simplicity and efficiency of the new mesh generation method for standard and arbitrary shaped polygonal domains. We have appended MATLAB programs which incorporate the mesh generation scheme developed in this paper. These programs provide valuable output on the nodal coordinates ,element connectivity and graphic display of the all triangular and quadrangular meshes for application to finite element analysis.

A simple and robust quadrilateral non-conforming element with a special 5-point quadrature

Abstract A 4-node, 8-DOF non-conforming quadrilateral element with four internal modes, denoted as iQ8, is developed. It takes the four edges' midpoints of the linear element Q4 as the internal virtual nodes, whose shape functions are the same as those of the mid-side nodes of the quadratic element Q8. In order to deal with the rank deficiency of the new element stiffness matrix under 2 × 2 Gaussian quadrature, a new reduced 5-point integration scheme is constructed. By both the eigenvalue analysis of the element stiffness matrix and the example for the patch test, the weight of the central integration point is determined to a value close to 4, correspondingly the 5-point quadrature converges to 1-point quadrature. The non-central integration points are far away from the element domain, which is much different from conventional quadratures. Finally, some typical numerical examples are tested, and the results show that, except for only passing the weak patch test, the new element achieves the best performance of other 4-node quadrilateral elements in the aspects of anti-distortion, being free of trapezoidal locking, and exactness in pure bending.

Improved nine-node shell element MITC9i with reduced distortion sensitivity

The 9-node quadrilateral shell element MITC9i is developed for the Reissner-Mindlin shell kinematics, the extended potential energy and Green strain. The following features of its formulation ensure an improved behavior: 1. The MITC technique is used to avoid locking, and we propose improved transformations for bending and transverse shear strains, which render that all patch tests are passed for the regular mesh, i.e. with straight element sides and middle positions of midside nodes and a central node. 2. To reduce shape distortion effects, the so-called corrected shape functions of Celia and Gray (Int J Numer Meth Eng 20:1447–1459, 1984) are extended to shells and used instead of the standard ones. In effect, all patch tests are passed additionally for shifts of the midside nodes along straight element sides and for arbitrary shifts of the central node. 3. Several extensions of the corrected shape functions are proposed to enable computations of non-flat shells. In particular, a criterion is put forward to determine the shift parameters associated with the central node for non-flat elements. Additionally, the method is presented to construct a parabolic side for a shifted midside node, which improves accuracy for symmetric curved edges. Drilling rotations are included by using the drilling Rotation Constraint equation, in a way consistent with the additive/multiplicative rotation update scheme for large rotations. We show that the corrected shape functions reduce the sensitivity of the solution to the regularization parameter gamma of the penalty method for this constraint. The MITC9i shell element is subjected to a range of linear and non-linear tests to show passing the patch tests, the absence of locking, very good accuracy and insensitivity to node shifts. It favorably compares to several other tested 9-node elements.

Analysis of strain gradient Reissner–Mindlin plates using a C0 four-node quadrilateral element

The weak form of Reissner–Mindlin plate model in the context of the classical continuum theory requires C0-continuity of the lateral deflection as well as rotation fields. However, due to the presence of higher-order derivatives, the same plate model based on strain gradient elasticity demands C1-continuity of interpolation functions for a standard finite element (FE) formulation. On the other hand, it is known that C0 interpolation functions result in more stable and cost-effective elements. Accordingly, a four-node quadrilateral gradient-enhanced plate element using C0 interpolation of the field variables is introduced. The main idea is to start with an eight-node quadrilateral element on which any field variable and its spatial derivatives are independently interpolated via standard C0 shape functions. By introducing appropriate algebraic constraints, the degrees of freedom corresponding to the midside nodes of the eight-node element are then eliminated and a quadrilateral element with four nodes on its corners is constructed. Accuracy and performance of the proposed element in several examples and for a wide range of parameters is investigated. It is shown that the new element can successfully capture the size-dependent behaviour of plates at small scales, has a proper rank (contains no spurious zero-energy modes), passes the patch test for thin as well as thick plates in an arbitrary mesh, and is free of shear locking. Furthermore, the introduced element can reproduce the results of the Reissner–Mindlin plate model based on the classical continuum theory when the plate thickness is far greater than the material length scale parameter.

An efficient algebraic multigrid method for quadratic discretizations of linear elasticity problems on some typical anisotropic meshes in three dimensions

SummaryThe quality of the mesh used in the finite element discretizations will affect the efficiency of solving the discreted linear systems. The usual algebraic solvers except multigrid method do not consider the effect of the grid geometry and the mesh quality on their convergence rates. In this paper, we consider the hierarchical quadratic discretizations of three‐dimensional linear elasticity problems on some anisotropic hexahedral meshes and present a new two‐level method, which is weakly independent of the size of the resulting problems by using a special local block Gauss–Seidel smoother, that is LBGS_v iteration when used for vertex nodes or LBGS_m iteration for midside nodes. Moreover, we obtain the efficient algebraic multigrid (AMG) methods by applying DAMG (AMG based on distance matrix) or DAMG‐PCG (PCG with DAMG as a preconditioner) to the solution of the coarse level equation. The resulting AMG methods are then applied to a practical example as a long beam. The numerical results verify the efficiency and robustness of the proposed AMG algorithms. Copyright © 2015 John Wiley & Sons, Ltd.

Testing of the Kriging-based Finite Element to Shell Structures with Varying Thickness

A variant of the finite element method with Kriging basis functions has been recently developed and applied to plane, plate bending, and shell elastostatic problems. The main advantage of this novel method is that high degree of basis functions can be easily constructed without additional finite element nodes (such as mid-side and inner nodes). This paper revisits the formulation of the Kriging-based finite element method for analysis of curved shell structures, referred to as K-Shell, and presents new numerical tests to shell structures with varying thickness. The K-Shell was formulated based on degenerated 3D elasticity theory. The basis functions were constructed using a set of nodes covering several layers of triangular elements and were employed to approximate both the displacement and geometry. The quartic polynomial basis was chosen to alleviate shear and membrane locking. The K-Shell has been tested using a series of shell benchmark problems with constant shell thickness. The tests showed that K-Shell performed very well in analyzing moderately thick smooth shells. In this paper the K-Shell is tested using two modified shell benchmark problems with varying thickness, i.e. the modified pinched cylinder and the modified hemispherical shell with 18 degrees cut-off. The converged results of three-dimensional finite element model using commercial software Abaqus were used to assess the convergence of the results. The tests show that the performance of K-Shell in analyzing shells with varying thickness is satisfactory.

An Efficient Finite Element Modeling of Composite Stiffened Shells

This paper presents an efficient finite element modeling technique for stiffened composite shells having different stiffening arrangements. The laminated shell skin is modeled with a triangular degenerated curved shell element having 3 corner nodes and 3 mid-side nodes. An efficient curved beam element compatible with the shell element is developed for the modeling of stiffeners which may have different lamination schemes. The formulation of the 3 nod degenerated beam element may be considered as one of the major contributions. The deformation of the beam element is completely defined in terms of the degrees of freedom of shell elements and it does not require any additional degrees of freedom. As the usual formulation of degenerated beam elements overestimates their torsional rigidity, a torsion correction factor is introduced for different lamination schemes. Numerical examples are solved by the proposed finite element technique to assess its performance.

Fatigue Crack Growth under Mode I, II and III for Plane-strain and Plane-stress Conditions

Fatigue crack growth (FCG) could be encountered in many mechanical components, which can be made from either thin or thick steel plates (or shells) and, therefore, be subjected to a plane-stress or a plane-strain condition, respectively. The loads applied in a solid body containing a narrow notch or a sharp crack will induce a yield zone near its tip with a dimension that will depend on the mechanical properties of the material, as well as on the thickness of the body, the crack length and the magnitude of the loads applied. Crack propagation can then occur under mode I, II, III or mixed-mode for general loading.The paper presents JI, JII and JIII integral functions, which were correlated with the elastic stress intensity factors KI, KII and KIII, for thin and thick CT specimens. The evaluation of J-Integral values was carried out for different crack lengths, along the crack front, and using the Finite Element Method (FEM), with collapsed nodes and midside nodes dislocated to ¼ of the edge's length, in order to simulate the crack tip singularity. Interaction between in-plane, in-plane sliding and out-of-plane modes are also discussed in the paper. In addition, FCG rates under mode I, mode III and a mixed-mode (mode I+III) were experimentally determined, at room temperature, for a high-strength Cr-Mn austenitic stainless steel.

C0-continuous triangular plate element for laminated composite and sandwich plates using the {2, 2} – Refined Zigzag Theory

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 C0 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.

Analysis of Multilayered/Sandwich Shells Using a New Discrete Finite Element Model

The model DDM (Discrete Displacement Mindlin), leads to a finite element which is geometrically simple: 4-node quadrilateral with 5 doffs per node for a shell and efficient. The mid-side rotational nodes, derived from a quadratic interpolation of normal rotations, are eliminated using a combination of local discrete kinematic and constitutive Mindlin hypotheses. The derived 4-node element is free of shear locking and passes all patch tests for thick and thin plates in arbitrary mesh. The applications concern the static and dynamic analysis of sandwich and multilayered shells.

Numerical Evaluation of Stress Intensity Factor for Inclined-Edge Crack Geometry using Singularity Elements

Edge cracks are more dangerous than interior cracks. Free edge close to the crack influences the stress field near the crack tip (since the free edge is traction free).In case of edge crack, the free edge is not only close to the crack, but it intersects the crack. Evaluation of Stress Intensity factor for oblique edge crack geometry is done using commercial FEM software ANSYS and compared with analytical results. The conventional elements always underestimate the sharply rising stress-displacement gradients near the crack tip. Therefore, in order to produce this singularity in stresses and strains Barsoum elements were employed which involves shifting the mid-side nodes to the quarter point locations. It observed that stress intensity factor found by FEM method has good agreement with analytical results.. The singular order of stresses near the interface is a good way of understanding of failure initiation; however, in engineering applications usually the knowledge of singular orders is not enough for the prediction of failure initiation. As an example in the case of homogeneous cracks, the singular order is -1/2 which remains constant irrespective of the surrounding environment and outside loading of the crack. These influential factors are reflected through the associated stress intensity factor of the cracks. Hwu and Kuo [1] have demonstrated several different kinds of examples such as cracks in homogenous isotropic or anisotropic materials, central or edge notches in isotropic materials, interface cracks and interface corners between two dissimilar materials. It is also shown that KI Evaluation Using Displacement etrapolation technique under adaptive dense mesh with Parallel Finite Element gives fairly accurate results [2]. Based upon the analytical solutions obtained previously for the multi-bonded anisotropic wedges and the well-known path-independent H-integral, Hwo and Kuo [3] provided a unified definition and a stable computing approach for the stress intensity factors of interface corners. The stress intensity factor calculations are usually limited to Linear Elastic Fracture Mechanics (LEFM). For a linear elastic material the stress and strain fields ahead of the crack tip are expressed as [4]. The singular order of stresses near the interface is a good way of understanding of failure initiation; however, in engineering applications usually the knowledge of singular orders is not enough for the prediction of failure initiation. As an example in the case of homogeneous cracks, the singular order is -1/2 which remains constant irrespective of the surrounding environment and outside loading of the crack. These influential factors are reflected through the associated stress intensity factor of the cracks. Hwu and Kuo [1] have demonstrated several different kinds of examples such as cracks in homogenous isotropic or anisotropic materials, central or edge notches in isotropic materials, interface cracks and interface corners between two dissimilar materials. It is also shown that KI Evaluation Using Displacement etrapolation technique under adaptive dense mesh with Parallel Finite Element gives fairly accurate results [2]. Based upon the analytical solutions obtained previously for the multi-bonded anisotropic wedges and the wellknown path-independent H-integral, Hwo and Kuo [3] provided a unified definition and a stable computing approach for the stress intensity factors of interface corners. The stress intensity factor calculations are usually limited to Linear Elastic Fracture Mechanics (LEFM). For a linear elastic material the stress and strain fields ahead of the crack tip are expressed as [4].