Quadratic Triangular Elements Research Articles

Peristaltic mechanism in an inclined asymmetric channel soaked with porous media and under magnetic effects: numerical simulation equipped with finite element method

This study is proposed for the numerical simulation of the peristaltic mechanism using the finite element method with the Galerkin residual technique to explore peristaltic transportation under the action of an externally applied magnetic field passing through an inclined asymmetric channel saturated with porous media. The Navier-stokes equations are solved for the peristaltic mechanism non-linearly by the exemption of lubrication theory to study the inertial effects. The numerical simulations and results are obtained by applying the finite element method based on quadratic triangular elements. It is observed that pressure rise per wavelength is enhanced by increasing the angle of inclination.

Auto Mesh generation algorithm for the convex domain with the triangular elements

Mesh creation is one of the primary tasks in implementing the Finite Element Method (FEM) to solve two- and three-dimensional boundary value problems. However, the whole procedure becomes tedious and problematic when higher-order finite elements are employed to construct mesh and prepare corresponding element data. In this study, we strive to develop a versatile algorithm for discretizing the two-dimensional domain using linear, quadratic, and cubic triangular finite elements. The algorithm is developed based on n vertices (actual or more in number) that constitute the boundary of the domain, along with a computed (n+1)-th point such as the centroid. The algorithm, using this input, generates an initial mesh consisting of n triangular elements. Then, in every subsequent step, the algorithm increases the number of triangles in the mesh fourfold compared to the previous step. The algorithm we present here can employ (a) straight-sided (linear, quadratic, and cubic) triangular elements to generate the mesh for domains with polygonal boundaries and (b) both straight-sided and curved (quadratic and cubic) triangular elements with two straight sides and one curved side to generate meshes for domains with curved boundaries. Thus, the algorithm generates the desired meshes if the vertices are specified once at the initial step. Additionally, the inclusion of the mathematical expression of the curved boundary enables the algorithm to generate the fine mesh for the curved domain utilizing higher-order curved and straight-sided triangular elements. We also present a computer code in MATLAB incorporating the algorithm to create the mesh, prepare the element's data, and determine the element's connectivity.
 GANIT J. Bangladesh Math. Soc. 43.1 (2023) 017- 035

On an efficient octic order sub-parametric finite element method on curved domains

An efficient finite element method using the sub-parametric transformations of the curved boundaries for the 45-node octic order triangular element is presented in this paper. The one-sided curved and two-sided straight triangle in the universal coordinate system (x,y) is mapped onto an isosceles triangle (ξ,η)∈[0,1] and ξ+η≤1 in the (ξ,η)-coordinate system by using an isoparametric coordinate transformation. Using such a transformation, the curved edge of the triangular element is altered implicitly by a newly defined octic curve. Such an octic curve is obtained by using the topology of the curved edge. With an apparent condition of arc being a parabola, a new transformation is developed to find a unique curve for each edge that passes through the points on the original curved arc. Overall, the new sub-parametric transformations are developed, these include the transformation for the curved edge on the original boundary, in addition to the parameters from the curved triangle's interior. Several numerical experiments are presented to show the efficacy of the proposed method. The computational results obtained for a boundary value problem using octic order basis functions outperform the results obtained using the existing higher order basis functions such as quadratic (6-node), cubic (10-node), quartic (16-node), quintic (21-node), sextic (28-node), and septic (36-node) order triangular elements (TEs) with curves. The method implemented in this paper can be easily extendable to quasi-static and dynamic crack evolution problems within linear and nonlinear elasticity.

Dynamic Response of Heat Transfer in Magnetohydrodynamic Blood Flow Through a Porous Bifurcated Artery with Overlapping Stenosis

The influence of heat transfers on magnetohydrodynamic blood flow treated as a Newtonian rheological model in permeable bifurcated arteries with a magnetic field applied in a transverse direction has been presented in this study. The blood flow is considered steady, incompressible, electrically conducting and porous as it consists of fatty accumulation on blood cells. A mathematical model governing such flow phenomena is developed by the Navier-Stokes and energy equations. The governing equations with appropriate boundary conditions are solved numerically by using a stabilized form of the finite element method, called the Galerkin least-squares method. The implementation of this method managed to overcome two sources of undesirable pathologies suffered by the Classical Galerkin method in solving the incompressible flow. This involves the requirement to satisfy the Babuška-Brezzi stability condition and the inherent instability arising from the approximations of highly advective flows. As a result, stable and converged solutions are generated for the velocity, temperature, pressure, wall shear stress and heat transfer coefficient in the constricted bifurcated artery by employing an equal order (quadratic triangular element) of interpolation function for velocity, pressure and temperature subspaces. The developed algorithm is validated with reported findings from previous literature where a close agreement has been achieved. Effects of the porosity parameter and Hartmann number in the presence of heat on blood flow characteristics at several locations along the branch artery are computed and presented graphically. The finding obtained from this work may be beneficial for cryosurgery and hyperthermia treatment in analysing the temperature distributions inside the vessel.

Peristaltic Phenomenon in an Asymmetric Channel Subject to Inclined Magnetic Force and Porous Space.

This research is engaged to explore biological peristaltic transport under the action of an externally applied magnetic field passing through an asymmetric channel which is saturated with porous media. The set of governing partial differential equations for the present peristaltic flow are solved in the absence of a low Reynolds number and long wavelength assumptions. The governing equations are to be solved completely, so that inertial effects can be studied. The numerical simulations and results are obtained by the help of a finite element method based on quadratic six-noded triangular elements equipped with a Galerkin residual procedure. The inertial effects and effects of other pertinent parameters are discussed by plotting graphs based on a finite element (FEM) solution. Trapped bolus is discussed using the graphs of streamlines. The obtained results are also compared with the results given in the literature which are highly convergent. It is concluded that velocity and the number of boluses is enhanced by an increase in Hartmann number and porosity parameter Increasing inertial forces increase the velocity of flow but increasing values of the porosity parameter lead to a decrease in the pressure gradient. The study elaborates that magnetic field and porosity are useful tools to control the velocity, pressure, and boluses in the peristaltic flow pattern.

Shape Optimisation of a Heat Source in a Thermal Convection Field Considering Perimeter Constraint Condition

In this study, we present an investigation of shape optimisation analysis for a heat convection problem taking into account perimeter constraint condition. The incompressible Navier–Stokes equation using the Boussinesq approximation, the equation of continuity and the energy equation are employed for the governing equations in the heat convection field. The mixed interpolation method is applied to solve the flow field, and the quadratic and linear triangular elements are, respectively, employed for the velocity and the pressure. The quadratic triangular element is applied to interpolate the temperature. The purpose of this study is to find the optimal shape of a heat source so as to maximise the quantity of radiation on the outer boundary. The adjoint variable method is applied to obtain the optimal shape, and the perimeter constraint condition for the heat source is considered in this optimisation problem. The perimeter constraint condition is adapted in the traction method.

Numerical simulation and experimental verification of pitting corrosion propagation in sweet pipeline service

This paper has numerically simulated and experimentally verified the pitting corrosion propagation inside a low-alloy carbon steel pipeline for sweet (CO2) petroleum service. In this study, a Finite-Element-Analysis-based mechanistic model was first developed to predict the transient dissolution rate of iron ion (Fe2+) from a pre-existing pit through solving the Nernst-Planck Equation. Specifically, the computational domain combines a hemispherical-shaped pit and a thin laminar boundary layer of an electrolyte solution. The mesh was generated using quadratic triangular elements in the Cartesian Coordinate System, and a moving mesh method was deployed to track the dynamic pitting propagation. The velocity distribution of the electrolyte solution was computed through solving the Navier-Stokes Equations. Distribution of electrochemical potentials was determined based on the Poisson Equation in consideration of electroneutrality whereas a Debye-Hückel approximation was applied to describe the variation of the potentials at the metal-solution interfaces by reason of the existence of the Electrical Double Layer. The distribution of the ionic concentrations of each participating chemical species was obtained through solving Fick’s Second Law. To verify the developed pitting propagation model, a laboratory testing system was established and a series of experimental tests were performed. The results demonstrate that the predicted pitting corrosion growth rates agree well with the experimental observations. The model described herein is able to predict pitting corrosion rates and induction times for the onset of pitting attack or passivation in a given sweet pipeline system set of operating conditions.

Numerical Solution of the Wave Propagation Problem in a Plate

In this work, we use the phase velocity method in combination with finite element method to compute the dispersion curve for phase velocity of an ultrasonic pulse traveling in a thin isotropic plate. This method is based on the numerical solution of the wave propagation equations for several selected frequencies. To solve these equations, a second order difference scheme is used to discretize the temporal variable, while spatial variables are discretized using the finite element method. The variational formulation of the problem corresponding to a fixed value of time is obtained and the existence and uniqueness of the solution is proved. A priori error estimates in the energy norm and in the [Formula: see text] norm are also obtained. The open software FreeFem++ is used with quadratic triangular elements to compute the displacements. Numerical experiments show that the velocities computed from the approximated displacements for different frequency values are in good agreement with analytical dispersion curve. This confirms that the proposed symbiosis between finite element and phase velocity method is suitable for computing dispersion curves in more general wave propagation problems, where the geometry is complex and the material is anisotropic.

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.

Incompatible Graded Finite Elements for Orthotropic Nonhomogeneous Media

Orthotropic materials or composites are challenging to model due to different elastic properties in two in-plane directions, which requires sufficient mesh refinement to attain desired accuracy. In this regard, an accurate and efficient incompatible graded element is developed for modeling orthotropic functionally graded materials. Properties of Graded finite elements such as Young’s moduli (E11, E22), shear modulus (G12) and Poisson’s ratio (v12) vary. A new incompatible graded element is developed using user subroutines in Abaqus and are compared to available exact solutions. In this study, performance (QM6) graded element is compared with lower-order and higher-order compatible elements (Q4 and Q8) as well as linear triangular (T3) and quadratic triangular (T6) elements with aid of several numerical examples. Additionally, orthotropic graded plate with carbon fibers properties as well as a circular disc with orthotropic properties under internal pressure is included. Furthermore, a curved beam with radially graded properties under the bending moment is studied. Additionally, the performance of various graded elements is compared with analytical solutions using Russell error.

A moving mesh method for modelling defects in nematic liquid crystals

The properties of liquid crystals can be modelled using an order parameter which describes the variability of the local orientation of rod-like molecules. Defects in the director field can arise due to external factors such as applied electric or magnetic fields, or the constraining geometry of the cell containing the liquid crystal material. Understanding the formation and dynamics of defects is important in the design and control of liquid crystal devices, and poses significant challenges for numerical modelling. In this paper we consider the numerical solution of a Q-tensor model of a nematic liquid crystal, where defects arise through rapid changes in the Q-tensor over a very small physical region in relation to the dimensions of the liquid crystal device. The efficient solution of the resulting six coupled partial differential equations is achieved using a finite element based adaptive moving mesh approach, where an unstructured triangular mesh is adapted towards high activity regions, including those around defects. Spatial convergence studies are presented using a stationary defect as a model test case, and the adaptive method is shown to be optimally convergent using quadratic triangular finite elements. The full effectiveness of the method is then demonstrated using a challenging two-dimensional dynamic Pi-cell problem involving the creation, movement, and annihilation of defects.

Micro-mechanical investigation of fatigue behavior of Al alloys containing surface/superficial defects

A crystal plasticity finite element (CPFE) simulation framework has been proposed in this study for the prediction of combined detrimental effect of mean stress and defects on the fatigue behavior of aluminum alloy. Experimental data for mean-stress effect on fatigue life and crack growth behavior was obtained on metal inter gas (MIG) welded joints of Al-5083/Al-5.8%Mg alloy plates and has been detailed in authors’ previous work, referred later. The present study focuses on its prediction using computational framework only. A 2D representative model for material’s microstructure was used for the simulations, generated using an anisotropic tessellation algorithm using the EBSD measurements data. A total of 10 different microstructure models were generated for each loading condition using six-node plane strain type quadratic triangular (CPE6) elements of mesh size 6 μm. Two different types of cases were investigated: one without defect and other with a semi-circular surface defect. The simulated loadings at different stress ranges and stress ratios (R-ratio) were similar to the experimental conditions for the better comparison of the results. Significant heterogeneity in the distribution of R-ratios and the far-field applied R-ratio was observed. When defect was not considered, a clear deviation in the predicted fatigue lives from the experimental data was observed at different R-ratios: the predicted fatigue lives were higher than the experimentally observed fatigue lives. This was probably because of not considering the detrimental effect of defects on fatigue lives. But, when the defects were considered, the predicted results for different R-ratios were consistent with the experimental fatigue lives. The proposed CPFE simulation framework not only predicted well the effect of defects and mean stress on the fatigue lives, but also the scatter induced in them due to the defects.

Application of the particle finite element method for large deformation consolidation analysis

Purpose Large deformation problems are frequently encountered in various fields of geotechnical engineering. The particle finite element method (PFEM) has been proven to be a promising method to solve large deformation problems. This study aims to develop a computational framework for modelling the hydro-mechanical coupled porous media at large deformation based on the PFEM. Design/methodology/approach The PFEM is extended by adopting the linear and quadratic triangular elements for pore water pressure and displacements. A six-node triangular element is used for modelling two-dimensional problems instead of the low-order three-node triangular element. Thus, the numerical instability induced by volumetric locking is avoided. The Modified Cam Clay (MCC) model is used to describe the elasto-plastic soil behaviour. Findings The proposed approach is used for analysing several consolidation problems. The numerical results have demonstrated that large deformation consolidation problems with the proposed approach can be accomplished without numerical difficulties and loss of accuracy. The coupled PFEM provides a stable and robust numerical tool in solving large deformation consolidation problems. It is demonstrated that the proposed approach is intrinsically stable. Originality/value The PFEM is extended to consider large deformation-coupled hydro-mechanical problem. PFEM is enhanced by using a six-node quadratic triangular element for displacement and this is coupled with a four-node quadrilateral element for modelling excess pore pressure.

Numerical modeling of rock failure under dynamic loading with polygonal elements

SummaryPolygonal finite elements are gaining an increasing attention in the computational mechanics literature, but their application in rock mechanics is very rare. This paper deals with numerical modeling of rock failure under dynamic loading based on polygonal finite elements. For this end, a damage‐viscoplastic constitutive model for rock based on the Mohr‐Coulomb criterion with the Rankine criterion as a tensile cutoff is employed and implemented with the polygonal finite element method. Moreover, the mineral mesostructure or rock is described by randomly mapping groups of polygonal elements representing the constituent minerals into a global mesh and assigning these groups with the corresponding mineral material properties. The performance of the polygonal elements is compared with that of the linear and quadratic triangular and bilinear quadrilateral elements in numerical simulations of controlled shear band formation under uniaxial compression and lateral splitting failure in the dogbone tension test. Numerical simulations of uniaxial tension and compression tests as well as dynamic Brazilian disc test under increasing loading rates demonstrate that the present approach predicts the correct failure modes as well as the dynamic increase in strength of rock.

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.

Novel quadratic Bézier triangular and tetrahedral elements using existing mesh generators: Applications to linear nearly incompressible elastostatics and implicit and explicit elastodynamics

SummaryIn this paper, we present novel techniques of using quadratic Bézier triangular and tetrahedral elements for elastostatic and implicit/explicit elastodynamic simulations involving nearly incompressible linear elastic materials. A simple linear mapping is proposed for developing finite element meshes with quadratic Bézier triangular/tetrahedral elements from the corresponding quadratic Lagrange elements that can be easily generated using the existing mesh generators. Numerical issues arising in the case of nearly incompressible materials are addressed using the consistent B‐bar formulation, thus reducing the finite element formulation to one consisting only of displacements. The higher‐order spatial discretization and the nonnegative nature of Bernstein polynomials are shown to yield significant computational benefits. The optimal spatial convergence of the B‐bar formulation for the quadratic triangular and tetrahedral elements is demonstrated by computing error norms in displacement and stresses. The applicability and computational efficiency of the proposed elements for elastodynamic simulations are demonstrated by studying several numerical examples involving real‐world geometries with complex features. Numerical results obtained with the standard linear triangular and tetrahedral elements are also presented for comparison.

Displacement and Stress Function-based Linear and Quadratic Triangular Elements for Saint-Venant Torsional Problems

Torsional problems commonly arise in frame structural members subjected to unsym­metrical loading. Saint-Venant proposed a semi inverse method to develop the exact theory of torsional bars of general cross sections. However, the solution to the problem using an analytical method for a complicated cross section is cumbersome. This paper presents the adoption of the Saint-Venant theory to develop a simple finite element program based on the displacement and stress function approaches using the standard linear and quadratic triangular elements. The displacement based approach is capable of evaluating torsional rigidity and shear stress distribution of homogeneous and nonhomogeneous; isotropic, orthotropic, and anisotropic materials; in singly and multiply-connected sections. On the other hand, applications of the stress function approach are limited to the case of singly-connected isotropic sections only, due to the complexity on the boundary conditions. The results show that both approaches converge to exact solutions with high degree of accuracy.

Use of independent volume parameters in the development of new large displacement ANCF triangular plate/shell elements

In this paper, a new unified kinematic description, obtained from Bezier geometry using linear mapping and position vector gradients associated with three independent parameters, is used to develop large displacement plate/shell finite elements (FE). Contrary to the conventional FE method, in the approach developed in this paper based on the absolute nodal coordinate formulation (ANCF), no distinction is made between plate and shell structures. The proposed ANCF triangular plate/shell elements have 12 coordinates per node: three position coordinates and nine position gradient coordinates that define vectors tangent to coordinate lines at the nodes. The fundamental differences between the conventional FE and the new ANCF parameterizations are highlighted. In this investigation, two different parameterizations, each of which employs independent coordinates, are used. In the first parameterization, called volume parameterization, coordinate lines along the sides of the triangular element in the straight (un-deformed) configuration are used in order to facilitate the development of closed-form cubic shape functions. In the second parameterization, called Cartesian parameterization, coordinate lines along the global axes of the structure (body) coordinate system are used to facilitate the element assembly. The element transformation between the volume and the Cartesian parameterizations is developed and used to define the structure inertia and elastic forces. Three new fully parameterized ANCF triangular plate/shell elements are developed in this investigation: a four-node mixed-coordinate element (FNMC) and two three-node elements (TN1 and TN2). All the elements developed in this investigation lead to a constant mass matrix and zero Coriolis and centrifugal forces. A non-incremental total Lagrangian procedure is used for the numerical solution of the nonlinear equations of motion. The performance of the proposed ANCF triangular plate/shell elements is analyzed by comparison with the ANCF rectangular plate element and conventional three-node linear (TNL) and six-node quadratic (SNQ) triangular plate elements.

A three-dimensional coupled thermo-hydro-mechanical model for deformable fractured geothermal systems

A fully coupled thermal-hydraulic-mechanical (THM) finite element model is presented for fractured geothermal reservoirs. Fractures are modelled as surface discontinuities within a three-dimensional matrix. Non-isothermal flow through the rock matrix and fractures are defined and coupled to a mechanical deformation model. A robust contact model is utilised to resolve the contact tractions between opposing fracture surfaces under THM loadings. A numerical model has been developed using the standard Galerkin method. Quadratic tetrahedral and triangular elements are used for spatial discretisation. The model has been validated against several analytical solutions, and applied to study the effects of the deformable fractures on the injection of cold water in fractured geothermal systems.Results show that the creation of flow channelling due to the thermal volumetric contraction of the rock matrix is very likely. The fluid exchanges heat with the rock matrix, which results in cooling down of the matrix, and subsequent volumetric deformation. The cooling down of the rock matrix around a fracture reduces the contact stress on the fracture surfaces, and increases the fracture aperture. Stress redistribution reduces the aperture, as the area with lower contact stress on the fracture expands. Stress redistribution reduces the likelihood of fracture propagation under pure opening mode, while the expansion of the area with lower contact stress may increase the likelihood of shear fracturing.

On the formulation of the planar ANCF triangular finite elements

In this paper, new planar isoparametric triangular finite elements (FE) based on the absolute nodal coordinate formulation (ANCF) are developed. The proposed ANCF elements have six coordinates per node: two position coordinates that define the absolute position vector of the node and four gradient coordinates that define vectors tangent to coordinate lines (parameters) at the same node. To shed light on the importance of the element geometry and to facilitate the development of some of the new elements presented in this paper, two different parametric definitions of the gradient vectors are used. The first parametrization, called area parameterization, is based on coordinate lines along the sides of the element in the reference configuration, while the second parameterization, called Cartesian parameterization, employs coordinate lines defined along the axes of the structure (body) coordinate system. The fundamental differences between the ANCF parameterizations used in this investigation and the parametrizations used for conventional finite elements are highlighted. The Cartesian parameterization serves as a unique standard for the triangular FE assembly. To this end, a transformation matrix that defines the relationship between the area and the Cartesian parameterizations is introduced for each element in order to allow for the use of standard FE assembly procedure and define the structure (body) inertia and elastic forces. Using Bezier geometry and a linear mapping, cubic displacement fields of the new ANCF triangular elements are systematically developed. Specifically, two new ANCF triangular finite elements are developed in this investigation, namely four-node mixed-coordinate and three-node ANCF triangles. The performance of the proposed new ANCF elements is evaluated by comparison with the conventional linear and quadratic triangular elements as well as previously developed ANCF rectangular and triangular elements. The results obtained in this investigation show that in the case of small and large deformations as well as finite rotations, all the elements considered can produce correct results, which are in a good agreement if appropriate mesh sizes are used.