Constant Strain Triangular Elements Research Articles

Thermal-Mechanical Coupling Model Based on the Hybrid Finite Element Method for Solving Bipolar the Plate Deformation of Hydrogen Fuel Cells

New energy is the focus of attention all over the world, and research into new energy can inject new vitality into the industrial system. Hydrogen fuel cells are not only environmentally friendly, but also rich in reserves that can be used as a strategic resource for the entire country. The difficulty lies in the safe design of application equipment and the batch generation and storage of hydrogen. In addition, fuel cells have the disadvantage of a slow start-up. Based on the above problems, this paper proposes a hybrid-element method to solve the thermal-mechanical coupling model of fuel cell plate, which can effectively solve the thermal stress change, temperature field distribution and displacement change of the battery plate when working. Firstly, the hybrid-element algorithm is given for 2D plate deformation. Then, the deformation application of a 3D fuel cell plate is given. The 2D numerical results show that the hybrid finite element method (FEM) is more flexible for realizing the flexible combination of sub-mesh and finite element basis functions, and has a better mesh quality compared to the traditional constant strain triangular element (CST) adaptive FEM and quadrilateral isoparametric element (Q4) adaptive FEM. This method achieves a balance between numerical accuracy and solving efficiency for the multi-porous elastic plate. In addition, a deformation control formula is given which can display the displacement deformation and stress merge to same graph, since it is convenient to quickly compare the regions where the displacement and stress extremum appear. In short, the hybrid finite element method proposed in this paper has good mesh evaluation results, and when the number of discrete elements is equivalent, the hybrid element converges faster and the solution efficiency is higher. This paper also provides a good numerical theory and simulation reference for industrial mechanics and new energy applications.

Implementing the Node Based Smoothed Finite Element Method as User Element in Abaqus for Linear and Nonlinear Elasticity

In this paper, the node based smoothed-strain Abaqus user element (UEL) in the framework of finite element method is introduced. The basic idea behind of the node based smoothed finite element (NSFEM) is that finite element cells are divided into subcells and subcells construct the smoothing domain associated with each node of a finite element cell [Liu, Dai and Nguyen-Thoi (2007)]. Therefore, the numerical integration is globally performed over smoothing domains. It is demonstrated that the proposed UEL retains all the advantages of the NSFEM, i.e., upper bound solution, overly soft stiffness and free from locking in compressible and nearly-incompressible media. In this work, the constant strain triangular (CST) elements are used to construct node based smoothing domains, since any complex two dimensional domains can be discretized using CST elements. This additional challenge is successfully addressed in this paper. The efficacy and robustness of the proposed work is obtained by several benchmark problems in both linear and nonlinear elasticity. The developed UEL and the associated files can be downloaded from https://github.com/nsundar/NSFEM.

Two-Dimensional Numerical Model of the Fracture Process in Steel Fibre Reinforced Concrete with the Continuum Strong Discontinuity Approach and Functional Data Analysis

Abstract This paper presents the formulation of a two-dimensional numeri-cal model able to describe the fracture process in structural mem-bers of steel fibre reinforced concrete (SFRC) from the volume ratio of the fibres and the mechanical properties of the compo-nents: a concrete matrix and a set of steel fibres with a random orientation. The relationship between the stress and the strain fields of the composite material is obtained using the mixture theory with a compatibility strain of its component materials. The concrete matrix is represented with a scalar damage constitutive model with a softening strain and a different strength in tension and compression. The mechanical strain of an insulated fibre and the slip between the fibre and the matrix are simultaneously de-scribed with a one-dimensional plasticity constitutive model. The cracking of the composite material indicates a jump in the dis-placement field and non-bounded values of the strain field, which are represented by the Continuum Strong Discontinuity Ap-proach. The model has been implemented in the framework of the nonlinear analysis with the Finite Element Method, using con-stant strain triangular elements. Moreover, the fibres distribution and orientation change randomly in each finite element and each simulation or observation. The structural responses of the simula-tions are treated as curves and analysed by tools from the Func-tional Data Analysis. Confidence intervals for the structural re-sponse are built using bootstrap methodology. Finally, experi-mental tests of SFRC members subjected to tension and bending are simulated. The structural response and the cracking patterns obtained from the numerical simulation are satisfactory.

Crack propagation in dynamics by embedded strong discontinuity approach: Enhanced solid versus discrete lattice model

In this work we propose and compare the two models for crack propagation in dynamics. Both models are based on embedded strong discontinuities for localized cohesive type crack description and both provide the advantage to not to require tracking algorithms. The first one is based on discrete lattice approach, where the domain is discretized with Voronoi cells held together prior to crack occurrence by cohesive links represented in terms of Timoshenko beams. The second one is based on constant strain triangular solid element. In both models, propagation of cracks activates enhancements in the displacement field leading to embedded strong discontinuities. The latter remain localized inside the element, regulated by the localized traction separation behavior defined through exponential softening law. Thus, the both models provide the result that remain mesh-independent, with fracture energy as the model parameter. We show that implementation in dynamics framework can be obtained by adding inertial effects without modifying the existing quasi-statics models. In order to provide reliable results, classical implicit Newmark algorithm can be used for time integration. The two presented models are subjected to dynamic crack propagation benchmarks, where detailed analysis on strain, kinetic, plastic free and dissipated energy during simulation is verified by comparison to the amount of total work which is introduced into the system. The main strength of the proposed approach is that cracks initiation, propagation, their coalescence, merging and branching are automatically obtained without any tracking algorithms. In addition, since the discontinuities remain localized inside elements, accurate results can be obtained even with coarser grids, leading to efficient methodology capable of capturing complex crack patterns in dynamics.

GraFEA: a graph-based finite element approach for the study of damage and fracture in brittle materials

In this paper the conventional finite element method for linear elastic response is reformulated in such a way that makes it favorable for the study of damage and fracture in brittle materials. This modified finite element framework is based on the idea presented by Reddy and Srinivasa (Finite Elem Anal Des 104:35–40, 2015), where it was shown that for discretized hyperelastic materials, the magnitude of the nodal forces (in the discretized form) can be written in terms of the axial strains along the edges of the elements and that the equilibrium equations at each node can be written in terms of the forces along the edges alone. Using this concept and by exploiting the fact that FEM discretization leads to an undirected cyclic graph with nodes and edges whose connectivity is related to the elements of the FEM, one can reformulate the displacement-based finite element framework with constant strain triangular elements to represent the continuum as a nonlocal network. The network representation of the continuum is “nonlocal” in the sense that the force along any given edge doesn’t only depend on the strain along that edge, but on a collective behavior of the strains along the edges neighboring the edge of interest. This method is named as GraFEA (for graph-based finite element analysis). Damage is introduced using a nonlocal damage criterion originating from the idea of the weakest links statistics proposed by Lin, Evans, and Ritchie (J Mech Phys Solids 34(5):477–497, 1986). This idea, which was very successful in studying cleavage fracture of mild steel at very low temperatures, can be used to impose a damage criterion to the nolocal network. GraFEA has the major advantage that one can impose an edge-based failure criterion using the weakest link thoery directly on the discretized body, and potentially simulating crack initiation, crack growth, and branching without the need for extra enrichment functions (as with other methods). The simplicity of the method and the fact that it is based on conventional finite element method makes it suitable for integration into commercial softwares. The governing equations for this approach are derived and applied to two simple crack growth simulations (as a proof of concept) in two-dimensional regions with a hole.

The stress analysis of a shear wall with matrix displacement method

Finite element method (FEM) is an effective quantitative method to solve complex engineering problems. The basic idea of FEM for a complex problem is to be able to find a solution by reducing the problem made simple. If mathematical tools are inadequate to obtain precise result, even approximate result, FEM is the only method that can be used for structural analyses. In FEM, the domain is divided into a large number of simple, small and interconnected sub-regions called finite elements. FEM has been used commonly for linear and nonlinear analyses of different types of structures to give us accurate results of plane stress and plane strain problems in civil engineering area. In this paper, FEM is used to investigate stress analysis of a shear wall which is subjected to concentrated loads and fundamental principles of stress analysis of the shear wall are presented by using matrix displacement method in this paper. This study is consisting of two parts. In the first part, the shear wall is discretized with constant strain triangular finite elements and stiffness matrix and load vector which is attained from external effects are calculated for each of finite elements using matrix displacement method. As to second part of the study, finite element analysis of the shear wall is made by ANSYS software program. Results obtained in the second part are presented with tables and graphics, also results of each part is compared with each other, so the performance of the matrix displacement method is demonstrated. The solutions obtained by using the proposed method show excellent agreements with the results of ANSYS. The results show that this method is effective and preferable for the stress analysis of shell structures. Further studies should be carried out to be able to prove the efficiency of the matrix displacement method on the solution of plane stress problems using different types of structures.

Bouc-Wen Type Hysteretic Plane Stress Element

In this work, a plane-stress element is proposed for the elastoplastic dynamic analysis of two-dimensional structures exhibiting hysteretic behavior. The constant-strain triangular element formulation for the elastic case is modified by introducing the Bouc-Wen hysteretic model, properly defined in the 2D stress-strain space. Solutions are obtained by simultaneously solving two sets of governing equations, namely the global equilibrium equations and the local constitutive equations, using a predictor-corrector differential equation solver. In following the proposed method, the linearization of the constitutive relations usually performed in step-by-step solution approaches is avoided. The proposed element is capable of modeling cyclic induced phenomena such as stiffness degradation and strength deterioration. Furthermore, the solution method implemented improves the accuracy of the results without increasing the computational cost of the analysis. Examples are presented which demonstrate the efficiency of...

A SIMPLE CO-ROTATIONAL GEOMETRICALLY NON LINEAR MEMBRANE FINITE ELEMENT WRINKLING ANALYSIS

Thin pre-tensioned membranes are often used in civil architecture as well as in marine and space technologies. Lacking in-plane compression stiffness, membranes will wrinkle at certain states of stress. This paper presents a geometrically nonlinear analysis of membranes in the presence of wrinkling using a unique incremental formulation that accounts for equilibrium in the deformed state and the current wrinkling state by iterations. The membranes with wrinkling are simulated by a geometrically nonlinear upgraded version of the constant strain triangular (CST) membrane finite element. Each load step is comprised of two iteration cycles: the geometrically nonlinear cycle that ensures equilibrium in the deformed sate using the Newton–Raphson iterations and the wrinkling cycle that identifies the location and direction of the wrinkles and redistributes stresses accordingly. Finally, validation and verification of the proposed analysis is made by comparing the present results with those existing for the benchmark examples.

Stress Analysis for Rock Mass Failure with Offset Joints

Two-dimensional finite element model was created in this work to investigate the stress distribution within rock-like samples with offset open non-persistent joints under uniaxial loading. The results of this study have explained the fracture mechanisms observed in tests on rock-like material with open non-persistent offset joints (Mughieda and Alzo’ubi, Geotech Geol Eng J 22:545–562, 2004). Finite element code SAP2000 was used to study the stresses distribution within the specimens. Four-noded isoperimetric plain strain element with two degrees of freedom per node, and the three-noded constant strain triangular element with two degree of freedom per node were used in the present study. The results of the present study showed that the tensile stress in the bridge area caused coalescence for specimens with overlapped preexisting cracks (joints) while the coalescence of the secondary cracks, due to shear stress, caused the failure of specimens with non-overlapping cracks.

Motion analysis of 3D membrane structures by a vector form intrinsic finite element

In this paper, a constant strain triangular membrane element, which is a new member of the VFIFE (vector form intrinsic finite element) family, is proposed to analyze space motion of arbitrary 3D membrane structures. A description of kinematics to dissect rigid body and deformation displacements, and a set of deformation coordinates for each time increment to describe deformation and internal nodal forces are provided. Similar to other members of the VFIFE family, a convected material frame and explicit time integration for the solution procedures are also adopted. Five numerical examples are presented to demonstrate the performance and applicability of the proposed element on the motion analysis of 3D membrane structures. It was found that the proposed element could go through the patch tests and gives stable, convergent and accurate results.

Allman's triangle, rotational DOF and partition of unity

AbstractA simplification of the 1984 Allman triangle (one of historically most important elements with rotational dofs) is presented. It is found that this old element takes a typical form of the partition of unity approximation. The Allman's rotation presented in the partition of unity form offers merits and convenience in formulation and practical applications. The stiffness matrix of the 1984 Allman triangle, which is originally computed from the linear strain triangular element (the 6 nodes quadratic triangle), can be obtained instead in a cheaper way from that of the constant strain triangular element. The constraint of the rotational terms during essential boundary treatment, which remains equivocal and ambiguous, is understood to be mandatory. The partition of unity notion enables a straightforward extension of the Allman's rotational dof to meshfree approximations. In numerical examples, we discuss suppression of spurious zero‐energy modes and patch tests. Standard benchmarks are carried out to assess performance of the newly formulated triangle and a meshfree approximation with the rotational dofs. Copyright © 2006 John Wiley & Sons, Ltd.

Study on the degeneration of quadrilateral element to triangular element

AbstractIn this paper, the problems involved in the process of degeneration of quadrilateral element into triangular element are thoroughly analysed. The contents include the formulation of the geometry mapping induced by collapsing one side of the quadrilateral element and the construction of the shape functions. The study focuses first on a 4‐node bilinear quadrilateral (Q4) element to 3‐node constant strain triangular (CST) element, and then on a 8‐node serendipity (Q8) element to 6‐node triangular element (T6). In the analysis, the quadrilateral element and degenerate triangular element are assumed to be enclosed by straight edges. The theoretical results show that there is another better approach to realize the degeneration, and that even for conventional approach of degeneration we can give more reasonable explanation to the unclear problems like the CST property in degenerate CST element and the necessity of the additional terms in degenerate T6 element. Copyright © 2004 John Wiley & Sons, Ltd.

Analysis of Tube Free Hydroforming Using an Inverse Approach With FLD-Based Adjustment of Process Parameters1

This paper describes an inverse approach (IA) formulation for the analysis of tubes under free hydroforming conditions. The IA formulation is derived from that of Guo et al. established for flat sheet hydroforming analysis using constant strain triangular membrane elements. First, an incremental analysis of free hydroforming for a hot-dip galvanized (HG/Z140) DP600 tube is performed using the Marc finite element code. The deformed geometry obtained at the last converged increment is then used as the final configuration in the inverse analysis. This comparative study allows an assessment of the predictive capability of the inverse analysis. The results are compared with the experimental values determined by Asnafi and Skogsga¨rdh. After that, a procedure based on a forming limit diagram (FLD) is described as a means to adjust the process parameters such as the axial feed and internal pressure. Finally, the adjustment process is illustrated through a re-analysis of the same tube using the inverse approach.

Warpage of Injection-Molded Thermoplastics Parts: Numerical Simulation and Experimental Validation

A mathematical model and the integrated simulation program for predicting part warpage, which arises during injection molding of thermoplastic polymers, were developed. To build the model, the finite element method (FEM) was used and the theory of shells, represented as an assembly of flat elements formed by combining the constant strain triangular element and the discrete Kirchhoff triangular element, was applied. This shell theory is well suited for thin injection molded products of complex shape. Furthermore, in this work, experimental research on flat plates was performed to study the effect of the plastic material, the mold structure, and certain key processing parameters on warpage. The investigated processing parameters include injection pressure, packing pressure, injection time, packing time, cooling time, and melting temperature. Different examples were presented to examine the developed simulation software, and results predicted from the simulation program were verified in the above experiments and showed reasonable accuracy compared with experiment data.

Automatic Plotting of Principal Stress Trajectories by Direction Function Contouring.

We propose a new algorithm to plot principal stress trajectories in plane systems. Much time and labor with experience are required in order to draw trajectories from the result of the photoelastic investigations. The possibility of branching or joining of trajectories cannot be disregarded, when the method of tracking principal stress direction from element to element is used. Our algorithm, on the contrary, plots the principal stress trajectories as the contours of the direction function which is evaluated from the result of FEM using the constant strain triangular elements. Direction function is a two dimensional scalar field whose value is constant along principal stress trajectories. Two function values for the first and the second principal stress of each node are computed by solving the simultaneous liner equations with rectangular sparse matrix. The unit direction vectors of principal stress given by FEM determine the matrix. When the scalar field forms a spiral surface, the unique value of the direction potential of each node is obtained by putting the discontinuity along a polygonal line in the finite element mesh. The line density of the trajectories with broken line pattern in the output figure is read as the magnitude of the principal stress declining toward the free boundary.

Failure mode transition speeds in an impact loaded prenotched plate with four thermoviscoplastic relations

Four different thermoviscoplastic relations, namely, the Litonski–Batra, Johnson–Cook, Bodner–Partom and the power law are used to model the thermoviscoplastic response of a material. Each one of these relations accounts for strain hardening, strain-rate hardening and thermal softening of the material. The material parameters in these relations are found by solving an initial-boundary-value problem corresponding to simple shearing deformations so that the computed effective stress vs. the effective plastic strain curves match closely with the experimental data of Marchand and Duffy who tested thin-walled HY-100 steel tubes in torsion. These four viscoplastic relations are used to analyze dynamic thermomechanical deformations of a prenotched plate impacted on the notched side by a cylindrical projectile made of the same material as the plate. The impact loading on the contact surface is simulated by prescribing the time history of the normal component of velocity and null tangential tractions. A plane strain state of deformation is assumed to prevail in the plate and its deformations are studied for different values of the impact speed. The in-house developed finite element code employs constant strain triangular elements, one point integration rule, and a lumped mass matrix. The Lagrangian description of motion is used to describe deformations of the plate. The coupled nonlinear partial differential equations are first reduced to coupled nonlinear ordinary differential equations (ODEs) by using the Galerkin approximation. The ODEs are integrated by using the stiff solver, LSODE, which adaptively adjusts the time step size and computes the solution within the prescribed accuracy. Results computed with the four constitutive relations are found to be qualitatively similar to each other and the general trends agree with the experimental observations in the sense that at low speed of impact, a brittle failure ensues at a point on the upper surface of the notch tip. However, at high impact speeds, a ductile failure in the form of a shear band initiates first from a point on the lower surface of the notch tip. The predicted speed at which the failure mode transitions from brittle to ductile is different for the four viscoplastic relations. The effects of the notch tip radius and the yield strength of the material in a quasistatic simple tension/compression test on the failure-mode transition speed have been ascertained.

Refined non-conforming triangular elements for analysis of shell structures

Based on the refined non-conforming element method, simple flat triangular elements with standard nodal displacement parameters are proposed for the analysis of shell structures. For ensuring the convergence of the elements a new coupled continuity condition at the inter-element has been established in a weaker form. A common displacement for the inter-element, an explicit expression of refined constant strain matrix, and an adjustable constant are introduced into the formulation, in which the coupled continuity requirement at the inter-element is satisfied in the average sense. The non-conforming displacement function of the well-known triangular plate element BCIZ [1] and the membrane displacement of the constant strain triangular element CST [2] are employed to derive the refined flat shell elements RTS15, and the refined flat shell elements RTS18 is derived by using the element BCIZ and the Allman's triangular plane element [3] with the drilling degrees of freedom. A simple reduced higher-order membrane strain matrix is proposed to avoid membrane locking of the element RTS18. An alternative new reduced higher-order strain matrix method is developed to improve the accuracy of the elements RTS15 and RTS18. Numerical examples are given to show that the present methods have improved the accuracy of the shell analysis. Copyright © 1999 John Wiley & Sons, Ltd.

Finite-element analysis and design of thin sheet superplastic forming

A 3-dimensional finite-element method using a constant strain triangular membrane element for modeling non-Newtonian sheet flow has been developed. Meanwhile, microstructure variation (such as grain growth and cavity growth) during superplastic forming is considered. An adaptive algorithm is presented for the calculation of the back pressure and the bulging pressure to control cavity growth and distribution and to maintain the optimum deformation mode. Then a simple and effective pre-forming design method is presented and applied to control the sheet forming and to obtain components with a more reasonable wall-thickness distribution. Finally, examples are presented for the forming of a square box and the cover of an electric device, with pre-forming design.

Evaluation of the FRANCO finite element fuel rod analysis code

Knowledge of the temperature distribution in a nuclear fuel rod is required to predict the thermal and mechanical response of the strongly temperature-dependent fuel elements. In this research, the FRANCO ( F inite Element Fuel R od AN alysis CO de) computer code was developed for use on an IBM-PC to predict thermal and mechanical fuel performance characteristics for reactor operating conditions. A cross-sectional area of a fuel rod is discretized using constant strain triangular elements. To simplify the time-dependent fuel performance analysis, this code uses quasi-steady state conditions or slow power changes for time-independent problems; thus, time-dependent cases can be represented by quasi-static “snap shots.” The development and testing of the FRANCO program for deformation and heat transfer problems is described. The fuel performance models and solution procedures are provided in detail. The accuracy of FRANCO is examined through selected benchmark problems with other nuclear industry finite difference and finite element fuel analysis codes and actual Halden experimental measurements. These benchmarks show that FRANCO performs well except for the fuel-clad contact cases when compared to results from other fuel performance codes, such as FREY and ESCORE. The FRANCO program developed in this research effort is easy to run, fast, and yields comparable results.

Upper-bound limit analysis of a rigid-plastic body with frictional interfaces

A finite element formulation of the upper-bound theorem for rigid-plastic solids, generalized to include interfaces with finite friction, is described. As proved by Collins [ J. Mech. Phys. Solids 17, 323 (1969)], the usual definition of a kinematically admissible velocity field is unnecessarily restrictive when the upper-bound theorem is applied to many practical problems. This paper shows that a relaxed inequality can be used successfully to derive upper bounds in the presence of Coulomb friction on interfaces, provided one considers a wide enough class of “admissible” velocity fields. One of the major advantages of using a numerical formulation of the upper-bound theorem is that both complex loading geometry and inhomogeneous material behaviour can be easily dealt with. Using a suitable linear approximation of the yield surface, the application of the necessary boundary conditions, the plastic flow rule and the yield criterion lead to a large linear programming problem. The numerical procedure uses constant-strain triangular elements with the unknown velocities as the nodal variables. An additional set of unknowns, the plastic multiplier rates, is associated with each element. Kinematically admissible velocity discontinuities are permitted along specified planes within the finite element mesh. During the solution phase, an active set algorithm is used to solve the linear programming problem.