Lagrangian Element Research Articles

Comparison of weak and strong formulations for 3D stress predictions of composite beam structures

Accurate full field stress responses are often necessary to predict the structural performance of composite structures. The Unified Formulation (UF) is a promising approach to realise efficient and accurate 3D stress predictions of beam-like structures by using recently proposed hierarchical Serendipity Lagrange Elements (SLE) for capturing the 2D response of the beam cross-section while the 1D behaviour along the beam axis is captured using the Finite Element Method (FEM). Despite the computational merits of SLE elements, the performance of UF-SLE-FEM model is strongly influenced by FEM mesh discretisation along the beam's longitudinal axis. With respect to multi-layered beam structures, a high density FEM mesh may lead to loss of efficiency of the UF-SLE-FEM model due to a significant increase in the number of degrees of freedom required to obtain convergence. This study proposes high-order refined formulations of the 1D structure based on strong-form, Differential Quadrature Method (DQM) and weak form, FEM for 3D stress analysis of composite structures. The proposed high-order strong-form DQM and weak-form FEM models, which are benchmarked against exact solutions, lead to significant computational advantages over UF-SLE-FEM models of similar accuracies. However, the symmetry and positive definiteness of the stiffness matrices of FEM models offer a numerical advantage over the strong-form DQM model with non-symmetric, non-positive definite stiffness matrices.

Large elastic deformation of micromorphic shells. Part II. Isogeometric analysis

In Part I of this study, a variational formulation was presented for the large elastic deformation problem of micromorphic shells. Using the novel matrix-vector format presented for the kinematic model, constitutive relations, and energy functions, an isogeometric analysis (IGA)-based solution strategy is developed, which appropriately estimates the macro- and micro-deformation field components. Due to the capability of constructing exact geometries and the powerful mesh refinement tools, IGA can be successfully applied to solve the equilibrium equations with dominant nonlinear terms. It is known that different types of locking phenomena take place in the conventional finite element analysis of thin shells based on low-order elements. Non-standard finite element models with mixed interpolation schemes and additional degrees of freedom (DOFs) or the ones used the high-order Lagrangian shell elements which require high computational costs, are the available solutions to tackle locking issues. The present 16-DOFs IGA is found to be efficient because of possessing a good rate of convergence and providing locking-free stable responses for micromorphic shells. Such a conclusion is found from several comparative studies with available data in the well-known macro-scale benchmark problems based on the classical elasticity as well as the corresponding numerical examples studied in nano-scale beam-, plate-, cylindrical shell- and spherical shell-type structures on the basis of the micromorphic continuum theory.

Quadrilateral 2D linked-interpolation finite elements for micropolar continuum

Quadrilateral finite elements for linear micropolar continuum theory are developed using linked interpolation. In order to satisfy convergence criteria, the newly presented finite elements are modified using the Petrov–Galerkin method in which different interpolation is used for the test and trial functions. The elements are tested through four numerical examples consisting of a set of patch tests, a cantilever beam in pure bending and a stress concentration problem and compared with the analytical solution and quadrilateral micropolar finite elements with standard Lagrangian interpolation. In the higher-order patch test, the performance of the first-order element is significantly improved. However, since the problems analysed are already describable with quadratic polynomials, the enhancement due to linked interpolation for higher-order elements could not be highlighted. All the presented elements also faithfully reproduce the micropolar effects in the stress concentration analysis, but the enhancement here is negligible with respect to standard Lagrangian elements, since the higher-order polynomials in this example are not needed.

Natural Frequency Analysis of Functionally Graded Orthotropic Cross-Ply Plates Based on the Finite Element Method

This paper aims to present a finite element (FE) formulation for the study of the natural frequencies of functionally graded orthotropic laminated plates characterized by cross-ply layups. A nine-node Lagrange element is considered for this purpose. The main novelty of the research is the modelling of the reinforcing fibers of the orthotropic layers assuming a non-uniform distribution in the thickness direction. The Halpin–Tsai approach is employed to define the overall mechanical properties of the composite layers starting from the features of the two constituents (fiber and epoxy resin). Several functions are introduced to describe the dependency on the thickness coordinate of their volume fraction. The analyses are carried out in the theoretical framework provided by the first-order shear deformation theory (FSDT) for laminated thick plates. Nevertheless, the same approach is used to deal with the vibration analysis of thin plates, neglecting the shear stiffness of the structure. This objective is achieved by properly choosing the value of the shear correction factor, without any modification in the formulation. The results prove that the dynamic response of thin and thick plates, in terms of natural frequencies and mode shapes, is affected by the non-uniform placement of the fibers along the thickness direction.

Nonlinear transient analysis of delaminated curved composite structure under blast/pulse load

The nonlinear time-dependent displacement values of the curved (single/doubly) composite debonded shell structure are examined under different kinds of pulse loading in this research. The structural curved panel model is derived mathematically using the higher-order displacement theories containing the thickness stretching effect, whereas the sub-laminate approach is adopted for the inclusion of delamination between the subsequent layers. The structural geometry distortion under variable loading has been included in the current theoretical analysis through Green–Lagrange type of strain kinematics. Further, the governing differential equation order has been reduced with the help of 2D finite element formulation via the nine-noded isoparametric Lagrangian elements with variable degrees of freedom (eighty-one and ninety) for two different higher-order kinematics, respectively. The final equation of motion is solved computationally to evaluate the transient responses through an original computer code including the direct iterative technique and Newmark’s average acceleration method. The convergence criteria of the current numerical solution are established as a priori and the subsequent validity is demonstrated via comparing the current responses with available published data. Further, the comprehensive behavior of the debonded structure under the influence of the variable loads (time and area dependent) is evaluated by solving different numerical illustrations for variable geometrical configuration and described in detail.

Performance of a Two-stages Gas Gun: Experimental, Analytical and Numerical Analysis

Two stages gas guns are used for various purposes, particularly for mechanical characterization of materials at high rate of deformations. The performance of a two stages gas gun is studied in this work using the theory of the two-stage gas gun proposed by Rajesh, numerical simulation using combined Eulerian/ Lagrangian elements in Autodyna commercial code and experiment using a two stage gas gun developed by the authors of this study. Equations governing the motion of the piston and projectile are solved using Runge-Kutta method. The effects of parameters such as chamber pressure, pump tube pressure and piston mass on the performance of gun are explored. The results of numerical simulation and analytical methods are validated by experiment. Finally, a comparison between the analytical, numerical and experimental results shows that the theory proposed by Rajesh, yields reasonable predictions for the two stage gas gun performance in the first place, and Autodyn software, using combined Eulerian/ Lagrangian elements, gives accurate estimations for gas gun parameters, in the second place. A 3-D working diagram is provided for prediction of projectile velocity for any state of pump and chamber pressures which are the most important variables for a gas gun with a fixed geometry. Two stages gas guns are used for various purposes, particularly for mechanical characterization of materials at high rate of deformations. The performance of a two stages gas gun is studied in this work using the theory of the two-stage gas gun proposed by Rajesh, numerical simulation using combined Eulerian/ Lagrangian elements in Autodyna commercial code and experiment using a two stage gas gun developed by the authors of this study. Equations governing the motion of the piston and projectile are solved using Runge-Kutta method. The effects of parameters such as chamber pressure, pump tube pressure and piston mass on the performance of gun are explored. The results of numerical simulation and analytical methods are validated by experiment. Finally, a comparison between the analytical, numerical and experimental results shows that the theory proposed by Rajesh, yields reasonable predictions for the two stage gas gun performance in the first place, and Autodyn software, using combined Eulerian/ Lagrangian elements, gives accurate estimations for gas gun parameters, in the second place. A 3-D working diagram is provided for prediction of projectile velocity for any state of pump and chamber pressures which are the most important variables for a gas gun with a fixed geometry.

Modeling Ice Shelf Cavities and Tabular Icebergs Using Lagrangian Elements

AbstractMost ocean climate models do not represent ice shelf calving in a physically realistic way, even though the calving of icebergs is a major component of the mass balance for Antarctic ice shelves. The infrequency of large calving events together with the difficulty of placing observational instruments around icebergs means that little is known about how calving icebergs affect the ocean. In this study we present a novel model of an ice shelf coupled to an ocean circulation model, where the ice shelf is constructed of Lagrangian elements that allow simulation of iceberg calving. The Lagrangian ice shelf model is used to simulate the flow beneath a static idealized ice shelf, to verify that it can reproduce the results of an existing Eulerian model simulation with an identical configuration. The Lagrangian model is then used to simulate the ocean's response to a calved iceberg drifting away from the ice shelf. The results show how a calving event and subsequent iceberg drift affect the ocean. At the ice front, the calving event leads to a warming of the ocean surface and cooling of the water column at depth, allowing cooler waters to enter the ice shelf cavity, leading to reduced melt rates within the cavity. A Taylor column is observed below the iceberg, which moves with the iceberg as it drifts into the open ocean. As the iceberg drifts further from the ice shelf, the circulation within the ice shelf cavity tends toward a new steady state, consistent with the new ice shelf geometry.

Analysis of finite element methods for vector Laplacians on surfaces

Abstract We develop a finite element method for the vector Laplacian based on the covariant derivative of tangential vector fields on surfaces embedded in ${\mathbb{R}}^3$. Closely related operators arise in models of flow on surfaces as well as elastic membranes and shells. The method is based on standard continuous parametric Lagrange elements that describe a ${\mathbb{R}}^3$ vector field on the surface, and the tangent condition is weakly enforced using a penalization term. We derive error estimates that take into account the approximation of both the geometry of the surface and the solution to the partial differential equation. In particular, we note that to achieve optimal order error estimates, in both energy and $L^2$ norms, the normal approximation used in the penalization term must be of the same order as the approximation of the solution. This can be fulfilled either by using an improved normal in the penalization term, or by increasing the order of the geometry approximation. We also present numerical results using higher-order finite elements that verify our theoretical findings.

A highly-accurate finite element method with exponentially compressed meshes for the solution of the Dirichlet problem of the generalized Helmholtz equation with corner singularities

In this study, a highly-accurate, conforming finite element method is developed and justified for the solution of the Dirichlet problem of the generalized Helmholtz equation on domains with re-entrant corners. The k − t h order Lagrange elements are used for the discretization of the variational form of the problem on exponentially compressed polar meshes employed in the neighbourhood of the corners whose interior angle is α π , α ≠ 1 ∕ 2 , and on the triangular and curved mesh formed in the remainder of the polygon. The exponentially compressed polar meshes are constructed such that they are transformed to square meshes using the Log-Polar transformation, simplifying the realization of the method significantly. For the error bound between the exact and the approximate solution obtained by the proposed method, an accuracy of O ( h k ) , h mesh size and k ≥ 1 an integer, is obtained in the H 1 -norm. Numerical experiments are conducted to support the theoretical analysis made. The proposed method can be applied for dealing with the corner singularities of general nonlinear parabolic partial differential equations with semi-implicit time discretization.

A Fully-Mixed Finite Element Method for then-Dimensional Boussinesq Problem with Temperature-Dependent Parameters

AbstractIn this paper, we introduce and analyze a high-order, fully-mixed finite element method for the free convection ofn-dimensional fluids,n∈{2,3}{n\in\{2,3\}}, with temperature-dependent viscosity and thermal conductivity. The mathematical model is given by the coupling of the equations of continuity, momentum (Navier–Stokes) and energy by means of the Boussinesq approximation, as well as mixed thermal boundary conditions and a Dirichlet condition on the velocity. Because of the dependence on the temperature of the fluid properties, several additional variables are defined, thus resulting in an augmented formulation that seeks the rate of strain, pseudostress and vorticity tensors, velocity, temperature gradient and pseudoheat vectors, and temperature of the fluid. Using a fixed-point approach, smallness-of-data assumptions and a slight higher-regularity assumption for the exact solution provide the necessary well-posedness results at both continuous and discrete levels. In addition, and as a result of the augmentation, no discrete inf-sup conditions are needed for the well-posedness of the Galerkin scheme, which provides freedom of choice with respect to the finite element spaces. In particular, we suggest a combination based on Raviart–Thomas, Lagrange and discontinuous elements for which we derive optimal a priori error estimates. Finally, several numerical examples illustrating the performance of the method and confirming the theoretical rates of convergence are reported.

A family of optimal Lagrange elements for Maxwell’s equations

In this paper we propose and study a new Lagrange finite element method for the two-dimensional Maxwell’s equations. Its solution may be singular because of the nonsmooth domain with reentrant corners. The proposed method allows the standard Lagrange elements on barycentric refinements of any order greater than or equal to two. We analyze the proposed method for the eigenvalue problem and the indefinite source problem, obtaining the well-posedness and optimal error estimates. Numerical results are presented for confirming the theoretical results.

Continuous mixed finite elements for the second order elliptic equation with a low order term

We propose a mixed finite element, where the velocity (in terms of Darcy’s law) is approximated by the continuous P k Lagrange elements and the pressure (the prime variable) is approximated by the discontinuous P k − 1 elements, for solving the second order elliptic equation with a low-order term. We show the quasi-optimality for this mixed finite element method. When a low order term is present, the traditional inf–sup condition is no longer required. But the inclusion condition, that the divergence of the discrete velocity space is a subspace of the discrete pressure space, is required. Thus the Taylor–Hood element and most other continuous-pressure mixed elements do not converge. Numerical tests are provided on the new elements and most other popular mixed elements.

Deformation characteristics of sinusoidally-corrugated laminated composite panel – A higher-order finite element approach

In this article, the deformation characteristics of sinusoidally-corrugated laminated composite panels are examined under uniform and sinusoidally distributed transverse loads. The sinusoidal corrugation in laminate structure is achieved using the general curvature method. The displacement field is based on the third-order shear deformation mid-plane kinematics theory with nine degrees-of-freedom. The minimum total potential energy method is employed to obtain the governing equation. The solutions are obtained using two-dimensional finite element approximation via nine-noded isoparametric quadrilateral Lagrangian element. The convergence and comparison studies reveal the consistency and accuracy of present model. The new numerical illustrations reflect the significance of corrugation ratios, edge constraints, side-to-thickness ratios and aspect ratios on the deformation behavior of the corrugated and non-corrugated laminated composite panels.

Numerical analysis of a time domain elastoacoustic problem

AbstractThis paper deals with the numerical analysis of a system of second order in time partial differential equations modeling the vibrations of a coupled system that consists of an elastic solid in contact with an inviscid compressible fluid. We analyze a weak formulation with the unknowns in both media being the respective displacement fields. For its numerical approximation, we propose first a semidiscrete in space discretization based on standard Lagrangian elements in the solid and Raviart–Thomas elements in the fluid. We establish its well-posedness and derive error estimates in appropriate norms for the proposed scheme. In particular, we obtain an $\mathrm L^{\infty }(\mathrm L^2)$ optimal rate of convergence under minimal regularity assumptions of the solution, which are proved to hold for appropriate data of the problem. Then we consider a fully discrete approximation based on a family of implicit finite difference schemes in time, from which we obtain optimal error estimates for sufficiently smooth solutions. Finally, we report some numerical results, which allow us to assess the performance of the method. These results also show that the numerical solution is not polluted by spurious modes as is the case with other alternative approaches.

Numerical Approximation of a Non-Newtonian Flow with Effect Inertial

V iscoplastic fluids are materials of great interest both in industry and in our daily lives. These applications range from food and cosmetics products to industrial applications such as plastics in the industry of polymers and drilling muds the oil industry. This class of material is characterized by having a yield stress that must be exceeded to the material starts to flow. These fluids are classically predicted by purely viscous models with yield stress. In the last decade, however, there some experimental visualizations has reported that the unyielded regions exhibit elasticity inside. This work is an attempt to investigate the effect of elasticity and inertia in those materials. We will studied, therefore, inertia flow of elastic-viscoplastic materials with no thixotropic behavior, according to the material equation introduced in de Souza Mendes (2011). The mechanical model is approximated by a stabilized finite element method in terms of extra stress, pressure and velocity. Due to its fine convergence feature, the method allows the use of equal-order finite elements and generates stable solutions in high advective-dominated flows. In this study is considered the geometry of a biquadratic cavity, in which the top wall moves to the right at constant velocity. In all computations is used biquadratic Lagrangian (Q1) elements. Results focuses in determining the influence of elasticity and inertia on the position and shape of unyielded. These results proved to be physically meaningful, indicating a strong interlace between elasticity and inertia on determining of the topology of yield surfaces.

New Mixed Elements for Maxwell Equations

New inf-sup stable mixed elements are proposed and analyzed for solving the Maxwell equations in terms of electric field and Lagrange multiplier. Nodal-continuous Lagrange elements of any order on simplexes in two- and three-dimensional spaces can be used for the electric field. The multiplier is compatibly approximated always by the discontinuous piecewise constant elements. A general theory of stability and error estimates is developed; when applied to the eigenvalue problem, we show that the proposed mixed elements provide spectral-correct, spurious-free approximations. Essentially optimal error bounds (only up to an arbitrarily small constant) are obtained for eigenvalues and for both singular and smooth solutions. Numerical experiments are performed to illustrate the theoretical results.

A higher-order conformal decomposition finite element method for plane B-rep geometries

A higher-order accurate conformal decomposition finite element method (CDFEM) is proposed, using an explicit boundary description of the domain. A level-set concept is introduced which converts the explicit B-rep geometry based on NURBSs to an implicit representation. This concept also applies to higher-order fictitious domain methods (FDMs) which are often based on implicit descriptions. In the proposed CDFEM, an arbitrary, non-conforming background mesh is introduced and manipulated to capture non-smooth features of the boundary such as corners and to ensure shape-regular elements. Elements that are cut by the boundary are decomposed into conforming sub-elements. The final mesh composed by higher-order Lagrange elements enables higher-order accurate approximations in the context of the FEM. Special emphasis is on the fact that NURBSs feature reduced continuity between knot spans which must be properly considered in the mesh generation, not only at corners.

A new nine-node element for analysing plates with varying thickness using basic displacement functions

The capability of the Finite Element Method in producing accurate and efficient results largely depends on the shape functions adopted to frame the displacement field inside the element. In this paper, a new nine-node Lagrangian element was developed to analyse thin plates with varying cross-sections using the shape functions obtained for non-prismatic straight beams with minimum number of elements. The formulated shape functions, which represent vertical displacements and rotations throughout elements, are rooted from a purely mechanical functions called Basic Displacement Functions (BDFs). These functions are obtained by implementing the force method in Euler–Bernoulli beam theory, which ensures that equilibrium equation is satisfied in all interior points of elements. To verify the competency of the proposed element, solutions for the static analysis of isotropic rectangular plates under various loading conditions, together with free vibration analysis of plates with linear thickness variation were obtained and compared with the previous literature. Results showed that the proposed nine-node Lagrangian element was computationally more cost-effective compared to other competing methods when small number of elements is employed.

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.

Use of explicit finite-element formulation to predict the rolling radius and slip of an agricultural tire during travel over loose soil

Theoretically, there is zero slip between two bodies when there is no relative motion in their contact points. In the contact between a wheel and a surface, zero slip can be obtained only in the case of a single contact point. In this case, the wheel and the surface must be rigid. The theoretical zero-slip condition can’t be obtained in the contact between tire and terrain surface. In much of the scientific literature, two alternatives are suggested for a practical definition of the zero-slip condition: the point at which the gross traction force is equal to zero, or the point at which the net traction force is equal to zero. In the ASABE (2013), there is still no unique definition for the practical zero-slip condition. According to the definition of zero-slip condition, the rolling radius is not constant and depends on the slip.A detailed finite-element model using Lagrangian elements was built for each tire, taking into account the effect of all tire materials and their arrangement, lug shape, and inflation pressure. The soil model was built with Eulerian elements, which allow a large degree of deformation and flow of the soil. The initial verification experiments of the tire models were conducted by pressing the tires against a rigid plane. Each tire was examined under several different inflation pressures. Very good correlations were obtained between the experimental and model results. The verification test for the gross and net traction forces was performed in the soil-bin laboratory at the Technion. Special equipment was built, including a heavy dragging platform and a cell to hook the tire. This equipment allows control of the tire slip. The net traction force, gross traction force, and vertical load were measured in each test. Good correlations were obtained between the experimental and model results.Using the FEM model developed, some definitions for zero-slip condition were examined. The results indicate that the best criterion for zero-slip condition is definition of the point at which the gross traction force is equal to zero.