Lagrangian Element Research Articles

A Morley–Wang–Xu Element Method for a Fourth Order Elliptic Singular Perturbation Problem

A Morley–Wang–Xu (MWX) element method with a simply modified right hand side is proposed for a fourth order elliptic singular perturbation problem, in which the discrete bilinear form is standard as usual nonconforming finite element methods. The sharp error analysis is given for this MWX element method. And the Nitsche’s technique is applied to the MXW element method to achieve the optimal convergence rate in the case of the boundary layers. An important feature of the MWX element method is solver-friendly. Based on a discrete Stokes complex in two dimensions, the MWX element method is decoupled into one Lagrange element method of Poisson equation, two Morley element methods of Poisson equation and one nonconforming \(P_1\)–\(P_0\) element method of Brinkman problem, which implies efficient and robust solvers for the MWX element method. Some numerical examples are provided to verify the theoretical results.

Review of the Science behind Oil Spill Fate Models: Are Updates Needed?

AbstractOil spill models play an important role in the oil spill response decision making and contingency planning processes. The current generation of spill models mostly use Lagrangian based particle tracking random walk methods for oil transport processes combined with individual algorithms for oil fate processes (Spaulding, 2017). The fate of near surface oil movement is modeled using algorithms describing oil spreading, evaporation, emulsification, entrainment, dissolution, and biodegradation. These fate processes are applied to oil in the Lagrangian particle tracking elements to alter the physical and chemical properties of the oil, and subsequently the oil behavior.In this paper, we review the major algorithms used in oil spill models and identify the science and physics underpinning them. For each, we evaluated how far the science has advanced since the algorithms were developed to identify those that could be upgraded based on current understanding. We also identified algorithms where future research is needed because the physical and chemical behaviors are not fully understood. These areas include the spreading behavior of surface slicks, surface-slick emulsification, and the physical transport of small oil droplets near the air-water interface.

Gradient recovery-based adaptive stabilized mixed FEM for the convection–diffusion–reaction equation on surfaces

In this article, we consider a gradient recovery-based adaptive stabilized mixed finite element method (FEM) for solving the convection–diffusion–reaction equation on surfaces. By defining a new total flux on the surface, we firstly transform the original equation into an equivalent mixed first-order formulation. Then, by using both the convection–convection term and the div–div term to stabilize the weak formulation, we prove the discrete inf–sup condition and the error estimate for the mixed FEM. In this new method, the equal order Lagrangian elements are allowed for the primal variable and the total flux, and a high accurate numerical solution for the flux in the H1 space is obtained. The impact of the curvature on the algorithm is also investigated. Additionally, to overcome the low resolution and the non-physical oscillation in simulating the convection-dominated problem, we study a gradient recovery-based adaptive stabilized mixed FEM. Finally, we show some numerical experiments to verify the theoretical prediction and efficiency of the studied method.

Geometrically nonlinear flexural analysis of multilayered composite plate using polynomial and non-polynomial shear deformation theories

In this paper, the geometrically nonlinear bending analysis of multilayered composite plates is carried out through a rigorous comparative study employing Green-Lagrange and von Kármán nonlinearity. The responses are analyzed for various transverse loads and boundary conditions, and emphasized the impending structural condition, which requires the consideration of full geometric nonlinearity. The study is conducted using a C0 finite element plate model using third-order and non-polynomial shear deformation theories (TSDT and NPSDT). The principle of virtual work is utilized to formulate the weak-form of governing equations, and discretized using nine-noded Lagrange elements with seven degrees-of-freedom per node. The performance of the present model has been validated by comparing present results with those available in the literature and obtained ANSYS results. The results reveal that the consideration of Green-Lagrange nonlinearity is essential for plates with all sides subjected to movable simply supported boundary conditions or with one free edge. The present study also provides a clear idea about the utilization of TSDT and NPSDT for the anti-symmetric and symmetric cross-ply plate, respectively, to get more accurate solution. Further, the effect of penalty for various theories and problems is also highlighted, and its prominent impact is asserted for some cases.

Development of the CSOMIO Coupled Ocean-Oil-Sediment- Biology Model

The fate and dispersal of oil in the ocean is dependent upon ocean dynamics, as well as transformations resulting from the interaction with the microbial community and suspended particles. These interaction processes are parameterized in many models limiting their ability to accurately simulate the fate and dispersal of oil for subsurface oil spill events. This paper presents a coupled ocean-oil-biology-sediment modeling system developed by the Consortium for Simulation of Oil-Microbial Interactions in the Ocean (CSOMIO) project. A key objective of the CSOMIO project was to develop and evaluate a modeling framework for simulating oil in the marine environment, including its interaction with microbial food webs and sediments. The modeling system developed is based on the Coupled Ocean-Atmosphere-Wave-Sediment Transport model (COAWST). Central to CSOMIO’s coupled modeling system is an oil plume model coupled to the hydrodynamic model (Regional Ocean Modeling System, ROMS). The oil plume model is based on a Lagrangian approach that describes the oil plume dynamics including advection and diffusion of individual Lagrangian elements, each representing a cluster of oil droplets. The chemical composition of oil is described in terms of three classes of compounds: saturates, aromatics, and heavy oil (resins and asphaltenes). The oil plume model simulates the rise of oil droplets based on ambient ocean flow and density fields, as well as the density and size of the oil droplets. The oil model also includes surface evaporation and surface wind drift. A novel component of the CSOMIO model is two-way Lagrangian-Eulerian mapping of the oil characteristics. This mapping is necessary for implementing interactions between the ocean-oil module and the Eulerian sediment and biogeochemical modules. The sediment module is a modification of the Community Sediment Transport Modeling System. The module simulates formation of oil-particle aggregates in the water column. The biogeochemical module simulates microbial communities adapted to the local environment and to elevated concentrations of oil components in the water column. The sediment and biogeochemical modules both reduce water column oil components. This paper provides an overview of the CSOMIO coupled modeling system components and demonstrates the capabilities of the modeling system in the test experiments.

Effect of material heterogeneity on the deformation behaviour of multidirectional (1D/2D/3D) functionally graded composite panels

PurposeThe purpose of this study is the comprehensive numerical assessment of multidirectional (1D/2D/3D) functionally graded composite panel structures with different material gradation patterns and degrees of material heterogeneity. Here, deformation characteristics are obtained under different loading and support conditions.Design/methodology/approachThe finite element solutions of multidirectional functionally graded composite panels subjected to uniform and sinusoidal transverse loads are presented under different support conditions. Here, different functionally graded composites, such as unidirectional (1D) and multidirectional (2D/3D), are considered by distributing constituent materials in one, two and three directions, respectively, using single and multivariable power-law functions. A constitutive model with fully spatial-dependent elastic stiffness is developed, whereas the kinematics of the present structure is defined using equivalent single-layer higher-order theory. The weak form, based on the principle of virtual work, is established and solved consequently using isoparametric finite element approximations via quadrilateral Lagrangian elements.FindingsThe appropriate mesh-refinement process is carried out to achieve the mesh convergence; whereas, the correctness of proposed heterogeneous model is confirmed through a verification test. The comprehensive numerical assessment of multidirectional functionally graded panels under various loading and support conditions depicts the importance of degree of material heterogeneity with different gradation patterns and volume-fraction exponents.Originality/valueA comprehensive analysis on the deformation behaviour of 1D-functionally graded materials (FGMs) (X-FGM, Y-FGM and Z-FGM), 2D-FGMs (XY-FGM, YZ-FGM and XZ-FGM) and 3D-FGM composite panels FGM structures is presented. Multifaceted heterogeneous FGMs are modelled by varying constituent materials in one, two and three directions, using power-law functions. The constitutive model of multi-directional FGM is developed using fully spatial-dependent elastic matrix and higher-order kinematics. Isoparametric 2D finite element formulation is adopted using quadrilateral Lagrangian elements to model 1D/2D/3D-FGM structures and to obtain their deflection responses under different loading and support conditions.

Comparison of second-order serendipity and Lagrange tetrahedral elements for nonlinear explicit methods

This paper evaluates the performances of second-order finite elements for nodal lumped-mass explicit methods in nonlinear solid dynamics, with a particular emphasis on 10-node “serendipity” and 15-node “Lagrange” tetrahedral elements. Historically, many nonlinear explicit finite element codes have exclusively used first-order elements, until a fairly recent flurry of activity that has resulted in higher-order elements becoming available in explicit codes including the authors' in-house one, ParaAble, and the production software EPIC, IMPETUS, LS-DYNA, and Abaqus. A major attractiveness of tetrahedrons is their ease in meshing and higher-order elements can facilitate the avoidance of severe volumetric locking with unstructured C0 meshes, which are generally used with these codes for the discontinuities of inelasticity, contact, etc. They also can improve modeling of flexure and curved shapes as well as eliminate spurious modes without artificial stabilization. The inclusion of face and body centroid nodes with Lagrange interpolants, including the 15-node tetrahedron, has proven to provide robust overall performance with lumped-mass explicit methods and with contact. Nevertheless, versions of the 10-node tetrahedron have also emerged in lumped-mass explicit software. In contrast to hexahedrons, an important observation about tetrahedrons is that the 10-node serendipity version uses about four times fewer quadrature points and a larger time increment than their 15-node Lagrange counterpart, which could result in tremendous computational differences. Serendipity elements, however, notoriously do not nodal mass lump well and tetrahedron versions have not been rigorously evaluated/documented for their effectiveness, as will be done herein with comparisons of those using 15-node tetrahedrons. Using row-summation lumping for Lagrange elements and the ad hoc HRZ scheme for serendipity ones, performances are assessed in common benchmark problems and practical applications using various elastic and inelastic material models and involving large strains/deformations/rotations and severe distortions. Whereas the 10-node tetrahedrons were found to perform much better than their 20-node serendipity hexahedral counterparts, specifically with substantial computational reductions and reasonable predictions, they are not generally quite as accurate or robust as the 15-node tetrahedral elements. The results thus indicate benefits of including both 10- and 15-node tetrahedrons in an explicit code's element library.

Approximation of singular solutions and singular data for Maxwell’s equations by Lagrange elements

Abstract A Lagrange finite element method is proposed for Maxwell’s equations in Lipschitz domains. The method is suitable for the approximation of singular solutions lying outside $(H^1(\varOmega ))^3$, with nonhomogeneous singular boundary data in the tangential trace space of $H({\mathbf{curl}}; \varOmega )$ and a singular right-hand-side source term in $(H_0({\mathbf{curl}}; \varOmega ))^{\prime}$ (the dual space of $H_0({\mathbf{curl}}; \varOmega ))$. Numerical results are presented to illustrate performance and the theoretical results.

On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elements

We consider the use of extreme learning machines (ELM) for computational partial differential equations (PDE). In ELM the hidden-layer coefficients in the neural network are assigned to random values generated on $[-R_m,R_m]$ and fixed, where $R_m$ is a user-provided constant, and the output-layer coefficients are trained by a linear or nonlinear least squares computation. We present a method for computing the optimal value of $R_m$ based on the differential evolution algorithm. The presented method enables us to illuminate the characteristics of the optimal $R_m$ for two types of ELM configurations: (i) Single-Rm-ELM, in which a single $R_m$ is used for generating the random coefficients in all the hidden layers, and (ii) Multi-Rm-ELM, in which multiple $R_m$ constants are involved with each used for generating the random coefficients of a different hidden layer. We adopt the optimal $R_m$ from this method and also incorporate other improvements into the ELM implementation. In particular, here we compute all the differential operators involving the output fields of the last hidden layer by a forward-mode auto-differentiation, as opposed to the reverse-mode auto-differentiation in a previous work. These improvements significantly reduce the network training time and enhance the ELM performance. We systematically compare the computational performance of the current improved ELM with that of the finite element method (FEM), both the classical second-order FEM and the high-order FEM with Lagrange elements of higher degrees, for solving a number of linear and nonlinear PDEs. It is shown that the current improved ELM far outperforms the classical FEM. Its computational performance is comparable to that of the high-order FEM for smaller problem sizes, and for larger problem sizes the ELM markedly outperforms the high-order FEM.

Thermal frequency analysis of FG sandwich structure under variable temperature loading

The thermal eigenvalue responses of the graded sandwich shell structure are evaluated numerically under the variable thermal loadings considering the temperature-dependent properties. The polynomial type rule-based sandwich panel model is derived using higher-order type kinematics considering the shear deformation in the framework of the equivalent single-layer theory. The frequency values are computed through an own home-made computer code (MATLAB environment) prepared using the finite element type higher-order formulation. The sandwich face-sheets and the metal core are discretized via isoparametric quadrilateral Lagrangian element. The model convergence is checked by solving the similar type published numerical examples in the open domain and extended for the comparison of natural frequencies to have the final confirmation of the model accuracy. Also, the influence of each variable structural parameter, i.e. the curvature ratios, core-face thickness ratios, end-support conditions, the power-law indices and sandwich types (symmetrical and unsymmetrical) on the thermal frequencies of FG sandwich curved shell panel model. The solutions are helping to bring out the necessary influence of one or more parameters on the frequencies. The effects of individual and the combined parameters as well as the temperature profiles (uniform, linear and nonlinear) are examined through several numerical examples, which affect the structural strength/stiffness values. The present study may help in designing the future graded structures which are under the influence of the variable temperature loading.

Implicit and semi-implicit second-order time stepping methods for the Richards equation

This study concerns numerical methods for efficiently solving the Richards equation where different weak formulations and computational techniques are analyzed. The spatial discretizations are based on standard or mixed finite element methods. Different implicit and semi-implicit temporal discretization techniques of second-order accuracy are studied. To obtain a linear system for the semi-implicit schemes, we propose second-order techniques using extrapolation formulas and/or semi-implicit Taylor approximations for the temporal discretization of nonlinear terms. A numerical convergence study and a series of numerical tests are performed to analyze efficiency and robustness of the different schemes. The developed scheme, based on the proposed temporal extrapolation techniques and the mixed formulation involving the saturation and pressure head and using the standard linear Lagrange element, performs better than other schemes based on the saturation and the flux and using the Raviart-Thomas elements. The proposed semi-implicit scheme is a good alternative when implicit schemes meet convergence issues.

Frequency function and damping function in assessment of dynamic processes in mechanical oscillatory systems with symmetry

Introduction. A new approach to the formation of the methodological basis of system analysis in the application to the problems on mechanical oscillatory structure dynamics is considered. The study objective is to develop a method forevaluating properties of the mechanical oscillatory systems with account for viscous friction forces based on frequency functions and damping functions that depend on the so-called coefficient of connection forms, which is the ratio of characteristics of generalized coordinates. Materials and Methods. The graphoanalytical methods used for evaluating the dynamic properties of mechanical oscillatory two-degree-of-freedom systems are based on determining the extreme values of the frequency functions and the damping function, which are determined from the relations between the kinetic, potential energy and the values of the energy dissipation function. Mathematical models are based on Lagrange formalism, matrix methods, and elements of the theory of functions of a complex variable. Results. A method is proposed for constructing frequency functions and damping functions for a class of mechanical oscillatory two-degree-of-freedom systems based on the analytical expressions that reflect features of the ratio of the potential and kinetic energy of the system considering viscous friction forces represented by the dissipative function. General analytical expressions for the frequency function and the damping function are derived. Graphoanalytical analysis of extreme properties of the corresponding frequency functions and damping functions is performed for mechanical vibrational systems with elastic-damping elements with symmetry properties. The results of numerical experiments are presented. A criterion for classifying frequency functions and damping functions based on the topological features of the graphs of the corresponding functions is proposed. Discussion and Conclusions. The developed method for constructing frequency functions and damping functions can be used to display the dynamic features of mechanical oscillatory systems. The proposed matrix method for constructing a frequency-damping function for a two-degree-of-freedom system can be extended to the mechanical vibrational systems considered in different coordinate systems.

Coupled Fluid–Structure Interaction based Numerical Investigation on the Large Deformation Behavior of Thin Plates Subjected to under Water Explosion

Coupled fluid structure interaction based numerical investigation on rectangular mild steel plates subjected to underwater explosion is tested using Explosion Bulge Testing (EBT) is presented in this paper. The test plate along with the EBT box model fixture immersed in water domain subjected to small explosive charges of PEK I at a standoff distance of 0.15 m from centre of the plate is considered for the analysis using LSDYNA code. For the analysis, explosive charge is modelled using high explosive burn with JWL equation of state. The fluid is using null material model with Gruinesian equation of state, the EBT setup is using Lagrangian solid element with rigid material model and the test plate is modelled using Belytschko-Tsay shell element with piece-wise linear plasticity material model. The numerical analysis aims to predict the permanent deformation of the plate under various shock loading conditions. The results are then compared with experimental results available in the literature and numerical results based on Taylor’s plate theory. Parametric investigations on the large deformation behaviour of different plate thickness of various shock loading conditions are presented.

Interfacial Fatigue and Discrete Interfacial Damage in a Finite Strain Thermomechanical Framework

We present a stabilized finite element method for thermomechanical problems in the class of materials with discrete microstructural interfaces that undergo interfacial fatigue and dominant interfacial damage. This formulation is applicable to polycrystalline solids, fibrous composites, filled elastomers, and additively manufactured layered materials. A finite strain formulation for monolithically coupled thermomechanical fields is presented where interfacial kinematic models for low-cycle fatigue and for strong interfacial discontinuities are variationally embedded at the interfaces. Formulation is written in the spatial configuration to account for large local strains and finite rotations of the interfaces. The method is implemented employing the family of low-order 3D Lagrange elements comprised of linear hexahedra and linear tetrahedra. A set of benchmark problems is presented to show the mathematical and algorithmic attributes of the method.

Geometrically nonlinear dynamic analysis of laminated composite plate using a nonpolynomial shear deformation theory

A computationally efficient C0 finite element model in conjunction with the nonpolynomial shear deformation theory (NPSDT) is extended to examine the free and forced vibration behavior of laminated composite plates. The employed NPSDT assumes the nonlinear distribution of in-plane displacements which qualify the requirement of traction free boundary conditions at the top and bottom surfaces. The present formulation utilizes both von Kármán and Green–Lagrange type of strain–displacement relations to model the geometric nonlinearity. Using Hamilton’s principle, the nonlinear governing equation of motion is derived and then discretized based on the nine-noded Lagrange element. The obtained equations are solved by utilizing unconditionally stable Newmark’s scheme in conjunction with Newton–Raphson method. A damping effect in the transient analysis has been introduced in the framework of the Rayleigh damping model. The steady state forced vibration analysis has also been carried out by employing harmonic force with excitation frequency around the natural frequency. The arc-length continuation method is applied to obtained the frequency response. The present model has been validated for a wide range of problems and a detailed numerical study has been carried out for several types of boundary conditions under various types of loading with different magnitude of the load.

Turbulent flame-vortex dynamics of bluff-body premixed flames

Abstract To better understand the coupled dynamics of turbulence and premixed flames, this study experimentally investigates the effects of turbulence on reacting vorticity dynamics. In this investigation, a bluff-body stabilized, turbulent, premixed flame is analyzed in a high-speed combustor. A custom-designed turbulence generator is used to vary the inflow turbulence conditions and understand the subsequent impacts on the flame-flow field. The turbulent reacting flow-field is measured using simultaneous high-speed particle imaging velocimetry (PIV) and CH* chemiluminescence. A Lagrangian tracking method is implemented to study flame-vortex dynamics. Lagrangian fluid elements are tagged from the experimental data and are tracked as they convect through the flame, from reactants to products. The components of the vorticity transport equation are decomposed along the Lagrangian trajectories to determine their relative balance under various turbulence conditions. The results confirm that altering the turbulence intensity influences flame-vortex interactions and the relative balance of the vorticity components. At high turbulence intensities, the baroclinic torque and dilatation decrease, while the magnitude of viscous diffusion increases, and vortex stretching dominates the reacting flow field. These results suggest that increasing the turbulence intensity alters the dominant physical processes in the reacting domain as the combustion-driven phenomena become less significant.

A locking-free solver for linear elasticity on quadrilateral and hexahedral meshes based on enrichment of Lagrangian elements

This paper presents a new finite element solver for linear elasticity on quadrilateral and hexahedral meshes based on enrichment of the classical bilinear or trilinear Lagrangian elements. It solves the primal variable displacement in the strain–div formulation and can handle both displacement and traction boundary conditions. It is a locking-free solver based on conforming finite elements. The solver has second order accuracy in displacement and first order accuracy in stress and dilation (divergence of displacement), as validated by theoretical analysis and illustrated by numerical experiments on benchmarks. deal.II implementation is also discussed.

Dynamic analysis of folded laminated composite plate using nonpolynomial shear deformation theory

In this paper, an efficient C0 finite element modeling based on higher-order nonpolynomial shear deformation theory (NPSDT) is proposed for the dynamic analysis of folded laminated composite plate. The theoretical formulations are based on nine-noded Lagrange isoparametric finite element and inverse hyperbolic shear deformation theory (IHSDT) as NPSDT. The employed theory, IHSDT, assumes the nonlinear and realistic distribution of transverse shear stresses, and also satisfies the traction-free boundary conditions at the top and bottom surfaces of the plate. Hamilton's principle has been adopted to derive the system's governing equation. A penalty approach has been used to take into account the artificial constraints generated due to incorporation of C0 Lagrange element. The free vibration pertaining to eigenvalue problem is solved using subspace iteration method. The transient analysis subjected to pressure load and vertical load is carried out using Newmark's direct integration scheme. The formulation has been validated with the available solution in the literature, and several novel solutions have been proposed to address the various practical aspects of folded plate. Numerical illustrations are presented to investigate the effect of various parameters such as crank angle, fiber angle, lamination scheme, fold location, and boundary conditions on the natural frequency and transient response of laminated composite plate.

Optimal error analysis of Crank–Nicolson lowest‐order Galerkin‐mixed finite element method for incompressible miscible flow in porous media

AbstractNumerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest‐order Galerkin‐mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest order Raviart–Thomas mixed element pair is used for the Darcy velocity and pressure. The existing error estimate of the method in L2‐norm is in the order in spatial direction, which however is not optimal and valid only under certain extra restrictions on both time step and spatial meshes, excluding the most commonly used mesh h = hp = hc. This paper focuses on new and optimal error estimates of a linearized Crank–Nicolson lowest‐order Galerkin‐mixed finite element method (FEM), where the second‐order accuracy for the concentration in both time and spatial directions is established unconditionally. The key to our optimal error analysis is an elliptic quasi‐projection. Moreover, we propose a simple one‐step recovery technique to obtain a new numerical Darcy velocity and pressure of second‐order accuracy. Numerical results for both two and three‐dimensional models are provided to confirm our theoretical analysis.

Convergence analysis of Crank-Nicolson extrapolated fully discrete scheme for thermally coupled incompressible magnetohydrodynamic system

In this paper, we consider the nonstationary magnetohydrodynamic coupled heat equation through the well-known Boussinesq approximation. The Crank-Nicolson extrapolation scheme is used for time derivative terms, and the mixed finite method is used for spatial discretization. We employ the Taylor-Hood finite elements to approximate Navier-Stokes equations, Nédélec edge element for the magnetic induction and the equal order Lagrange elements for the thermal equation. This fully discrete scheme only needs to solve a linear system at each time step, and the system is unique solvable. We prove the proposed scheme is unconditionally energy stable. Under a weak regularity hypothesis on the exact solution, we present optimal error estimates for the velocity, magnetic induction and temperature. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme.