High Order Mixed Finite Elements with Mass Lumping for Elasticity on Triangular Grids

A family of conforming mixed finite elements with mass lumping on triangular grids are presented for linear elasticity.The stress field is approximated by symmetric H(div)−Pk(k≥3)polynomial tensors enriched with higher order bubbles so as to allow mass lumping,and the displacement field is approximated by C−1−Pk−1 polynomial vectors enriched with higher order terms.For both the proposed mixed elements and their mass lumping schemes,optimal error estimates are derived for the stress and displacement in H(div)norm and L 2 norm,respectively.Numerical results confirm the theoretical analysis.

Simulation of anchor chain based on lumped mass method

In order to develop a anchoring operation simulation system and improve safety during anchoring operations,a relatively accurate mathematical model of anchoring operations needs to be established.In this paper,the stress condition of anchor chain under environmental and subsea geological conditions is further studied and the stress condition of anchor chain is analyzed based on the previous research.In this paper,a quasi-static model based on catenary method is used as the basis of dynamic analysis,and the dynamic model of anchor chain is established based on the concentrated mass method,which fully considers the influence of anchor chain weight,hydrodynamic force,ocean current and interaction with the seabed.The fourth-order Runge Kutta method was used to solve the model numerically,and a calculation procedure was developed.The accuracy of the model was verified by comparing the calculated results with the experimental results,indicating that the constructed anchor chain dynamics model has a high accuracy.

A lumped mass Chebyshev spectral element method and its application to structural dynamic problems

A diagonal or lumped mass matrix is of great value for time-domain analysis of structural dynamic and wave propagation problems,as the computational efforts can be greatly reduced in the process of mass matrix inversion.In this study,the nodal quadrature method is employed to construct a lumped mass matrix for the Chebyshev spectral element method(CSEM).A Gauss-Lobatto type quadrature,based on Gauss-Lobatto-Chebyshev points with a weighting function of unity,is thus derived.With the aid of this quadrature,the CSEM can take advantage of explicit time-marching schemes and provide an efficient new tool for solving structural dynamic problems.Several types of lumped mass Chebyshev spectral elements are designed,including rod,beam and plate elements.The performance of the developed method is examined via some numerical examples of natural vibration and elastic wave propagation,accompanied by their comparison to that of traditional consistent-mass CSEM or the classical finite element method(FEM).Numerical results indicate that the proposed method displays comparable accuracy as its consistent-mass counterpart,and is more accurate than classical FEM.For the simulation of elastic wave propagation in structures induced by high-frequency loading,this method achieves satisfactory performance in accuracy and efficiency.

Modal analysis on transverse vibration of axially moving roller chain coupled with lumped mass

The modal characteristics of the transverse vibration of an axially moving roller chain coupled with lumped mass were analyzed.The chain system was modeled by using the multi-body dynamics theory and the governing equations were derived by means of Lagrange's equations.The effects of the parameters,such as the axially moving velocity of the chain,the tension force,the weight of lumped mass and its time-variable assign position in chain span,on the modal characteristics of transverse vibration for roller chain were investigated.The numerical examples were given.It is found that the natural frequencies and the corresponding mode shapes of the transverse vibration for roller chain coupled with lumped mass change significantly when the variations of above parameters are considered.With the movement of the chain strand,the natural frequencies present a fluctuating phenomenon,which is different from the uniform chain.The higher the order of mode is,the greater the fluctuating magnitude and frequency are.

A lumped mass nonconforming finite element method for nonlinear parabolic integro-differential equations on anisotropic meshes

A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is derived on anisotropic meshes without referring to the traditional nonclassical elliptic projection.

Simplified lateral free field boundary based on a lumped mass system

After reviewing the studies on the lateral artificial boundaries in dynamic soil-structure interactions,the free field boundary was theoretically analyzed in asymmetric-and symmetric-matrix forms.First,the lumped mass system was combined with viscous or viscoelastic elements to obtain a lumped mass-free field boundary.Second,typical examples were implemented using the finite element software ABAQUS.The incident shear wave was taken to be perpendicular to the bottom to verify the effectiveness of the lumped mass-free field under various sites:underground structures,uniform sites,and layered sites.Finally,the accuracy of the lumped mass-free field boundary was compared with those of the viscoelastic and roller boundaries on different calculation scales,soil damping ratios,structure sizes,and ground motion characteristics.Subsequently,the engineering design values for different damping ratios are given.The results show that the precision of the lumped mass-free field boundary was reasonable,and the operation was simple within the same engineering application.

Lumped Mass Finite Element Method of the Nonlinear BBM Equation

In this paper, the full-discrete approximation scheme of the lumped mass nonconforming finite element method for BBM equation is discussed. Without the Riesz projection used in the traditional finite element analysis, the optimal error estimations are derived based on interpolation technique and special properties of element.

An Unsaturated Soil Water Flow Problemand Its Numerical Simulation

A numerical model for the unsaturated flow equation with moisture content as prognostic variable is established in order to simulate liquid moisture flow in an unsaturated zone with homogeneous soil, and different initial and boundary conditions. For an infiltration or evaporation problem, its numerical solution by using a finite difference method is very sensitive to its upper boundary condition and the related soil parameters, and using a traditional finite element method usually yields oscillatory non-physics profiles. However, we obtain a nonoscillatory solution and evade a non-physics solution for the problem by using the mass-lumped finite element method. This kind of boundary conditions is handled very well. Numerical simulations for certain soils show that the numerical scheme can be used in simulation of liquid moisture flow for infiltration, evaporation, re-distribution and their alternate appearances. It can be also applied to a high-resolution land surface model.

Numerical simulation of the influences of sinker weight on the deformation and load of net of gravity sea cage in uniform flow

A numerical model for determining fishing net configuration and load is developed, based on the lumped masses method. The model is used to analyze the influences on the net of gravity sea cage by different sinker weights in uniform flow. The net of gravity sea cage is simulated under different current velocities and sinker weights. In order to verify the validity of the numerical results, model test results made by Lader and Enerhaug are cited and compared with the numerical ones. The results of numerical simulation agree well with the experimental ones and the agreement is within an error range of 13% under different velocities and sinker weights. The numerical results indicate that increasing the sinker weight can effectively reduce the net deformation, but the total drag force on the net will increase accordingly. Results will give references for better knowledge of the hydrodynamic behavior of gravity sea cage.

Study on A New Mooring System Integrating Catenary with Taut Mooring

Analyzed are the merits and demerits of catenary mooring system and taut mooring system, which are commonly used nowadays. As falling somewhere between these two systems, a new mooring system integrating catenary with taut mooring is proposed. In order to expound and prove the advantages of this new system, the motion performance of a semi-submersible platform is simulated by employing full time domain coupled analysis method. A comparison of the result of new mooting system with that of taut mooring system shows that the movement of the platform using the new type mooting system is smaller than that using the taut mooring system, which ensures a better working condition. Furthermore, the new mooring system is also compatible with the characteristics of catenary mooting system, which eliminates the requirement of anti-uplift capacity of the anchors.

Global Analysis of a Flexible Riser

The mechanical performance of a flexible riser is more outstanding than other risers in violent environmental conditions. Based on the lumped mass method, a steep wave flexible riser configuration attached to a Floating Production Storage and Offioading （FPSO） has been applied to a global analysis in order to acquire the static and dynamic behavior of the flexible riser. The riser was divided into a series of straight massless line segments with a node at each end. Only the axial and torsional properties of the line were modeled, while the mass, weight, and buoyancy were all lumped to the nodes. Four different buoyancy module lengths have been made to demonstrate the importance of mode selection, so as to confirm the optimum buoyancy module length. The results in the sensitivity study show that the flexible riser is not very sensitive to the ocean current, and the buoyancy module can reduce the Von Mises stress and improve the mechanical performance of the flexible riser. Shorter buoyancy module length can reduce the riser effective tension in a specific range of the buoyancy module length when other parameters are constant, but it can also increase the maximum curvature of the riser. As a result, all kinds of the riser performances should be taken into account in order to select the most appropriate buoyancy module length.

Numerical Simulation of Dynamic Response of Supple Nets

A simulation method based on the lumped mass model is proposed for determining the dynamic behavior of nets exposed to a uniform current. Every mesh bar is modeled by a linear bar clement. The lumped mass point is set at the ends of each element. The net can be simulated by a discretized model consisting of many point masses and elements without mass. 3D shapes and the distribution of tensions of the net at different moments can be found from time integration of a set of motion equations with a computer program. Two nets are simulated according to reference experiments. Calculated results are in accordance with experimental results. The method is applicable and can be applied to improving design of, and research into other flexible structures, such as net cages.

Influence of Wave Direction on the Dynamic Response of A Submarine Equipment Launch and Recovery System

A launch and recovery system for a seafloor drill was studied using a dynamic model that considered the influences of seawater resistance and the elastic deformation of the cable based on the lumped mass method.The influence of wave direction angle on heave,roll,and pitch motions of the ship was analyzed,and those motion characteristics were then used to assess the tension response of the armored umbilical cable at the lifting point under different wave direction angles.By analyzing the different wave direction angles we found that,when a ship experiences longitudinal waves it will express longitudinal movement.When a ship encounters transverse waves,it will have transverse movement.Under oblique waves from bow or stern,a ship will have both longitudinal and transverse movement,exhibiting obvious heave and pitch movements.Oblique waves,in this study,produced the most obvious impact on armored umbilical cable tension.However,the tension of the armored umbilical cable will change based on the weight of the armored umbilical cable and the seafloor drill in the water.This analysis has provided a useful reference for the study of heave compensation and the constant tension automatic control.

Axial Vibration Analysis of the Mud Recovery Line on Deepwater Riserless Mud Recovery Drilling System

The series connection of multistage pumping module is the common concept of deepwater riserless mud recovery drilling system. In this system, the influence of the mass of pumping module on the vibration of mud recovery line cannot be ignored, and the lumped mass method has been utilized to discretize the mud recovery line. Based on the analysis of different boundary conditions, the paper establishes the axial forced vibration model of the mud recovery line considering the seawater damping, and the vibration model analysis provides the universal solution to the vibration model. An example of the two-stage pumping system has been used to analyze the dynamic response of mud recovery line under different excited frequencies. This paper has the important directive significance for the application of riserless mud recovery drilling technology in deepwater surface drilling.

A stable one-point quadrature rule for three-dimensional numerical manifold method

We present a numerically stable one-point quadrature rule for the stiffness matrix and mass matrix of the three-dimensional numerical manifold method(3D NMM).The rule simplifies the integration over irregularly shaped manifold elements and overcomes locking issues,and it does not cause spurious modes in modal analysis.The essential idea is to transfer the integral over a manifold element to a few moments to the element center,thereby deriving a one-point integration rule by the moments and making modifications to avoid locking issues.For the stiffness matrix,after the virtual work is decomposed into moments,higher-order moments are modified to overcome locking issues in nearly incompressible and bending-dominated conditions.For the mass matrix,the consistent and lumped types are derived by moments.In particular,the lumped type has the clear advantage of simplicity.The proposed method is naturally suitable for 3D NMM meshes automatically generated from a regular grid.Numerical tests justify the accuracy improvements and the stability of the proposed procedure.

A lumped mass finite element formulation with consistent nodal quadrature for improved frequency analysis of wave equations

A lumped mass finite element formulation with consistent nodal quadrature is presented for improved frequency analysis of wave equations with particular reference to the Lagrangian elements with Lobatto nodes.For the Lagrangian Lobatto elements,a lumped mass matrix can be conveniently constructed by employing the nodal quadrature rule that takes the Lobatto nodes as integration points.In the conventional finite element analysis,this nodal quadrature-based lumped mass matrix is usually accompanied by the stiffness matrix computed via the Gauss quadrature.In this work,it is shown that this combination is not optimal regarding the frequency accuracy of finite element analysis of wave equations.To elevate the frequency accuracy,in addition to the lumped mass matrix formulated by the nodal quadrature,a frequency accuracy measure is established as a function of the quadrature rule used in the stiffness matrix integration.This accuracy measure discloses that the frequency accuracy can be optimized if both lumped mass and stiffness matrices are simultaneously computed by the same nodal quadrature rule.These theoretical results are well demonstrated by two-and three-dimensional numerical examples,which clearly show that the proposed consistent nodal quadrature formulation yields much higher frequency accuracy than the conventional finite element analysis with nodal quadrature-based lumped mass and Gauss quadrature-based stiffness matrices for wave equations.

WAVE EQUATION MODEL FOR NUMERICAL SOLUTIONS OF ADVECTION－DOMINANT HEAT TRANSFER

In this paper the wave equation model (WEM) [1] is extended to solve advection-dominant heat transfer problems in multi-dimensional space. Based on the operator-splitting method the heat transfer equation is divided into an advection equation and a diffusion equation which are solved separately. In advection stage the first order advection equation is transferred to a second order wave equation first, than the wave equation is solved by FEM with mass lumping. The diffusion equation can be solved accurately without many difficulties. A number of numerical examples of multi-dimensional advection are presented in which the advection velocities are non-uniform in space and unsteady in time. The numerical results are quite accurate in comparison with the exact solutions.The mass lumping saves computational effort greatly.

High-Order Low Dissipation Conforming Finite-Element Discretization of the Maxwell Equations

In this paper,we study high order discretization methods for solving the Maxwell equations on hybrid triangle-quad meshes.We have developed high order finite edge element methods coupled with different high order time schemes and we compare results and efficiency for several schemes.We introduce in particular a class of simple high order low dissipation time schemes based on a modified Taylor expansion.

Dynamic response of towed line array

Three-dimensional motion equations with consideration of the bending and torsional effects of a marine towed cable system under large elastic deformation conditions are formulated using the lumped parameter model. The lumped mass model is used to simulate a circular maneuver of a towed horizontal array that was first presented by Ablow. The results of this paper are in a good agreement with those obtained by finite difference schemes, and the minimum depth is closer to the experimental result. Although the calculations take more computation time, the lumped mass model is a good method, which can also be used to solve problems of towed line array, especially multi-branched towed line array, because of its flexibility.

NUMERICAL SIMULATION OF THE EFFECTS OF WEIGHT SYSTEM ON THE HYDRODYNAMIC BEHAVIOR OF 3-D NET OF GRAVITY CAGE IN CURRENT

In this article, a model of 3-D net is set up by using lumped mass method. Model test results made by Lader and Enerhaug are cited to verify the numerical model. The aim of this paper is to investigate the effects of weight system on the hydrodynamic behavior of 3-D net of gravity cage in current. Using the 3-D net model, with different styles and masses of weight system, hydrodynamic behavior of gravity cage net in current is simulated. In this article, two styles of common weight system are used, which include： （1） sinker system , （2） bottom collar-sinker system. Under each style, three different masses of weight system are adopted. The numerical results indicate that the bottom collar-sinker system is practically feasible in improving the cage net volume deformation. Results of this study will give references for better knowledge of hydrodynamic behavior of gravity cage.