Nodal Velocities Research Articles

Finite Element Simulation of a Full-Scale Crash Test of a Composite Helicopter

Finite Element Simulation of a Full-Scale Crash Test of a Composite Helicopter

Full-scale crash test and simulation of a composite helicopter

A finite element model of the Sikorsky Advanced Composite Airframe Program (ACAP) helicopter was developed using the non-linear, explicit transient dynamic code, MSC.Dytran. Analytical predictions were correlated with experimental data obtained from a full-scale crash test of the Sikorsky ACAP helicopter flight test article that was conducted in June 1999 at the Impact Dynamics Research Facility of NASA Langley Research Centre, Hampton, Virginia, USA. The helicopter was impacted at 11.58 m/s vertical and 9.9-m/s forward velocity with an attitude of 6.25° pitch (nose up) and 3.5° left roll. Due to the relatively long crash pulse duration, a rigid-body helicopter model with an energy absorbing landing gear model was executed initially. Prior to fuselage contact, a deformable structural model was executed with the rigid-body nodal displacements and velocities used as initial conditions.

Analyses of plane-strain upsetting by the rigid–plastic domain-boundary element method

In this paper, the rigid–plastic domain-boundary element method is formulated. This method does not need iterative calculations in any computing step. Therefore, the rigid–plastic domain-boundary element method possesses a merit in that there is no possibility of divergence of the iterative calculations and the computing costs are less than those of the rigid–plastic finite element methods which are a kind of iterative solution. On the other hand, the rigid–plastic domain-boundary element method is formulated with mixed variables which are nodal velocities and derivatives of nodal velocity. Therefore, nodal velocities and derivatives of nodal velocity can be calculated with the same precision, whilst the rigid–plastic finite element methods are formulated with single variables which are nodal velocities. Therefore, the rigid–plastic finite element methods cannot meet the need for compatibility of the nodal velocity’s derivative, and nodal velocities and derivatives of nodal velocity cannot be calculated with the same precision for these methods. In this paper a plane-strain metal forming (cold upsetting) problem is analyzed by the rigid–plastic domain-boundary element method. Contours of effective strain rate, effective strain and shear stress, etc. are obtained successfully.

A coupled FE‐BE analysis of acoustic cavities confined inside laminated composite enclosures

The structural acoustic problem, wherein an acoustic domain is confined within a partly flexible laminated composite enclosure is presented. From the finite element free vibration analysis of the laminated folded plate structure a mobility relation is derived between the normal velocity of the structure and normal pressure on the structure. A boundary element solver for the Helmholtz equation with quadratic isoparametric elements is developed using pressure‐velocity formulation. Velocity is known over certain parts of the boundary, the rest being the interactive boundary, where the mobility relation correlates nodal pressures and velocities, neither explicitly known. The pressure boundary values are solved from the boundary element and the mobility relations, while the nodal particle velocities and domain pressures are computed at desired points thereafter. New results presented here reveal the effects of the variation in magnitude of structural damping, fiber angles and the thickness of walls.

Three-dimensional rigid–plastic finite element method using diagonal matrix for large-scale simulation of metal-forming processes

A rigid–plastic finite element method using a diagonal matrix is proposed to perform large-scale three-dimensional simulation of metal forming processes in a realistic computing time. Uncoupled equilibrium equations of forces at one nodal point are repeatedly calculated in turn without solving the large global matrix. In the equilibrium equations, three corrective velocities at the nodal point are determined in order to eliminate imbalanced forces for trial velocities. This formulation leads to reduction in the increase rate of the computing time with the number of nodal point. In addition, the program for the developed simulator is much simpler than that of the conventional method and the memory size is very small because the global simultaneous equations are not solved at a time. To examine the convergence of solution in the present method, three-dimensional deformation in upsetting of a rectangular block with flat dies is simulated under non-sliding contact over the die surface. The effects of the initial nodal velocities, the order of nodal points for solving the equilibrium equations and the modification factor for the corrective velocities on the convergence are evaluated.

A comparison between the rigid–plastic finite-boundary element method and the penalty rigid–plastic finite element method

The rigid–plastic finite-boundary element method (FBEM) is formulated with mixed-type. Therefore, this method can meet the need for compatibility of not only the nodal velocity but also the deritives of the nodal velocity. Then nodal velocities and derivatives of the nodal velocity can be calculated with the same precision for the rigid–plastic FBEM. Whilst the penalty rigid–plastic finite element method (FEM) is formulated with single-type like the other conventional rigid–plastic FEMs, the penalty rigid–plastic FEM cannot meet the need for compatibility of the derivitive of nodal velocity. On the other hand, the rigid–plastic FBEM is a kind of solution in open form, so that this method possesses another merit in that it does not need repetitive calculations, and there is not any possibility of divergence with the repetitive calculations. However, the penalty rigid–plastic FEM is a kind of solution in closed form like the other conventional rigid–plastic FEMs, so that there are some divergence possibilities with the repetitive calculations for the penalty rigid–plastic FEM. This paper makes analyses of an axisymmetric forging process by the two methods. It is found that the outline of the workpiece calculated by the rigid–plastic FBEM is smoother than that calculated by the penalty rigid–plastic FEM from the calculated results of this paper.

Finite element simulation of the filling of thin moulds

In this work we present a finite element formulation to simulate the filling of thin moulds. The model has as starting point a 3-D approach using either div-stable elements, such as the Q2/P1 element (tri-quadratic velocities and piecewise linear, discontinuous pressures) or stabilized finite element formulations. The tracking of the free surface is based on the Volume-Of-Fluid (VOF) method. The velocity profile is assumed to be parabolic in the direction normal to the mid-plane, so that one element along the width of the mould is enough to reproduce this profile if this element is quadratic. The velocity is prescribed to zero on the upper and lower surfaces and the normal to the mid-plane is also prescribed to zero. In the case of div-stable elements, the pressure profile is prescribed to be constant along the width of the mould. This can be achieved by using as interpolation degrees of freedom the pressure values at the element centroid and its derivatives in the directions tangent and normal to the mid-plane, and prescribing the latter to zero. No modifications are needed when stabilized formulations are employed. To advance in time the function used in the VOF technique, we use a constant velocity across the width of the mould, which is taken as the projection on the tangent plane of each element of the nodal velocities. This is needed in order to have mass conservation. Copyright © 1999 John Wiley & Sons, Ltd.

Limit analysis of plates-a finite element formulation

A procedure for the computation of the load carrying capacity of perfectly plastic plates in bending is presented. The approach, based on the kinematic theorem of limit analysis, requires the evaluation of the minimum of a convex, but non-smooth, function under linear equality constraints. A systematic solution procedure is devised, which detects and eliminates the finite elements which are predicted as rigid in the collapse mechanism, thus reducing the problem to the search for the minimum of a smooth and essentially unconstrained function of nodal velocities. Both Kirchhoff and Mindlin plate models are considered. The effectiveness of the approach is illustrated by means of some examples.

Sensitivity analysis based preform die shape design for net-shape forging

A sensitivity analysis method for preform die shape design in net-shape forging processes is developed in this paper using the rigid viscoplastic finite element method. The preform die shapes are represented by cubic B-spline curves. The control points or coefficients of B-spline are used as the design variables. The optimization problem is to minimize the zone where the realized and desired final forging shapes do not coincide. The sensitivities of the objective function, nodal coordinates and nodal velocities with respect to the design variables are developed in detail. A procedure for computing the sensitivities of history-dependent functions is presented. The remeshing procedure and the interpolation/transfer of the history-dependent parameters, such as effective strain, are stated. The procedures of sensitivity analysis based preform die design are also described. In addition, a method for the adjustment of the volume loss resulting from the finite element analysis is given in order to make the workpiece volume consistent in each optimization iteration. The method developed in this paper is used to design the preform die shape of H-shaped forging processes, including plane strain and axisymmetric deformations. The results show that a flashless forging with a complete die fill is realized using the optimized preform die shape.

Sensitivity analysis based preform die shape design using the finite element method

This paper uses a finite element-based sensitivity analysis method to design the preform die shape for metal forming processes. The sensitivity analysis was developed using the rigid visco-plastic finite element method. The preform die shapes are represented by cubic B-spline curves. The control points or coefficients of the B-spline are used as the design variables. The optimization problem is to minimize the difference between the realized and the desired final forging shapes. The sensitivity analysis includes the sensitivities of the objective function, nodal coordinates, and nodal velocities with respect to the design variables. The remeshing procedure and the interpolation/transfer of the history/dependent parameters are considered. An adjustment of the volume loss resulting from the finite element analysis is used to make the workpiece volume consistent in each optimization iteration and improve the optimization convergence. In addition, a technique for dealing with fold-over defects during the forming simulation is employed in order to continue the optimization procedures of the preform die shape design. The method developed in this paper is used to design the preform die shape for both plane strain and axisymmetric deformations with shaped cavities. The analysis shows that satisfactory final forging shapes are obtained using the optimized preform die shapes.

A finite element method for simulating interface motion—I. Migration of phase and grain boundaries

This paper describes a finite element method for simulating migration of interfaces (e.g. phase and grain boundaries) in materials. The method is built on a classical theory. Each individual grain is in an equilibrium state; interface tension and bulk phase chemical potential constitute the free energy. An interface migrates—as atoms break from one grain, cross the interface, and attach to the other grain—at a velocity proportional to the free energy reduction per unit volume of atoms crossing the interface. We express this theory in a weak statement, model the interfaces with finite elements, and update nodal positions incrementally. The variations of the free energy, associated with the virtual motions of the nodes, define the generalized forces. The weak statement connects the generalized forces and the nodal velocities with a viscosity matrix. The method takes into account large interface shape changes, interface tension anisotropy, and non-equilibrium triple junctions (if present). We illustrate the method with examples including grooving on a polycrystal surface, grain growth in a thin film, and facet formation of a single crystal particle.

Modelling of the thermomechanical coupling during hot forming of viscoplastic materials

In the present paper, a strategy for solving transient thermoviscoplastic problems is described. The general thermal and mechanical coupling phenomena are discussed by assuming material properties to be temperature-dependent and accounting for the mechanical terms in the non-stationary heat conduction equation. Time integration is carried out with an implicit scheme. An interesting version of the three-field approach is presented to obtain an accurate solution for the stress field. Finally, a global product coupling algorithm, in which the unknown nodal velocities and temperatures are solved with a two-step solution procedure is suggested. The proposed fractional step method partitions the initial coupled problem into an isothermal viscoplastic problem followed by a purely heat conduction problem at fixed configuration. Applications in metal forming analyses involving full thermomechanical coupling conclude the paper.

Upper bound limit analysis using discontinuous velocity fields

A new method for computing rigorous upper bounds under plane strain conditions is described. It is based on a linear three-noded triangular element, which has six unknown nodal velocities and a fixed number of unknown plastic multiplier rates, and uses the kinematic theorem to define a kinematically admissible velocity field as the solution of a linear programming problem. Unlike existing formulations, which permit only a limited number of velocity discontinuities whose directions of shearing must be specified a priori, the new formulation permits velocity discontinuities at all edges shared by adjacent triangles and the directions of shearing are found automatically. The variation of the velocity jump along each discontinuity is described by an additional set of four unknowns. All of the unknowns are subject to the constraints imposed by an associated flow rule and the velocity boundary conditions. The objective function corresponds to the dissipated power, or some related load parameter of interest, and is minimised to yield the desired upper bound. Since plastic deformation may occur not only in the discontinuities, but also throughout the triangular elements as well, the method is capable of modelling complex velocity fields accurately and typically produces tight upper bounds on the true limit load. The formulation is applicable to materials whose strength is cohesive-frictional, purely cohesive and uniform, or purely cohesive and linearly varying, and thus, quite general. The new procedure is very efficient and always requires fewer elements than existing methods to obtain useful upper bound solutions. Moreover, because of the extra degrees of freedom introduced by the discontinuities, the linear elements no longer need to be arranged in a special pattern to model incompressible behaviour accurately.

An effective data parallel self‐starting explicit methodology for computational structural dynamics on the connection machine CM‐5

AbstractThis paper discusses the implementation aspects and our experiences towards a data parallel explicit self‐starting finite element transient methodology with emphasis on the Connection Machine (CM‐5) for linear and non‐linear computational structural dynamic applications involving structured and unstructured grids. The parallel implementation criteria that influence the efficiency of an algorithm include the amount of communication, communication routing, and load balancing. To provide simplicity, high level of accuracy, and to retain the generality of the finite element implementation for both linear and non‐linear transient explicit problems on a data parallel computer which permit optimum amount of communications, we implemented the present self‐starting dynamic formulations (in comparison to the traditional approaches) based on nodal displacements, nodal velocities, and elemental stresses on the CM‐5. Data parallel language CMFortran is employed with virtual processor constructs and with:SERIAL and:PARALLEL layout directives for arrays. The communications via the present approach involve only one gather operation (extraction of element nodal displacements or velocities from global displacement vector) and one scatter operation (dispersion of element forces onto global force vector) for each time step. These gather and scatter operations are implemented using the Connection Machine Scientific Software Library communication primitives for both structured and unstructured finite element meshes. The implementation aspects of the present self‐starting formulations for linear and elastoplastic applications on serial and data parallel machines are discussed. Numerical test models for linear and non‐linear one‐dimensional applications and a two‐dimensional unstructured finite element mesh are then illustrated and their performance studies are discussed.

Moving finite elements method applied to the solution of front reaction models: causticizing reaction

In this paper a Moving Finite Elements Method based on cubic Hermite polynomials is presented. PDE's solution is obtained by minimizing the square norm of discretized residuals all over the domain in order to the time derivatives of the solutions on the nodes and nodal velocities. Spatial discretization gives origin to a system of ODEs, that can degenerate when singularities occur. These can be of two kinds: parallelism and nodal coalescence. Nodal coalescence is avoided by fixing the minimum internodal distance while parallelism is dealt with the addition of penalty functions to the objective function resulting from the application of Galerkin Method. Boundary conditions are previously time derived originating constraints to the minimization system arising from the basic formulation. The solution of this problem is obtained via the Lagrange Multipliers Method. In this paper this methodology is applied to the solution of a front reaction model - the causticizing reaction. The results obtained are in close agreement with those published in the literature where orthogonal collocation in Finite Elements was used. The adequacy of the Moving Finite Elements Method for the solution of front reaction models is demonstrated in this paper.

The incremental flow formulation for the numerical analysis of 3-dimensional viscous deformation processes: continuum formulation and computational aspects

The incremental flow formulation recently presented (J. Bonet, The incremental flow formulation for the numerical analysis of plane stress and thin sheet forming processes, Comput. Methods Appl. Mech. Engrg., in press) is applied herein for the analysis of three dimensional viscous deformation processes. Using this technique the equilibrium equations become functions of the nodal positions instead of nodal velocities as in the case of the standard flow formulation. Consequently, geometrical constraints such as the incompressibility of the flow can be enforced in an exact manner. The discretization techniques essential for a successful implementation of this method are also discussed. In particular the element proposed is the 8-noded hexahedron with a mean quadrature integration technique. Finally, some applications relating to the superplastic forming of thick sheet are used to illustrate the effectiveness of the methods presented.

Corotational finite element analysis of planar flexible multibody systems

A computer-oriented method for the formulation and the solution of nonlinear constrained differential equations of motion is developed. The method is applicable to flexible muitibody systems with large displacements and rotations. A corotational finite element formulation is derived for the dynamic analysis of a planar flexible muitibody system. An inertial frame is used to define the nodal coordinates, velocities, accelerations, displacements, and rotations. The equations of motion are defined in terms of that inertial frame, while strains are measured in the corotational coordinate system of the element. This elemental coordinate system rotates and translates with each element but does not deform with it. The equations of motion are derived using Lagrange's equations. The numerical solution is obtained by employing an incremental-iterative method based on the Newmark direct integration algorithm and the Newton-Raphson method. The applicability and the accuracy of the method are demonstrated by studying some nonlinear flexible mechanical systems.

A cubic‐spline technique to calculate nodal Darcian velocities in aquifers

A continuous and reliable velocity field is required for the simulation of mass and heat transport in an aquifer system. In this paper, a cubic‐spline technique is proposed to calculate nodal Darcian velocities after the flow field is obtained. With this technique, a continuous velocity field is produced, and mass conservation is satisfied to an acceptable degree. Its requirements for computer storage are very small, and the calculation speed is very fast; it can be implemented in parallel. A synthetic diffusion problem is used to examine the accuracy of this technique. The results agree very well with the analytical solution. Application to another problem shows that this technique can be applied to any heterogeneous and anisotropic aquifer. The results of the two sample problems are also compared with the Galerkin finite element method [Yeh, 1981] and the dual mesh method [Batu, 1984]. It is observed that its agreement with the Galerkin method is much better than the dual mesh method.

An upper-bound approach to plane-strain problems using a general triangular element

This paper investigates an upper-bound elemental approach to plane-strain deformation of rigid, perfectly plastic material. A kinematically admissible velocity field based on nodal velocities is derived from a proposed arbitrarily triangular element. From the derived velocity field, the upper-bound forming load and deformation configuration are determined by optimizing independent parameters related to the velocity field. This new element formulation can be applied to diversified plane-strain metal-forming problems, even though only the application to strip forging is discussed in this paper.

A novel isoparametric finite element displacement formulation for axisymmetric analysis of nearly incompressible materials

AbstractIt is well accepted that severe numerical difficulties arise when using the conventional displacement method to analyse incompressible or nearly incompressible solids. These effects are caused by the kinematic constraints imposed on the nodal velocities by the constant volume condition. In elastic‐plastic analysis, these effects are due to a conflict between the plastic flow rule and the finite element discretization. Although several methods have been proposed to cope with this problem, none has been based on the appropriate choice of displacement interpolation functions to minimize the constraints. The theoretical formulation of a new six‐noded isoparametric displacement finite element, which is well suited for elastic‐plastic analysis of axisymmetric constrained solids by using a rational displacement interpolation function, is presented in this paper. The proposed displacement interpolation function implies that the displacement in the axial direction and the product of the displacement in the radial direction and the radius should be treated as two independent basic variables. Alternatively, the proposed displacement interpolation function can also be implemented in a conventional displacement formulation simply by using a modified shape function matrix. The suitability of the proposed formulations is first studied theoretically by assessing the number of degrees of freedom per constraint and then verified by performing numerical experiments on typical boundary value problems which involve incompressible behaviour.