Explicit Central Difference Research Articles

Time-domain analysis of Schrodinger equation for single electron in magnetic field

In this paper the time-dependent Schrodinger equation governing dynamics of a single electron in magnetic fields is solved by the finite difference method (FDM) with the explicit central difference scheme and Trotter-Suzuki method (TSM). Motion of a wave packet in rectangular and stadium-shaped domains immersed in uniform magnetic field is analyzed. The performance of the two numerical methods is compared. It is shown that FDM is superior over TSM for accuracy and computing efficiency while TSM is superior for numerical stability. Moreover, the probability density of an electron in the stadium is shown to form complex spatial patterns which suggest quantum chaos.

Numerical Simulation of Cylindrical Laminated Shells Under Impulsive Lateral Pressure

The finite element approach used to investigate the behavior of cylindrical laminated shells when acted upon by sudden dynamic loads is described. When solving the dynamic buckling problem, two major time-integration techniques are used, namely, the implicit integration operator in its Newmark method implementation and the explicit central difference scheme. Their performance and efficiency are compared, and the conclusion is made that, unlike some other types of dynamic problems for dynamic shell stability, the explicit finite element approach shows good accuracy of results combined with an excellent computational efficiency. Therefore, this approach is applied in the present investigation. The loading acting upon the shell structure consists of lateral suddenly applied fluid pressure, which remains normal to the shell surface as the shell deforms. Two different dynamic buckling criteria are used and combined with the finite element method in the proposed solution procedure. A new approach for assessing shell stability by using the variation of the volume enclosed by the shell is proposed. A demonstration example is presented. The performance of the present approach is compared with previously published results.

Spatio–temporal growth of nematic domains in liquid crystal polymer mixtures

Dynamics of phase separation and morphology development in mixtures of a low molar mass liquid crystal (LC) and a polymer have been investigated theoretically in comparison with experimental results. In the theoretical model, the combined free-energy densities of Flory–Huggins theory for isotropic mixing and Maier–Saupe theory for nematic ordering have been incorporated into the time-dependent Ginzburg–Landau equation (type C). The temporal evolution of the structure factor and the emergence of phase-separated liquid crystal domains have been simulated on the basis of an explicit central difference method based on a square lattice (128×128) with a periodic boundary condition. Of particular interest is the observed plateau (or inflection) region in the growth dynamic curve, which may be attributed to the breakdown of the interconnected domains caused by the nematic ordering. This unique behavior has been verified experimentally in terms of the growth of structure factor following several temperature quenches into a nematic+liquid region of the experimental phase diagram of an E7/poly(methyl methacrylate) mixture. Further, the emergence of LC domains in the metastable and unstable nematic–liquid spinodal regions has been investigated theoretically and compared with the reported experimental results.

Analysis of reinforced concrete structures subjected to dynamic loads with a viscoplastic Drucker–Prager model

The behavior of reinforced concrete structures subjected to dynamic loads is analyzed. The concrete material is modelled by an elasto-viscoplastic law, whose inviscid counterpart is the Drucker–Prager model. A viscous regularization is introduced in order to avoid the mesh dependency effects that usually appear when strain softening occurs. The model is implemented in a general finite element computer code for fast transient analysis of fluid-structure systems, based on an explicit central difference scheme. The model is activated to both continuum elements and layered shell elements. So, realistic numerical analyses of complex 3-D engineering problems are simple and efficient. Three examples, two of which are modelled with layered shell elements, are presented below.

Accuracy of numerical methods for solving the advection–diffusion equation as applied to spore and insect dispersal

Three algorithms for solving a simplified 3-D advection–diffusion equation were compared as to their accuracy and speed in the context of insect and spore dispersal. The algorithms tested were the explicit central difference (ECD) method, the implicit Crank–Nicholson (ICN) method, and the implicit Chapeau function (ICF) method. The three algorithms were used only to simulate the diffusion process. A hold-release wind shifting method was developed to simulate the wind advection process, which shifts the concentration an integer number of grids and accumulates the remaining wind travel distance (which is less than the grid spacing) to the next time step. The test problem was the dispersal of a cloud of particles (originally in only one grid cell) in a 3-D space. The major criterion for testing the accuracy was R 2, which represents the proportion of the total variation in particle distribution in all grid cells that is accounted for by the particle distribution through numerical solutions. Other criteria included total remaining mass, peak positive density, and largest negative density. High R 2 values were obtained for the ECD method with (Δ t K z)/(Δ z) 2≤0.5 (Δ t=time step; K z=vertical eddy diffusion coefficient; Δ z=vertical grid spacing), and for the two implicit methods with Δ t K z/(Δ z) 2≤5. The ICN method gave higher R 2 values than the ICF method when the concentration gradients were high, but its accuracy decreased more rapidly with the progress of time than the ICF method with a combination of a large grid spacing and a large time step. With very steep concentration gradients, the ICF method generated huge negative values, the ICN method generated negative values to a lesser extent, and the ECD method generated only small negative values. It was also found that good mass and/or peak preservation did not necessarily correspond to a higher R 2 value. Based on the R 2 value and the requirement for concentration positivity, for simulations with steep concentration gradients, the ECD method would be most appropriate, followed by the ICN method, and the ICF method would be least appropriate due to large negative values. For simulations with low concentration gradients, the ECD or ICF or ICN method could be used, but the ICN method would not be appropriate for use in a combination of a large time step and a large grid spacing. The results from this study could help selection and use of appropriate numerical methods in studying the spatial dynamics of spores and insects.

Numerical Instability due to Varying Time Steps in Explicit Wave Propagation and Mechanics Calculations

Explicit central-difference time integration is frequently used to solve the wave equation, and the classical criterion for numerical stability is the Courant–Friedrichs–Lewy condition. Similarly, explicit integration of a spring-mass mechanical system has a stability condition. These conditions are derived under the assumption of constant time steps. This paper demonstrates the new and perhaps surprising result that numerical instability may occur when time steps vary, even though all steps are substantially less than the constant step criterion.

Sensitivity analysis for failure and damage in dynamically loaded tensile bars

A computational procedure is presented for evaluating the sensitivity coefficients of porous viscoplastic solids under dynamic loading conditions. The effects of finite strains, material strain and strain-rate hardening, and the thermal softening due to adiabatic heating are incorporated into the formulation. A critical void volume fraction criterion is used to identify the initiation of failure. The equations of motion emanating from a finite element semi-discretization are integrated using an explicit central difference scheme. First- and second-order sensitivity coefficients of the response quantities (derivatives with respect to various material parameters) are evaluated using a direct differentiation approach in conjunction with an automatic differentiation software facility. Numerical results are presented for tensile specimens subjected to impact loading. Both axisymmetric and plane strain states are considered. The first- and second-order sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to various macroscopic and microscopic material parameters. Time histories of the response and sensitivity coefficients, and their spatial distributions at selected times are presented.

APPLICATION OF AN ASYMPTOTIC METHOD TO TRANSIENT DYNAMIC PROBLEMS

A new method to solve linear dynamics problems using an asymptotic method is presented. Asymptotic methods have been efficiently used for many decades to solve non-linear quasistatic structural problems. Generally, structural dynamics problems are solved using finite elements for the discretization of the space domain of the differential equations, and explicit or implicit schemes for the time domain. With the asymptotic method, time schemes are not necessary to solve the discretized (space) equations. Using the analytical solution of a single degree of freedom (DOF) problem, it is demonstrated, that the Dynamic Asymptotic Method (DAM) converges to the exact solution when an infinite series expansion is used. The stability of the method has been studied. DAM is conditionally stable for a finite series expansion and unconditionally stable for an infinite series expansion. This method is similar to the analytical method of undetermined coefficients or to power series method being used to solve ordinary differential equations. For a multi-degree-of-freedom (MDOF) problem with a lumped mass matrix, no factorization or explicit inversion of global matrices is necessary. It is shown that this conditionally stable method is more efficient than other conditionally stable explicit central difference integration techniques. The solution is continuous irrespective of the time segment (step) and the derivatives are continuous up to orderN-1 whereNis the order of the series expansion.

Frictional contact/impact response of textile composite structures

The results of a detailed study of the frictional contact/impact response of axisymmetric textile composite structures are presented. The structures are assumed to consist of an arbitrary number of perfectly bonded layers of woven fabric or braided preforms. The material of each layer is assumed to be hyperelastic and the effect of geometric nonlinearity is included. The equations of motion of the structure are established in the current configuration and a displacement finite element model is used for the spatial discretization. A Coulomb friction model is used and the temporal integration is performed by using an explicit central difference scheme. Both the dynamic response and the sensitivity coefficients are evaluated. The sensitivity coefficients measure the sensitivity of the response to variations in the textile preform architecture and constituent properties, as well as to variations in the effective layer properties. Numerical results are presented for the frictional contact/impact response of a spherical cap made of textile (woven and braided) composite material, impacting a rigid surface. Results are compared with those of a spherical cap made of tape laminate.

Large-scale contact/impact simulation and sensitivity analysis on distributed-memory computers

A parallel computational strategy is presented for simulating frictional contact/impact response and evaluating sensitivity coefficients for large-scale finite element models of composite structures. The explicit central difference scheme is used for the temporal integration of the semi-discrete governing equations. The two key elements of the strategy are: (a) an element-based domain decomposition technique; and (b) a robust exchange algorithm for communicating information across subdomain interfaces. An implementation of the strategy suitable for any distributed-memory computing system (including clusters of workstations) is described. Numerical results are presented for three structures impacting a rigid surface. The three structures are: an inclined composite cylinder, an inclined stiffened cylinder and an Advanced Composites Technology (ACT) program test panel. For each of these problems, the performance characteristics of the computational procedure are assessed on the IBM SP2 parallel computer system.

Assessment of temporal integration schemes for the sensitivity analysis of frictional contact/impact response of axisymmetric composite structures

An assessment is made of three temporal integration schemes for the sensitivity analysis of the frictional contact/impact response of axisymmetric composite structures. The structures considered consist of an arbitrary number of perfectly-bonded homogeneous anisotropic layers. The material of each layer is assumed to be hyperelastic, and the effect of geometric non-linearity is included. Sensitivity coefficients measure the sensitivity of the response to variations in the different material, lamination and geometric parameters of the structure. A displacement finite element model is used for the spatial discretization. Normal contact conditions are incorporated into the formulation by using a perturbed Lagrangian approach with fundamental unknowns consisting of both the nodal displacements and the Lagrange multipliers associated with the contact conditions. The Lagrange multipliers are allowed to be discontinuous at interelement boundaries. Tangential contact conditions are incorporated by using either a penalty method or a Lagrange multiplier technique, in conjunction with the classical Coulomb's friction model. The three temporal integration schemes considered are: the implicit Newmark and Houbolt schemes, and the explicit central difference method. In the case of the implicit methods, the Newton-Raphson iterative technique is used for the solution of the resulting non-linear algebraic equations, and for the determination of the contact region, contact conditions (sliding or sticking), and the contact pressures. Sensitivity coefficients are evaluated by using a direct differentiation approach in conjunction with the incremental equations. Numerical results are presented for the frictional contact of a composite spherical cap impacting a rigid plate, showing the effects of each of the following factors on the accuracy of the predicted response and sensitivity coefficients: (a) incorporating the normal contact conditions, (b) the magnitude of the penalty parameter in the normal direction (for the perturbed Lagrangian method), and (c) the time step size for the response and the sensitivity analyses.

Sensitivity analysis for the dynamic response of thermoviscoplastic shells of revolution

A computational procedure is presented for evaluating the sensitivity coefficients of the dynamic axisymmetric, fully-coupled, thermoviscoplastic response of shells of revolution. The analytical formulation is based on Reissner's large deformation shell theory with the effects of large-strain, transverse shear deformation, rotatory inertia and moments turning around the normal to the middle surface included. The material model is chosen to be viscoplasticity with strain hardening and thermal hardening, and an associated flow rule is used with a von Mises effective stress. A mixed formulation is used for the shell equations with the fundamental unknowns consisting of six stress resultants, three generalized displacements and three velocity components. The energy-balance equation is solved using a Galerkin procedure, with the temperature as the fundamental unknown. Spatial discretization is performed in one dimension (meridional direction) for the momentum and constitutive equations of the shell, and in two dimensions (meridional and thickness directions) for the energy-balance equation. The temporal integration is performed by using an explicit central difference scheme (leap-frog method) for the momentum equation; a predictor-corrector version of the trapezoidal rule is used for the energy-balance equation; and an explicit scheme consistent with the central difference method is used to integrate the constitutive equations. The sensitivity coefficients are evaluated by using a direct differentiation approach. Numerical results are presented for a spherical cap subjected to step loading. The sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to the thickness, mass density, Young's modulus, two of the material parameters characterizing the viscoplastic response and the three parameters characterizing the thermal response. Time histories of the response and sensitivity coefficients are presented, along with spatial distributions of some of these quantities at selected times.

Sensitivity analysis for the dynamic response of viscoplastic shells of revolution

A computational procedure is presented for evaluating the sensitivity coefficients of the dynamic axisymmetric response of viscoplastic shells of revolution. The analytical formulation is based on Reissner's large deformation shell theory with the effects of transverse shear deformation, rotatory inertia and moments turning around the normal to the middle surface included. The material model is chosen to be isothermal viscoplasticity, and an associated flow rule is used with a von Mises effective stress. A mixed formulation is used with the fundamental unknowns consisting of six stress resultants, three generalized displacements and three velocity components. Spatial discretization is performed using finite elements, with discontinuous stress resultants across element interfaces. The temporal integration is performed by using an explicit central difference scheme (leap-frog method) with an implicit constitutive update. The sensitivity coefficients are evaluated using a direct differentiation approach. Numerical results are presented for a spherical cap subjected to step loading, and a circular plate subjected to impulsive loading. The sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to the thickness, mass density, Young's modulus, and two of the material parameters characterizing the viscoplastic response. Time histories of the response and sensitivity coefficients are presented, along with spatial distributions of these quantities at selected times.

Validation of code using turbulence model applied to three-dimensional transonic channel

The flow induced by a swept bump mounted on the lower wall of a transonic channel has been thoroughly analyzed on the basis of surface pressure measurements, surface flow visualizations, and field measurements by a three-component laser Doppler velocimeter system. The same flow has been computed by a code solving the full time-averaged Navier-Stokes equations by a method using an explicit centered finite difference scheme applied to a finite volume approach. An algebraic model using the mixing length concept and the Jones-Launder [k, e] transport equation model have been considered. Both models give a faithful prediction of the essential features of the flow; in particular, the shock pattern forming in the channel is well predicted. However, the [k, e] model shows a clear superiority in the prediction of the behavior of the interacting dissipative layers

Sensitivity analysis of the non‐linear dynamic viscoplastic response of 2‐D structures with respect to material parameters

AbstractA computational procedure is presented for evaluating the sensitivity coefficients of the viscoplastic response of structures subjected to dynamic loading. A state of plane stress is assumed to exist in the structure, a velocity strain‐Cauchy stress formulation is used, and the geometric non‐linearities arising from large strains are incorporated. The Jaumann rate is used as a frame indifferent stress rate. The material model is chosen to be isothermal viscoplasticity, and an associated flow rule is used with a von Mises effective stress. The equations of motion emanating from a finite element semi‐discretization are integrated using an explicit central difference scheme with an implicit stress update. The sensitivity coefficients are evaluated using a direct differentiation approach. Since the domain of integration is the current configuration, the sensitivity coefficients of the spatial derivatives of the shape functions must be included. Numerical results are presented for a thin plate with a central circular cutout subjected to an in‐plane compressive loading. The sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to Young's modulus, and two of the material parameters characterizing the viscoplastic response. Time histories of the response and sensitivity coefficients, and spatial distributions at selected times are presented.

Geometrically non-linear transient analysis of laminated composite and sandwich shells with a refined theory and C0 finite elements

A C 0 continuous finite element formulation of a higher order shear deformation theory is presented for predicting the linear and geometrically non-linear, in the sense of von Karman, transient responses of composite and sandwich laminated shells. The displacement model accounts for the non-linear cubic variation of the tangential displacement components through the thickness of the shell and the theory requires no shear correction coefficients. In the time domain, the explicit central difference integrator is used in conjunction with the special mass matrix diagonalization scheme which conserves the total mass of the element and includes effects due to rotary inertia terms. Numerical results for central transverse deflection and stresses are presented for composite and sandwich laminated shells with various boundary conditions subjected to different types of loads and are compared with the results from other sources. Some new results are also included for future reference.

Large Deflection Elastic and Inelastic Transient Analyses of Composite and Sandwich Plates with a Refined Theory

A C° continuous finite element formulation of a higher order displacement theory is presented for predicting linear and non-linear transient responses of composite and sandwich plates. The geometric non-linearity is accounted for in the sense of von Kar man assumptions and the material behaviour is assumed as elasto-perfectly plastic. The elasto-perfectly plastic material behaviour is incorporated using the flow theory of plastic ity. In particular, the modified version of Hill's initial yield criterion with amsotropic pa rameters of plasticity is used. The layered approach is adopted to account for the plasticity through the thickness of the plate. The displacement model accounts for non-linear cubic variation of tangential displacement components through the thickness of the laminate and the theory requires no shear correction coefficients. In the time domain, the explicit cen tral difference integrator is used in conjunction with the special mass matrix diagonaliza tion scheme which conserves the total mass of the element and includes effects due to rotary inertia terms. The parametric effects of the time step, finite element mesh, lamina tion scheme and orthotropy on the linear and non-linear responses are investigated. Nu merical results are presented for composite and sandwich plates under various boundary conditions and loadings and these are compared with the results from other sources. Some new results are also presented for future reference.

Investigation of economical difference systems for dynamical problems in elasticity theory in a class of nonsmooth solutions

Using an explicit central difference scheme, an economical difference system is constructed for the first boundary-value problem of the dynamical theory of elasticity for velocities and stresses. Estimates of the rate of convergence of the difference scheme is obtained under weak hypotheses on the smoothness of solutions to the differential problem.

Geometrically non-linear transient C° finite element analysis of composite and sandwich plates with a refined theory

A <TEX>$C^{\circ}$</TEX> continuous finite element formulation of a higher order displacement theory is presented for predicting linear and geometrically non-linear in the sense of von Karman transient responses of composite and sandwich plates. The displacement model accounts for non-linear cubic variation of tangential displacement components through the thickness of the laminate and the theory requires no shear correction coefficients. In the time domain, the explicit central difference integrator is used in conjunction with the special mass matrix diagonalization scheme which conserves the total mass of the element and included effects due to rotary inertia terms. The parametric effects of the time step, finite element mesh, lamination scheme and orthotropy on the linear and geometrically non-linear responses are investigated. Numerical results for central transverse deflection, stresses and stress resultants are presented for square/rectangular composite and sandwich plates under various boundary conditions and loadings and these are compared with the results from other sources. Some new results are also tabulated for future reference.

Massively parallel computational methods for finite element analysis of transient structural responses

With the emphasis on the finitely damped system (e.g. control structure interaction) case, two fully implicit and two semi-implicit sets of finite element method-based numerical algorithms are formulated for transient response analysis of space frame and truss structures in a massively parallel processing (MPP) environment. All algorithm sets use an implicit force calculation/vector equation of motion assembly procedure. The semi-implicit algorithms are based on the explicit central difference (CD) and the fourth-order Runge-Kutta (RK4) schemes, respectively, in conjunction with the use of mass lumping techniques so that solution of the recurrence equations for unknown displacements is reduced to a trivial diagonal matrix inversion operation. The fully implicit algorithm sets are based on the Newmark Beta (NB) and CD schemes, respectively, in conjunction with use of the (iterative) preconditioned conjugate gradient (PCG) method for solving the linear algebraic recurrence equations. The semi-implicit algorithm sets are fully implemented and assessed on an MPP CM-2 computer. A preliminary assessment of the fully implicit sets of algorithms is made on a Sun Workstation. These numerical study results show that the newly formulated MPP algorithms are, to a varying degree, superefficient (or potentially superefficient) on the CM-2 compared with, and even highly competitive against, the conventional sequential algorithms on an advanced serial computer.