Coupled System Of Nonlinear Equations Research Articles

Thermo-mechanical analysis of industrial solidification processes

The paper presents an up-to-date finite element numerical model for fully coupled thermo-mechanical problems, focussing in the simulation of solidification processes of industrial metal parts. The proposed constitutive model is defined by a thermo-visco-elasto-(visco)plastic free energy function which includes a contribution for thermal multiphase changes. Mechanical and thermal properties are assumed to be temperature-dependent, and viscous-like strains are introduced to account for the variation of the elastic moduli during the cooling process. The continuous transition between the initial fluid-like and the final solid-like behaviour of the part is modelled by considering separate viscous and elasto-plastic responses as a function of the solid fraction. Thermo-mechanical contact conditions between the mould and the part are specifically considered, assuming that the heat flux is a function of the normal pressure and the thermal and mechanical gaps. A fractional step method arising from an operator split of the governing equations is used to solve the non-linear coupled system of equations, leading to a staggered product formula solution algorithm suitable for large-scale computations. Representative simulations of industrial solidification processes are shown, and comparison of computed results using the proposed model with available experimental data is given. Copyright © 1999 John Wiley & Sons, Ltd.

On the formulation of coupled thermoplastic problems with phase-change

This paper deals with a numerical formulation for coupled thermoplastic problems including phase-change phenomena. The final goal is to get an accurate, efficient and robust numerical model, allowing the numerical simulation of solidification processes in the metal casting industry. Some of the current issues addressed in the paper are the following. A fractional step method arising from an operator split of the governing differential equations has been used to solve the nonlinear coupled system of equations, leading to a staggered product formula solution algorithm. Nonlinear stability issues are discussed and isentropic and isothermal operator splits are formulated. Within the isentropic split, a strong operator split design constraint is introduced, by requiring that the elastic and plastic entropy, as well as the phase-change induced elastic entropy due to the latent heat, remain fixed in the mechanical problem. The formulation of the model has been consistently derived within a thermodynamic framework. The constitutive behavior has been defined by a thermoelastoplastic free energy function, including a thermal multiphase change contribution. Plastic response has been modeled by a J2 temperature dependent model, including plastic hardening and thermal softening. A brief summary of the thermomechanical frictional contact model is included. The numerical model has been implemented into the computational Finite Element code COMET developed by the authors. A numerical assessment of the isentropic and isothermal operator splits, regarding the nonlinear stability behavior, has been performed for weakly and strongly coupled thermomechanical problems. Numerical simulations of solidification processes show the performance of the computational model developed.

Translationally-invariant coupled-cluster method for finite systems

The translational invariant formulation of the coupled-cluster method is presented here at the complete SUB(2) level for a system of nucleons treated as bosons. The correlation amplitudes are solution of a non-linear coupled system of equations. These equations have been solved for light and medium systems, considering the central but still semi-realistic nucleon-nucleon S3 interaction.

Finite amplitude, free vibrations of an axisymmetric load supported by a highly elastic tubular shear spring

The finite amplitude, free vibrational characteristics of a simple mechanical system consisting of an axisymmetric rigid body supported by a highly elastic tubular shear spring subjected to axial, rotational, and coupled shearing motions are studied. Two classes of elastic tube materials are considered: a compressible material whose shear response is constant, and an incompressible material whose shear response is a quadratic function of the total amount of shear. The class of materials with constant shear response includes the incompressible Mooney-Rivlin material and certain compressible Blatz-Ko, Hadamard, and other general kinds of models. For each material class, the quasi-static elasticity problem is solved to determine the telescopic and gyratory shearing deformation functions needed to evaluate the elastic tube restoring force and torque exerted on the body. For all materials with constant shear response, the differential equations of motion are uncoupled equations typical of simple harmonic oscillators. Hence, exact solutions for the forced vibration of the system can be readily obtained; and for this class, engineering design formulae for the load-deflection relations are discussed and compared with experimental results of others'. For the quadratic material, however, the general motion of the body is characterized by a formidable, coupled system of nonlinear equations. The free, coupled shearing motion for which either the axial or the azimuthal shear deformation may be small is governed by a pair of equations of the Duffing and Hill types. On the other hand, the finite amplitude, pure axial and pure rotational motions of the load are described by the classical, nonlinear Duffing equation alone. A variety of problems are solved exactly for these separate free vibrational modes, and a number of physical results are presented throughout.

Illustration of an optimization procedure for fiber‐spinning operating conditions: Maximum draw ratio under a Maxwell thin‐filament model

To illustrate a procedure for optimizing an industrial fiber spinning process through modeling, we make two choices: (1) we adopt the viscoelastic fiber‐spinning model of Denn, Petrie, and Avenas; and (2) we choose draw ratio as the process variable to be optimized. We begin with an analysis of the time‐dependent generalization of the Denn, Petrie, and Avenas model to determine all well‐posed boundary and initial conditions for this coupled system of nonlinear equations. Next we give the general analytical solution to the steady dimensionless form of the problem, which we exploit to analyze the response of the solution to all possible steady‐state variations in the dimensionless process parameters available within the Denn, Petrie, and Avenas model. We then predict both the conditions which optimize draw ratio in this model for the fiber spinning process, and how to achieve the maximum. This determination is first obtained in the dimensionless formulation and then transferred to the more practical dimensional formulation.

Long-time quasilinear evolution of the free-electron laser instability for a relativistic electron beam propagating through a helical magnetic wiggler

The long-time quasilinear development of the free-electron laser instability is investigated for a tenuous electron beam propagating in the z direction through a helical wiggler field B0=−B̂ cos k0zêx−B̂ sin k0zêy. The analysis neglects longitudinal perturbations (δφ≂0) and is based on the nonlinear Vlasov–Maxwell equations for the class of beam distributions of the form fb(z,p,t) =n0δ(Px)δ(Py)G(z,pz,t), assuming ∂/∂x=0=∂/∂y. The long-time quasilinear evolution of the system is investigated within the context of a simple ‘‘water-bag’’ model in which the average distribution function G0( pz,t) =(2L)−1∫L−L dz G(z,pz,t) is assumed to have the rectangular form G0( pz,t) =[2Δ(t)]−1 for ‖pz−p0(t)‖ ≤Δ(t), and G0( pz,t) =0 for ‖pz−p0(t)‖ >Δ(t). Making use of the quasilinear kinetic equations, a coupled system of nonlinear equations is derived which describes the self-consistent evolution of the mean electron momentum p0(t), the momentum spread Δ(t), the amplifying wave spectrum ‖Hk(t)‖2, and the complex oscillation frequency ωk(t) +iγk(t). These coupled equations are solved numerically for a wide range of system parameters, assuming that the input power spectrum Pk(t=0) is flat and nonzero for a finite range of wavenumber k that overlaps with the region of k space where the initial growth rate satisfies γk(t=0) >0. To summarize the qualitative features of the quasilinear evolution, as the wave spectrum amplifies it is found that there is a concomitant decrease in the mean electron energy γ0(t)mc2=[m2c4+e2B̂2/k20 +p20(t)c2]1/2, an increase in the momentum spread Δ(t), and a downshift of the growth rate γk(t) to lower k values. After sufficient time has elapsed, the growth rate γk has downshifted sufficiently far in k space so that the region where γk >0 no longer overlaps the region where the initial power spectrum Pk(t=0) is nonzero. Therefore, the wave spectrum saturates, and γ0(t) and Δ(t) approach their asymptotic values.

Nonlinear superposition principles: A new numerical method for solving matrix Riccati equations

Previously derived superposition formulae, expressing the general solution of the matrix Riccati equation W ̇ W(t) = A + WB + CW + WDW (A(t), B(t), C(t), D(t), W(t) ϵ R n×n, n ≥ 2, t ϵ R) in terms of 5 particular solutions, are applied to obtain numerical solutions of this coupled system of nonlinear equations.

Hydromagnetic free convection effects on the oscillatory flow of an electrically conducting rarefied gas past an infinite vertical porous plate

Unsteady two-dimensional free convection flow of an electrically conducting, viscous, incompressible rarefied gas, past an infinite vertical porous plate in the presence of a transverse magnetic field is studied. The freestream velocity oscillates in time about a constant mean, while the suction velocity, normal to the porous plate, is constant. The magnetic Reynolds number of the flow is not taken to be small enough, so that the induced magnetic field is not negligible. The plate temperature is constant and the difference between the temperature of the plate and the freestream is moderately large causing the free convection currents. The flow field is described by a nonlinear coupled system of equations subjected to the first-order velocity slip and temperature jump boundary conditions. With viscous dissipative heat and Joule heating taken into account, approximate solutions of the problem are obtained for the velocity, temperature and induced magnetic field, as well as, for the related to them quantities of the skin friction, rate of the heat transfer and electric current density.

Radially separated monopole solutions in non-Abelian gauge models

We consider a Yang-Mills-Higgs Lagrangian invariant under local gauge transformations belonging to an arbitrary compact group $G$. The Higgs fields are assumed to belong to a real representation of $G$. We analyze in detail the conditions imposed on the fields due to the requirement that the static Hamiltonian or the total energy of the system be finite. We then seek static finite-energy solutions for which the radial dependence of the fields is factorized. We show that for the coupled system of nonlinear equations emerging from the Lagrangian the equations for angular functions which carry the internal-symmetry labels and the equations for the radial functions separate into two separate systems of coupled nonlinear equations. We solve the equations for angular functions completely and show that the gauge fields vanish outside a fixed SO(3) subgroup of $G$ and that inside the SO(3) group they reduce to the 't Hooft-Polyakov solution with unit magnetic charge in appropriate units. The Higgs fields may belong to any integer representation of this SO(3) group. The static Hamiltonian and consequently the total energy or mass of the monopole depend on the representation of the Higgs field. Thus we obtain in principle a mass formula for the monopoles, the one with the lowest mass corresponding to the 't Hooft-Polyakov case.