Bisection for parallel computing using Ritz and Fiedler vectors

Summary. In this article, an efficient algorithm is developed for the decomposition of large-scale finite element models. A weighted incidence graph with N nodes is used to transform the connectivity properties of finite element meshes into those of graphs. A graph G₀ constructed in this manner is then reduced to a graph G_n of desired size by a sequence of contractions G₀ G₁ G₂ G_n. For G₀, two pseudoperipheral nodes s₀ and t₀ are selected and two shortest route trees are expanded from these nodes. For each starting node, a vector is constructed with N entries, each entry being the shortest distance of a node n_i of G₀ from the corresponding starting node. Hence two vectors v₁ and v₂ are formed as Ritz vectors for G₀. A similar process is repeated for G_i (i=1,2,,n), and the sizes of the vectors obtained are then extended to N. A Ritz matrix consisting of 2(n+1) normalized Ritz vectors each having N entries is constructed. This matrix is then used in the formation of an eigenvalue problem. The first eigenvector is calculated, and an approximate Fiedler vector is constructed for the bisection of G₀. The performance of the method is illustrated by some practical examples.     

Flow of a generalized second grade fluid between heated plates

Summary We examine the fully developed flow of a generalized fluid of second grade between heated parallel plates, due to a pressure gradient along the plate. The constant coefficient of shear viscosity of a fluid of second grade is replaced by a shear dependent viscosity with an exponentm. If the normal stress coefficients are set equal to zero, this model reduces to the standard power-law model. We obtain the solution for the case when the temperature changes only in the direction normal to the plates for the two most commonly used viscosity models, i.e. (i) when the viscosity does not depend on temperature, and (ii) when the viscosity is an exponentially decaying function of temperature.

List of symbols

Alphanumeric A ₁,A ₂ Kinematical tensor - b Body force - C Dimensionless parameter related to the pressure gradient - h Separation between the plates - L Velocity gradient - m Power-law index - M Constant appearing in the Reynolds viscosity model - p Pressure field - Modified pressure field - q Heat flux vector - r Radiant heating - T Cauchy's stress tensor - l Unit tensor - v Velocity vector - V Characteristic velocity - x Axis along the plate - y Axis perpendicular to the plate Greek ₁, ₂ Normal stress coefficient - Specific internal energy - Dimensionless parameter related to the viscous dissipation - Conservative body force field - Specific entropy - Thermal conductivity - Coefficient of viscosity - ₀ Reference viscosity - Second invariant of the stretching tensor - Temperature - ₁ Temperature of the lower plate - ₂ Temperature of the upper plate - Density - Specific Helmholtz free energy Operators div Divergence - grad Gradient - tr Trace     

Finite element method for thermomechanical response of near-incompressible elastomers

Summary The present study addresses finite element analysis of the coupled thermomechanical response of near-incompressible elastomers such as natural rubber. Of interest are applications such as seals, which often involve large deformations, nonlinear material behavior, confinement, and thermal gradients. Most published finite element analyses of elastomeric components have been limited to isothermal conditions. A basic quantity appearing in the finite element equation for elastomers is thetangent stiffness matrix. A compact expression for theisothermal tangent stiffness matrix has recently been reported by the first author, including compressible, incompressible, and near-incompressible elastomers. In the present study a compact expression is reported for the tangent stiffness matrix under coupled thermal and mechanical behavior, including pressure interpolation to accommodate near-incompressibility. The matrix is seen to have a computationally convenient structure and to serve as a Jacobian matrix in a Newton iteration scheme. The formulation makes use of a thermoelastic constitutive model recently introduced by the authors for near-incompressible elastomers. The resulting relations are illustrated using a near-incompressible thermohyperelastic counterpart of the conventional Mooney-Rivlin model. As an application, an element is formulated to model the response of a rubber rod subjected to force and heat.Notation A _i n _i /c - C Cauchy strain tensor - c VEC (C) - C ₁,C ₂ constants in Mooney-Rivlin model for elastomer - c ₂ VEC (C ²) - c _i eigenvalues ofC - c _e ,c _e , _e specific heat at constant strain - D _nl stiffness matrix due to the geometric nonlinearity - D _T ,D _T isothermal tangent modulus matrices - e VEC () - e _d VEC ( _d ) - f, f(T) thermal expansion function, =1/[1+(T-T ₀)/3] - f combined vector of nodal forces and heat fluxes - f _M consistent nodal force vector - f _T ,f _1T ,f _2T ,f _3T ,f _4T consistent heat flux vector - F deformation gradient tensor - g related tof _T - h time step - I _i invariants ofC - I 9×9 identity tensor - I identity tensor - i vectorial counterpart ofI:VEC(I) - J the Jacobian matrix in Newton iteration scheme - J determinant ofF - J _i invariants ofI ₁ ^–1/3 C - k, k(T) thermal conductivity - K tangent stiffness matrix - K _MM ,K _MT ,K _MP tangent stiffness submatrices - L,L _M ,L _P ,L _S lower triangular matrices related toLU decomposition ofK - M ₁,M ₂ strain-displacement matrices - N interpolation matrix - n surface normal vector - n _i (I _i /c) ^T - P matrix arising in theLU decomposition ofK - P the tension applied to the rubber rod - p (true) pressure - Q heat flux - q heat flux vector - r,r _M ,r _T ,r residual vectors - R,R ₁,R ₂,R ₃ matrices from thermal boundary conditions - R _s 1/2(R+R ^T ) - R _a 1/2(R–R ^T ) - R –R ₁+R ₂+R - R _s 1/2(R+R ^T ) - R _a 1/2(R-R ^T ) - s VEC() - S surface in undeformed configuration - t time - t traction referred to undeformed configuration - T, T ₀ temperature, reference temperature - T upper-shelf temperature in the surface convective relation - U upper triangular matrix inLU decompositionK - u displacement vector - v combined vector of nodal parameters - v _n value ofv at thenth time step - V volume in undeformed configuration - w strain energy density per unit undeformed volume - x position vector in deformed configuration - X position vector in undeformed configuration - volumetric thermal expansion coefficient - _c coefficient in the surface convective relation - ₁ strain-displacement matrix - _T interpolation matrix for thermal gradient: T - vector of nodal displacements - Lagrangian strain tensor - _d deviatoric Lagrangian strain tensor - interpolation function for - entropy per unit mass in the undeformed configuration - vector of nodal temperatures - þ isothermal bulk modulus - interpolation function forT - temperature-adjusted pressure,p/f ³(T) - mass density in the undeformed configuration - matrix arising inLU decomposition ofK - 2nd Piola-Kirchhoff stress - Cauchy stress tensor - , _M , _T , _c , ₀ Helmholtz free energy density function per unit mass - _i - _ij - vector of nodal values of - matrix arising in theLU decomposition ofK - near-incompressibility constraint function - internal energy density per unit mass - (·) variational operator - VEC(·) vectorization operator - symbol for Kronecker product of two tensors - tr(·) trace of a tensor - det(·) determinant of a tensor     

Incremental finite element equations for thermomechanical response of elastomers: Effect of boundary conditions including contact

Summary The present investigation concerns the solution of nonlinear finite element equations by Newton iteration, for which the Jacobian matrix plays a central role. In earlier investigations [1], [2], a compact expression for the Jacobian matrix was derived for incremental finite element equations governing coupled thermomechanical response of near-incompressible elastomers. A fully Lagrangian formulation was adopted, with three important restrictions: (a) the traction and heat flux vectors were referred to theundeformed coordinates; (b) Fourier's law for heat conduction was expressed in terms of theundeformed coordinates; and (c) variable contact was not considered. In contrast, in the current investigation, the boundary conditions and Fourier's law of heat conduction are referred to thedeformed coordinates, and variablethermomechanical contact is modeled. A thermohyperelastic constitutive equation introduced by the authors [3] is used and is specialized to provide a thermomechanical, near-incompressible counterpart of the two-term Mooney-Rivlin model. The Jacobian matrix is now augmented with several terms which are derived in compact form using Kronecker product notation. Calculations are presented on a confined rubber O-ring seal submitted to force and heat.List of symbols A contact area - A _i - A _MM coefficient matrix for foundation model - a _TM coefficient vector for foundation model - a _MT coefficient vector for foundation model - a _TT coefficient scalar for foundation model - B _t,B _q matrices related to boundary terms due to large deformations - B _{c t} ,B _{c q} matrices related to boundary terms due to variable thermomechanical contact - B _{f t} ,B _{f q} matrices related to boundary terms due to nonlinear foundation model - B _{c MM} submatrix inB _{c t} - B _{c TM} submatrix inB _{c q} - B _{c TT} submatrix inB _{c q} - B _{f MM} submatrix inB _{f t} - B _{f TM} submatrix inB _{c q} - B _{f TT} submatrix inB _{f q} - B _{t MM} submatrix inB _t - B _{q TM} submatrix inB _q - B _{q TT} submatrix inB _q - C Cauchy-Green strain tensor - C ₁,C ₂ constants in strain energy density functions for the elastomer - c vec (C) - c ₂ vec (C ²) - c _e specific heat at constant strain - _e - c _hi parameters in contact heat conductance model - D _nl stiffness matrix due to geometric nonlinearity - D _T isothermal tangent modulus matrix - D _{T e} modulus matrix at constantT ande - D _T tangent modulus matrix at constantT and - d g _i nodal vectors related to prescribed traction and heat flux - d g _{c q} nodal vector related to thermal contact - d g _{c t} nodal vector related to contact traction - d g _{f q} nodal vector related to heat flux - d g _{f q 0} nodal vector related to heat flux - d g _{f q 0} nodal vector related to heat flux - d g _{f t} nodal vector related to traction - d g _{f t 0} nodal vector related to prescribed traction - d g _{f q 0} nodal vector related to prescribed traction - d g _{q 0} nodal vector related to heat flux - d g _{q 0} nodal vector related to heat flux - d g _{t 0} nodal vector related to traction - d g _{t 0} nodal vector related to traction - d(n ^TT} q) prescribed heat flux increment - d tt} prescribed traction increment - e vec() - e _d vec( _d ) - e _r shift parameter in elastic foundation model - f, f(T) thermal expansion function, = - F deformation gradient tensor - f _c nodal vector from the contact traction - g gap function - g _i nodal vectors related to mechanical and thermal loads - g _M nodal vector related to mechanical load - g _{i T} nodal vectors related to thermal terms - h time step - h _n - I _i invariants ofC - I ₉ 9×9 identity tensor - I 3×3 identity tensor - i vectorial counterpart ofI: vec(I) - J Jacobian matrix for Newton iteration - J determinant ofF - k thermal conductivity - k _H high stiffness in elastic foundation model - k _L low stiffness in elastic foundation model - K(g) stiffness function for elastic foundation model - K tangent stiffness matrix - K _MM tangent stiffness submatrix - K _MT tangent stiffness submatrix - K _MP tangent stiffness submatrix - K _PP tangent stiffness submatrix - K _PT tangent stiffness submatrix - K _TT tangent stiffness submatrix - M ₁ strain-displacement matrix - M ₂ strain-displacement matrix - m unit vector normal to target surface - N interpolation matrix - n vector normal to current surface - n ₀ vector normal to undeformed surface - n _i - p (true) pressure - Q heat rate across contact surface - q heat flux vector referred to the deformed configuration - q ₀ heat flux vector referred to the undeformed configuration - qq} prescribed heat flux - r residual vector in combined equilibrium equation - r _M residual vector from mechanical equilibrium - r _T residual vector from thermal equilibrium - r residual vector from near-incompressibility constraint - R matrix of heat conduction in domain - S surface in current configuration - s vec () - S ₀ surface in undeformed configuration - S _c candidate contact surface in current configuration - S _{c 0} candidate contact surface in undeformed configuration - S _{f M} surface corresponding to nonlinear foundation in current configuration - S _{f M 0} surface corresponding to nonlinear foundation in undeformed configuration - S _{f T} surface corresponding to nonlinear foundation in current configuration - S _{f M 0} surface corresponding to nonlinear foundation in undeformed configuration - S _T prescribed temperature boundary surface in current configuration - S _{T 0} prescribed temperature boundary surface in undeformed configuration - S _t prescribed traction boundary surface in current configuration - S _{t 0} prescribed traction boundary surface in undeformed configuration - S _q prescribed heat flux boundary surface in current configuration - S _{q 0} prescribed heat flux boundary surface in undeformed configuration - S _u prescribed displacement boundary surface in current configuration - S _{u 0} prescribed displacement boundary surface in undeformed configuration - T current temperature - T ₀ reference temperature - T _r temperature of rigid foundation - t time - t _n contact traction normal to contact surface - t _{n, n} solution value oft _n at thenth load step - t _ti components of tangential contact traction vector - t traction referred to current configuration - tt} prescribed traction - u displacement vector - v combined vector of nodal parameters - V ₀ volume in undeformed configuration - V volume in deformed configuration - w - x position vector in deformed configuration - X position vector in undeformed configuration - y possible contact point in the target surface - volumetric thermal expansion coefficient - _i parameters in metal-metal thermal contact models - _hi coefficients in thermal contact model - _k coefficient in elastic foundation model - interpolation matrix for strain field - _T interpolation matrix for thermal gradient: ₀ T - - vector of nodal displacements - Lagrangian strain tensor - _d deviatoric portion of Lagrangian strain tensor - interpolation function for - entropy density - vector of nodal temperatures - þ isothermal bulk modulus - surface area factor - interpolation function forT - temperature-adjusted pressure, - mass density in the deformed configuration - ₀ mass density in the undeformed configuration - 2nd Piola-Kirchhoff stress tensor - Helmholtz free energy density function - _M Helmholtz free energy density function - ₀ Helmholtz free energy density function - _i - _i - _ij - _ij - nodal vector for pressure field - near-incompressibility constraint function - the target surface equation: (y)=0 - (·) variational operator - vec(·) vectorization operator - symbol for Kronecker product of two tensors - tr(·) trace of a tensor - det(·) determinant of a tensor - divergence operator with respect to current configuration - ₀ divergence operator with respect to undeformed configuration - · the norm of vector     

Systematic Magnetization Measurements on Single Crystalline Bi2Sr2CaCu2O8+δ with Columnar Defects

We have performed systematic magnetization measurements on single crystalline Bi ₂ Sr ₂ CaCu ₂ O ₈₊ with columnar defects of B = 0.005 to 1 T by using a SQUID magnetometer. Magnetization hysteresis curves of the pristine sample show a weak irreversible behavior in the vortex liquid state, suggesting the existence of the new vortex state in the vortex liquid state. This weak irreversible region persists systematically in the samples with columnar defects even up to B = 1 T. It is shown that the weak hysteresis of magnetization is sensitive to the disorder level of the sample and shifts systematically to higher temperature and field region with increasing the number of columnar defects. This behavior clearly indicates that effective pinning mechanism exists even in the vortex liquid state and generates a finite critical current.     

Very Large Rashba Spin–Orbit Splitting in HgTe Quantum Wells

A large Rashba spin splitting has been observed in the first conduction subband of n-type modulation doped HgTe quantum wells with an inverted band structure via an investigation of Shubnikov–de Haas oscillations as a function of gate voltage. Self-consistent Hartree calculations of the band structure based on an 8 × 8 k p model quantitatively describe the experimental results. It has been shown that the heavy-hole nature of the H1 conduction subband greatly influences the spatial distribution of electrons in the quantum well and also enhances the Rashba spin splitting at large electron densities. These are unique features of type III heterostructures in the inverted band regime. The k ³ dispersion predicted by an analytical model is a good approximation of the self-consistent Hartree calculations for small values of the in-plane wave vector k . This is in contrast to the commonly used k dispersion for the conduction subband in type I heterojunctions.     

Magnetic Ordering of Dy3+ Ions in RbDy(WO4)2 Single Crystal

The specific heat C(T) of the monoclinic RbDy(WO ₄ ) ₂ crystal has been studied at very low temperatures 0.2T1.9 K and in magnetic fields 0H0.38 T. The Neel temperature was shown to be equal to T_N = 0.818 ± 0.005 K. The experimental value of the effective exchange parameter was obtained to be equal to J/k = – 0.798 K. The C T) dependence below Neel temperature 0.5T_N0.99T_N ) is well described by 2D Ising model, whereas in the temperature region close above T_N (1.01 T_N2T_N) it can be described by neither 2D, nor 3D Ising model. The experimental and theoretical H-T_N diagrams for field H a are in a reasonable agreement for simple quadratic lattice.     

Theory of thermohyperelasticity for near-incompressible elastomers

Summary Elastomers are often used in hot and confining environments in which thermomechanical properties are important. It appears that published constitutive models for elastomers are mostly limited to isothermal conditions. In this study, athermohyperelastic constitutive model for near-incompressible elastomers is formulated in terms of the Helmholtz free energy density . Shear and volume aspects of the deformation are decoupled. Thermomechanical coupling occurs mostly as thermal expansion. Criteria for thermodynamic stability are derived in compact form. As illustration, a particular expression for is presented which represents the thermomechanical counterpart of the conventional two-term incompressible Mooney-Rivlin model. It is used to analyze several adiabatic problems in a rubber rod.List of symbols A _i matrices appearing inD - c _e, _e, c _e specific heat at constant strain - C Cauchy strain tensor - C _R reduced Cauchy strain tensor - C ₁,C ₂ coefficients of Mooney-Rivlin model - c vectorial counterpart ofC: VEC (C) - c ₂ vectorial counterpart ofC ²: VEC (C ²) - D isothermal tangent stiffness matrix - e vectorial counterpart of : VEC () - deviatoric Lagrangian strain tensor - e _R reduced deviatoric Lagrangian strain tensor - e volume strain - e _T reduced volume strain - f thermal expansion function,=[1+(T–T ₀)/3]^–1 - F, F _T deformation gradient tensor - F _R reduced deformation gradient tensor - H Hessian matrix for the Gibbs free energy density - H related toH - I ₁,I ₂,I ₃ invariants ofC - I _1R,I _2R,I _3R invariants ofC _R - I, I ₉ identity matrix - i vectorial counterpart ofI: VEC (I) - J determinant ofF - J _R determinant ofF _R - J _T determinant ofF _T - J ₁,J ₂ invariants of /J _{R 2} ³ - J _1R,J _2R invariants of _R /R _J ^2/3 - k thermal conductivity coefficient - K ₁,K ₂,K ₃ invariants of /J ^2/3 - K _1R,K _2R,K _3R invariants ofe _R/J^2/3 - p hydrostatic pressure - s vectorial counterpart of stress : VEC () - s isotropic stress - T (absolute) temperature - T ₀ reference temperature - conventional (isothermal) strain energy density (per unit volume) - volumetric thermal expansion coefficient - thermal expansion vector - strain, Lagrangian strain - entropy density - isothermal bulk modulus - Lagrange multiplier - _i extension ratio - shear modulus - stability coefficient - mass density - stress, 2nd Piola-Kirchhoff stress - _i principal stress - Cauchy stress - _d deviatoric Cauchy stress - , _{M C T 0} Helmholtz free energy density - _i /I _i - _ij ²/I _i I _j - Gibbs free energy density - (.) variational operator - VEC (.) vectorization operator - operator for Kronecker product     

Interface shapes in a torsionally oscillating,layered medium of viscoelastic liquids

Summary The free surface motion of a layered medium of liquids in a gravitationally stable configuration, resting on top of a layer of mercury, driven by a torsionally oscillating, cylindrical outer wall is investigated. The non-linear problem in the unknown physical domain is expressed as a series of linear problems in the rest state by means of a domain perturbation method. The flow variables and the stress are expanded into series in terms of the amplitude of the oscillation of the cylinder. The shapes in the mean of the interfaces between layers and the flow field are determined up to second order in the perturbation parameter, the amplitude of the oscillation.Nomenclature Density - Modified pressure field - Amplitude of the oscillation - Frequency of the oscillation - Interfacial value of the surface tension - Dynamic viscosity - , , Material functions - Complex viscosity - Stream function - Position vector at timet= - ₁, ₂ The first two Rivlin-Ericksen constants - Quadratic shear relaxation modulus - ,t Time - u Velocity vector - u,v,w Velocity components - S Extra stress tensor - h Interface elevation - D Stretching tensor - G Strain history tensor - A ₁ The first Rivlin-Ericksen tensor - J Mean curvature - p Pressure - t Unit tangent vector - n Unit normal vector - G Shear relaxation modulus - X Position vector in the rest stateD ₀ - r, ,z Rest state coordinates - x Position vector in the physical spaceD - R, ,Z Physical space coordinates - r ₀ Radius of the oscillating cylinder - e _r ,e ,e _z Physical basis vectors inD ₀ - e _R ,e ,e _Z Physical basis vectors inD - Indicates the jump in the enclosed quantity across an interface With 1 FigurePresented at the Xth Canadian Congress of Applied Mechanics, The University of Western Ontario, London, Ontario, Canada, June 2–7, 1985.