Global System Of Equations Research Articles

A heterogeneous multiscale modeling framework for hierarchical systems of partial differential equations

AbstractThis paper presents a heterogeneous multiscale method with efficient interscale coupling for scale‐dependent phenomena modeled via a hierarchy of partial differential equations. Physics at the global level is governed by one set of partial differential equations, whereas features in the solution that are beyond the resolution capability of the coarser models are accounted for by the next refined set of differential equations. The proposed method seamlessly integrates different sets of equations governing physics at various levels, and represents a consistent top–down and bottom–up approach to multi‐model modeling problems. For the top–down coupling of equations, this method provides a variational residual‐based embedding of the response from the coarser or global system equations, into the corresponding local or refined system equations. To account for the effects of local phenomena on the global response of the system, the method also accommodates bottom–up embedding of the response from the local or refined mathematical models into the global or coarser model equations. The resulting framework thus provides a consistent way of coupling physics between disparate partial differential equations by means of up‐scaling and down‐scaling of the mathematical models. An integral aspect of the proposed framework is an uncertainty quantification and error estimation module. The structure of this error estimator is investigated and its mathematical implications are delineated. Copyright © 2010 John Wiley & Sons, Ltd.

Ritz analysis of discontinuous beams using local trigonometric functions

The objective of the current paper is to present a Ritz-type analytical model for predicting the behavior of discontinuous beams such as thin-walled beams with cracks and multiply-stepped beams. The beam is discretized in the cracked as well as the un-cracked domains for a cracked thin-walled beam and in uniform beams for a multiple-stepped beam. A set of local trigonometric trial functions is used to define the twist angle for the cracked domain and the un-cracked domains, as well as to define the displacement field for uniform domains. A global equation system of unknown Ritz coefficients is derived by minimizing the Lagrangian functional or the total potential energy. In the present Ritz model, the interface continuity conditions between sub-domains are investigated and enforced into the global equation system using the condensation procedure or the Lagrange multipliers. Examples are presented to illustrate the effectiveness of the current model for free vibration and torsional analysis. Results obtained from the current model are found to agree well with those obtained using a detailed finite element method or with existing results in literature. The proposed model offers an efficient approach to reduce the modeling efforts and computational time required to analyze complex beams with cracks or multiple steps.

A dual mortar approach for 3D finite deformation contact with consistent linearization

AbstractIn this paper, an approach for three‐dimensional frictionless contact based on a dual mortar formulation and using a primal–dual active set strategy for direct constraint enforcement is presented. We focus on linear shape functions, but briefly address higher order interpolation as well. The study builds on previous work by the authors for two‐dimensional problems. First and foremost, the ideas of a consistently linearized dual mortar scheme and of an interpretation of the active set search as a semi‐smooth Newton method are extended to the 3D case. This allows for solving all types of nonlinearities (i.e. geometrical, material and contact) within one single Newton scheme. Owing to the dual Lagrange multiplier approach employed, this advantage is not accompanied by an undesirable increase in system size as the Lagrange multipliers can be condensed from the global system of equations. Moreover, it is pointed out that the presented method does not make use of any regularization of contact constraints. Numerical examples illustrate the efficiency of our method and the high quality of results in 3D finite deformation contact analysis. Copyright © 2010 John Wiley & Sons, Ltd.

A Schur complement formulation for solving free-boundary, Stefan problems of phase change

A Schur complement formulation that utilizes a linear iterative solver is derived to solve a free-boundary, Stefan problem describing steady-state phase change via the Isotherm-Newton approach, which employs Newton's method to simultaneously and efficiently solve for both interface and field equations. This formulation is tested alongside more traditional solution strategies that employ direct or iterative linear solvers on the entire Jacobian matrix for a two-dimensional sample problem that discretizes the field equations using a Galerkin finite-element method and employs a deforming-grid approach to represent the melt-solid interface. All methods demonstrate quadratic convergence for sufficiently accurate Newton solves, but the two approaches utilizing linear iterative solvers show better scaling of computational effort with problem size. Of these two approaches, the Schur formulation proves to be more robust, converging with significantly smaller Krylov subspaces than those required to solve the global system of equations. Further improvement of performance are made through approximations and preconditioning of the Schur complement problem. Hence, the new Schur formulation shows promise as an affordable, robust, and scalable method to solve free-boundary, Stefan problems. Such models are employed to study a wide array of applications, including casting, welding, glass forming, planetary mantle and glacier dynamics, thermal energy storage, food processing, cryosurgery, metallurgical solidification, and crystal growth.

Finite deformation frictional mortar contact using a semi‐smooth Newton method with consistent linearization

AbstractA two‐dimensional, finite deformation frictional contact formulation with Coulomb's law is presented. The approach considers multibody contact and is based on a mortar formulation. The enforcement of contact constraints is realized with dual Lagrange multipliers. These alternative multiplier spaces are constructed in a way that the multipliers can easily be eliminated from the global system of equations by static condensation such that the system size does not increase. Friction kinematic variables are formulated in an objective way and enter non‐smooth complementarity functions for expressing the contact constraints. An active set strategy is derived by applying a semi‐smooth Newton method, which treats contact nonlinearities, material and geometrical nonlinearities in one single iterative scheme. By further carrying out a consistent linearization for both normal and frictional contact forces and constraints, a robust and highly efficient algorithm for linear and higher‐order (quadratic) interpolation is achieved. Efficiency of the proposed method and quality of results are demonstrated in several examples. Copyright © 2010 John Wiley & Sons, Ltd.

A Study of Load Capacity of Finite Slider Bearings with Slip Surfaces and Stokesian Couple Stress Fluids

A mathematical model for the hydrodynamic lubrication of finite slider bearings with velocity slip and couple stress lubricants is presented. A numerical solution for the mathematical model using finite element scheme is obtained using four node linearly interpolated quadrilateral elements. Stiffness integrals obtained from the weak form of the governing equations were solved using Gauss Quadrature to obtain a finite number of stiffness matrices. The global system of equations was obtained for the bearing and solved using Gauss Seidel iterative scheme. The converged pressure solution was used to obtain the load capacity of the bearing. Numerical experiments reveal the existence of an optimum velocity slip for which maximum benefit is obtained for the slider bearing in terms of bearing load. Increase in the slip parameter beyond this optimum value was shown not to augment the bearing load. Computations put forth also affirm that the bearing load is augmented with increase in couple stress parameter. An optimal film thickness ratio was also obtained for which load capacity is maximized with or without the application of slip to the bearing surfaces.

Mixed meshless formulation for analysis of shell-like structures

An efficient mixed meshless computational method based on the Local Petrov–Galerkin approach for analysis of plate and shell structures is presented. A concept of a three-dimensional solid is applied allowing the use of complete three-dimensional constitutive equations, and exact shell geometry can be described. Discretization is carried out by using both the moving least square approximation and the polynomial functions. Independent field variables are the strain and stress tensor components expressed in terms of the nodal values, which are then replaced by the nodal displacements by using the independent displacement interpolation. A closed global system of equations with only nodal displacements as unknown variables is derived. The undesired locking phenomena are fully suppressed. The proposed mixed formulation is numerically more efficient than the available meshless fully displacement approach, as demonstrated by the numerical examples.

An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection–diffusion equations

In this paper, we present hybridizable discontinuous Galerkin methods for the numerical solution of steady and time-dependent nonlinear convection–diffusion equations. The methods are devised by expressing the approximate scalar variable and corresponding flux in terms of an approximate trace of the scalar variable and then explicitly enforcing the jump condition of the numerical fluxes across the element boundary. Applying the Newton–Raphson procedure and the hybridization technique, we obtain a global equation system solely in terms of the approximate trace of the scalar variable at every Newton iteration. The high number of globally coupled degrees of freedom in the discontinuous Galerkin approximation is therefore significantly reduced. We then extend the method to time-dependent problems by approximating the time derivative by means of backward difference formulae. When the time-marching method is ( p + 1 ) th order accurate and when polynomials of degree p ⩾ 0 are used to represent the scalar variable, each component of the flux and the approximate trace, we observe that the approximations for the scalar variable and the flux converge with the optimal order of p + 1 in the L 2 -norm. Finally, we apply element-by-element postprocessing schemes to obtain new approximations of the flux and the scalar variable. The new approximate flux, which has a continuous interelement normal component, is shown to converge with order p + 1 in the L 2 -norm. The new approximate scalar variable is shown to converge with order p + 2 in the L 2 -norm. The postprocessing is performed at the element level and is thus much less expensive than the solution procedure. For the time-dependent case, the postprocessing does not need to be applied at each time step but only at the times for which an enhanced solution is required. Extensive numerical results are provided to demonstrate the performance of the present method.

A finite deformation mortar contact formulation using a primal–dual active set strategy

AbstractIn recent years, nonconforming domain decomposition techniques and, in particular, the mortar method have become popular in developing new contact algorithms. Here, we present an approach for 2D frictionless multibody contact based on a mortar formulation and using a primal–dual active set strategy for contact constraint enforcement. We consider linear and higher‐order (quadratic) interpolations throughout this work. So‐called dual Lagrange multipliers are introduced for the contact pressure but can be eliminated from the global system of equations by static condensation, thus avoiding an increase in system size. For this type of contact formulation, we provide a full linearization of both contact forces and normal (non‐penetration) and tangential (frictionless sliding) contact constraints in the finite deformation frame. The necessity of such a linearization in order to obtain a consistent Newton scheme is demonstrated. By further interpreting the active set search as a semi‐smooth Newton method, contact nonlinearity and geometrical and material nonlinearity can be resolved within one single iterative scheme. This yields a robust and highly efficient algorithm for frictionless finite deformation contact problems. Numerical examples illustrate the efficiency of our method and the high quality of results. Copyright © 2009 John Wiley & Sons, Ltd.

Meshfree co‐rotational formulation for two‐dimensional continua

AbstractIn this paper, a meshfree co‐rotational formulation for two‐dimensional continua is proposed. In a co‐rotational formulation, the motion of a body is separated into rigid motion and strain‐producing deformation. Traditionally, this has been done in the setting of finite elements for beams and shell‐type elements. In the present work every node in a meshfree discretized domain has its own co‐rotating coordinate system. Three key ingredients are established in order to apply the co‐rotational formulation: (i) the relationship between global and local variables, (ii) the angle of rotation of a typical co‐rotating coordinate system, and (iii) a variationally consistent tangent stiffness matrix. An algorithm for the co‐rotational formulation based on load control is provided. Maximum‐entropy basis functions are used to discretize the domain and stabilized nodal integration is implemented to construct the global system of equations. Numerical examples are presented to demonstrate the validity of the meshfree co‐rotational formulation. Copyright © 2009 John Wiley & Sons, Ltd.

An implicit high-order hybridizable discontinuous Galerkin method for linear convection–diffusion equations

We present a hybridizable discontinuous Galerkin method for the numerical solution of steady and time-dependent linear convection–diffusion equations. We devise the method as follows. First, we express the approximate scalar variable and corresponding flux within each element in terms of an approximate trace of the scalar variable along the element boundary. We then define a unique value for the approximate trace by enforcing the continuity of the normal component of the flux across the element boundary; a global equation system solely in terms of the approximate trace is thus obtained. The high number of globally coupled degrees of freedom in the discontinuous Galerkin approximation is therefore significantly reduced. If the problem is time-dependent, we discretize the time derivative by means of backward difference formulae. This results in efficient schemes capable of producing high-order accurate solutions in space and time. Indeed, when the time-marching method is ( p + 1 ) th order accurate and when polynomials of degree p ⩾ 0 are used to represent the scalar variable, the flux and the approximate trace, we observe that the approximations for the scalar variable, the flux and the trace of the scalar variable converge with the optimal order of p + 1 in the L 2 -norm. Finally, we introduce a simple element-by-element postprocessing scheme to obtain new approximations of the flux and the scalar variable. The new approximate flux, which has a continuous inter-element normal component, is shown to converge with order p + 1 in the L 2 -norm. The new approximate scalar variable is shown to converge with order p + 2 in the L 2 -norm. For the time-dependent case, the postprocessing does not need to be applied at each time-step but only at the times for which an enhanced solution is required. Moreover, the postprocessing procedure is less expensive than the solution procedure, since it is performed at the element level. Extensive numerical results are presented to demonstrate the convergence properties of the method.

Quadrilateral Subcell Based Finite Volume Micromechanics Theory for Multiscale Analysis of Elastic Periodic Materials

In this paper, we extend the finite volume direct average micromechanics to enable the use of quadrilateral subcells. To do this work, the quadrilateral subcells are used to discretize the repeating unit cells first. Then the average displacement and traction defined on the boundary of the subcell are evaluated by direct integral method. This contrasts with the original formulation in which all of the subcells are rectangular. Following the discretization, the cell problem is defined by combining the directly volume-average of the subcell stress equilibrium equations with the displacement and traction continuity in a surface-average sense across the adjacent subcell faces. In order to assemble the above equations and conditions into a global equation system, the global and local number systems, which index the boundary of subcell in different manners, are employed by the extended method. Finally, the global equation system is solved and the solutions give the formulations of the microstress field and the global elastic moduli of material. The introduction of quadrilateral subcells increases the efficiency of modeling the material’s microstructure and eliminates the stress concentrations at the curvilinear bimaterial corners. Herein, the advantage of the extension is presented by comparing the global moduli and local stress fields predicted by the present method with the corresponding results obtained from the original version.

New finite elements with embedded strong discontinuities in the finite deformation range

This paper develops new finite elements with embedded strong discontinuities for the modeling of the failure of solids in the finite deformation range. The cases of interest include cracks, shear bands, delamination in composites and similar failures modeled by a propagating discontinuity in the deformation mapping that incorporates a cohesive law in terms of the associated displacement jumps. The new elements accommodate the discontinuity in their interiors with a linear interpolation of the displacement jumps in contrast to the existing elements with constant jumps. A general methodology is proposed for the development of such finite elements, but the final focus is on quadrilateral elements for plane problems where a constant jump interpolation leads to a spurious transfer of stresses through the discontinuity (or stress locking) in certain common configurations. The key aspect of the new elements is the proper enhancement of the deformation gradient that can represent correctly the kinematics of linearly separating discontinuities, including a relative finite rotation across the discontinuity, and in a frame indifferent manner. The enhanced parameters modeling the displacement jumps are introduced independently for each element, allowing for their static condensation at the element level and, hence, to arrive at a global system of equations with the same number and connectivity of the original global degrees of freedom for the underlying finite element mesh. This situation leads to a particularly simple implementation of the proposed approach in an existing finite element code. The performance of the new elements is illustrated with several numerical simulations, including delamination in composite materials and propagating shear bands in a finitely deforming elastoplastic solid.

Jacobi galerkin spectral method for cylindrical and spherical geometries

The approximation of the convection–diffusion problem based on the Galerkin method in Cartesian, cylindrical and spherical coordinates is considered with emphasis in the last two cases. In particular, cylindrical and spherical coordinates can lead to a degeneracy in the global system of equations. This difficulty is removed by incorporating the factor r into the weight function which is introduced naturally by using Jacobi polynomials P k ( α , β ) with α = 0 and β = 1 , 2 . By doing this, an unified framework is obtained for handling the typical geometries required in chemical engineering. Examples are presented based on the Galerkin method for discussing the applicability of this approach.

A solution methodology for contacting domains in pressure die casting

Multi-domain elastostatic problems can often be efficiently solved using coupled single domain iterative techniques. A particular difficulty however is the decrease in convergence rate associated with the increase in number of solution domains. Further convergence difficulties are encountered for multi-domain problems where contacting domains suffer variable contact conditions. Problems of this type are evident in pressure die casting with die blocks coupled together prior and subsequent to thermal loading. Particular interest in this paper is the development of an efficient solution methodology for prediction of gaps at block interfaces in pressure die casting. The methodology developed incorporates a coarse preconditioner designed to enhance the overall system stability. The governing equations across domains are coupled by means of a multiplicative-Schwarz method for non-overlapping domains, as focus is on the use of serial processing. The coarse preconditioner is obtained from a crude representation of the global system of equations although high accuracy is generally required at the contacting interfaces. Attention is restricted to thermo-elastostatic problems arising in pressure die casting with domains connected through spring interfaces. The effect of lowering and increasing the interfacial stiffness coefficients between domains is investigated. In addition, it is demonstrated how variable contact arising through thermally induced distortion is accounted for by the removal and attachment of interface springs. The coarse preconditioner is shown to be particularly beneficial to variable contact problems providing stability to the overall assembly. Computation times are determined for the iterative procedures and for elimination techniques indicating the relative benefits for problems of this nature. In addition, predictions are compared against experiment results to demonstrate the practical worth of the method for predicting gaps at die interfaces and the consequential flashing that can often result.

A Second Look at the Higher-Order Theory for Periodic Multiphase Materials

In this communication, we present a reformulation, based on the local/global stiffness matrix approach, of the recently developed higher-order theory for periodic multiphase materials, Aboudi et al. [“Linear Thermoelastic Higher-Order Theory for Periodic Multiphase Materials,” J. Appl. Mech., 68(5), pp. 697–707]. This reformulation reveals that the higher-order theory employs an approximate, and standard, elasticity approach to the solution of the unit cell problem of periodic multiphase materials based on direct volume-averaging of the local field equations and satisfaction of the local continuity conditions in a surface-averge sense. This contrasts with the original formulation in which different moments of the local equilibrium equations were employed, suggesting that the theory is a variant of a micropolar, continuum-based model. The reformulation simplifies the derivation of the global system of equations governing the unit cell response, whose size is substantially reduced through elimination of redundant continuity equations employed in the original formulation, allowing one to test the theory’s predictive capability in most demanding situations. Herein, we do so by estimating the elastic moduli of periodic composites characterized by repeating unit cells obtained by rotation of an infinite square fiber array through an angle about the fiber axis. Such unit cells possess no planes of material symmetry in the rotated coordinate system, and may contain a few or many fibers, depending on the rotation angle, which the reformulated theory can easily accommodate. The excellent agreement with the corresponding results obtained from the standard transformation equations confirms the new model’s previously untested predictive capability for a class of periodic composites characterized by nonstandard, multi-inclusion repeating unit cells lacking planes of material symmetry. Comparison of the effective moduli and local stress fields with the corresponding results obtained from the original Generalized Method of Cells, which the higher-order theory supersedes, confirms the need for this new model, and dramatically highlights the original model’s shortcomings for a certain class of unidirectional composites.

A conservative flux for the continuous Galerkin method based on discontinuous enrichment

In this paper we develop techniques for computing elementwise conservative approximations of the flux on element boundaries for the continuous Galerkin method. The technique is based on computing a correction of the average normal flux on an edge or face. The correction is a jump in a piecewise constant or linear function. We derive a basic algorithm which is based on solving a global system of equations and a parallel algorithm based on solving local problems on stars. The methods work on meshes with different element types and hanging nodes. We prove existence, uniqueness, and optimal order error estimates. Lastly, we illustrate our results by a few numerical examples.

On zero‐thickness interface elements for diffusion problems

AbstractThe present study focuses on the hydraulic behaviour of joints, and, specially, on its numerical implementation in terms of the FEM analysis using a discrete fracture flow approach. Fluid flow through discontinuities has traditionally been modelled using special elements of zero‐thickness, which we can classify into single, double and triple‐nodded. Single node elements are the simplest and consist of ‘line’ or ‘pipe’ elements which are superimposed onto the standard continuum mesh and that can only model the longitudinal conductivity through the discontinuity. On the other hand, some authors have included the influence of a transversal conductivity, and the subsequent localized potential drop, by using triple node interface elements. In those, the two nodes of the adjacent continuum elements represent the potentials in the pore system on each side of the interface, and a third node in the middle represents the average potential of the fluid in the channel represented by the discontinuity. Finally, double node interface elements have also been proposed, which have the advantage of making it possible to use the same FE mesh for both mechanical and flow analysis. In some cases the influence of a transversal conductivity is not considered and, therefore, although geometrically double‐nodded, these elements belong to the single node type and when time comes to solve the system the two nodes must have the same potential, which can only be obtained by the ‘trick’ of prescribing the equivalence of these two d.o.f. before solving the global system of equations. This limitation may, however, be avoided by assuming that the potential in the channel is the average of the two sides of the interface. Based in this simple assumption, an alternative flow interface model has been recently developed and implemented, which preserves both longitudinal and transversal conductivities. An application example is developed and solved with the three types of interfaces described. The results offer useful information regarding the range of applicability and limitations of the new double‐nodded interface element proposed. Copyright © 2004 John Wiley & Sons, Ltd.

Interpolation with slackness and continuity control and convexity-preservation using singular blending

This paper presents a new interpolation method that enables the construction of C 2 cubic polynomial spline curves without solving a global system of equations, while providing slackness/continuity control and convexity preserving ability. The basic idea is to blend a cubic B-spline curve with a singularly parametrized sequence of connected line segments. A global slackness parameter controls the tautness, specifically the distance between the interpolating curve and the linear interpolant. The order of continuity at each knot is controlled via multiple knot insertions so that cusps and straight-line segments can be conveniently prescribed. In addition, a method for selecting local slackness values to produce G 1 convexity preserving curve is presented. With the low-degree polynomials and direct computation of control vertices, this local method is computationally simple and is useful for interactive shape design and computer graphics applications.

Application of the material force method to thermo-hyperelasticity

The numerical analysis of material forces in the context of thermo-hyperelasticity constitutes the central topic of the present paper. In contrast to classical spatial forces in the sense of Newton, material forces in the sense of Eshelby indicate the tendency of material inhomogeneities to move relative to their surrounding material. Material forces are thus considered of particular importance in the context of thermo-elasticity where thermal effects can be understood as a potential source of inhomogeneity. The relevant balance equations of thermo-elasticity, i.e. the balance of momentum and energy, which essentially govern the evolution of the deformation and the temperature field are thus illustrated for both, the classical spatial and the material motion context. Guided by arguments of duality, the corresponding weak forms are derived. Next, we carry out the finite element discretization of both problems. While the numerical solution of the spatial motion problem renders the discrete spatial deformation map and the temperature as nodal degrees of freedom, the solution of the material motion problem provides the discrete material node point forces. The former typically relies on the solution of a global system of equations whereas the latter is introduced as a mere post-processing procedure. Since we apply a simultaneous solution of the mechanical and the thermal problem with the deformation and the temperature interpolated in a C 0 -continuous way, all the relevant information for the material force method is readily available once the spatial motion problem has been solved. Selected examples from the field of fracture mechanics illustrate the additional insight that is provided by the results of the material force method.