block partitioned solvers for coupled poromechanics a unified framework abstract coupled poromechanical problems appear in a variety of disciplines from reservoir engineering to biomedical applications this work focuses on efficient strategies for solving the matrix systems that result from discretization and linearization of the governing equations these systems have an inherent block structure due to the coupled nature of the mass and momentum balance equations recently several iterative solution schemes have been proposed that exhibit stable and rapid convergence to the coupled solution these schemes appear distinct but a unifying feature is that they exploit the block partitioned nature of the problem to accelerate convergence this paper analyzes several of these schemes and highlights the fundamental connections that underlie their effectiveness we begin by focusing on two specific methods a fully implicit and a sequential implicit scheme in the first approach the system matrix is treated monolithically and a krylov iteration is used to update pressure and displacement unknowns simultaneously to accelerate convergence a preconditioning operator is introduced based on an approximate block factorization of the linear system next we analyze a sequential implicit scheme based on the fixed stress split in this method one iterates back and forth between updating displacement and pressure unknowns separately until convergence to the coupled solution is reached we re interpret this scheme as a block preconditioned richardson iteration and we show that the preconditioning operator is identical to that used within the fully implicit approach rapid convergence in both the richardson and krylov based methods results from a particular choice for a sparse schur complement approximation this analysis leads to a unified framework for developing solution schemes based on approximate block factorizations many classic fully implicit and sequential implicit schemes are simple sub cases the analysis also highlights several new approaches that have not been previously explored for illustration we directly compare the performance and robustness of several variants on a benchmark problem

stabilization of partitioned solution procedure for pore fluid soil interaction analysis on montre que la solution decalee conventionnelle des problemes d interaction fluide sol dans les pores souffre de restrictions sur la stabilite on presente une forme stabilisee de la procedure de solution decalee de ce probleme on recommande cette procedure pour l implementation des problemes d interaction sol fluide

diffusion in poro elastic media existence uniqueness and regularity theory is developed for a general initial boundary value problem for a system of partial differential equations which describes the biot consolidation model in poro elasticity as well as a coupled quasi static problem in thermoelasticity additional effects of secondary consolidation and pore fluid exposure on the boundary are included this quasi static system is resolved as an application of the theory of linear degenerate evolution equations in hilbert space and this leads to a precise description of the dynamics of the system

minimal surfaces in finite volume non compact hyperbolic 3 manifolds we prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic 3 manifold mathcal n we also obtain a least area incompressible properly embedded finite topology 2 sided surface we prove a properly embedded minimal surface of bounded curvature has finite topology this determines its asymptotic behavior some rigidity theorems are obtained

standard staggered and staggered newton schemes in thermo hydro mechanical problems the aim of the paper is to compare the rates of iteration convergence of two staggered numerical schemes one using successive substitutions and the other the newton method in the case of very weakly coupled problems the newton method is one order faster but there are specific cases when it is slower than the standard staggered algorithm in both cases a convergence condition must be verified the behaviour of the investigated staggered procedures in case of thermo hydro mechanical problems are shown on examples

unconditionally stable staggered solution procedure for soil pore fluid interaction problems this paper presents a simple procedure for modifying the dynamic coupled soil pore fluid equations in a manner which allows a stable and efficient staggered solution procedure to be used the solution method is simpler than an alternative suggested by park15 and has a direct physical interpretation this stabilized procedure is efficient in numerical time domain analysis of soil pore fluid occurring in earthquake and consolidation problems numerical examples of one and two dimensional saturated poroelastic media demonstrate the procedure

efficient solvers for hybridized three field mixed finite element coupled poromechanics abstract we consider a mixed hybrid finite element formulation for coupled poromechanics a stabilization strategy based on a macro element approach is advanced to eliminate the spurious pressure modes appearing in undrained incompressible conditions the efficient solution of the stabilized mixed hybrid block system is addressed by developing a class of block triangular preconditioners based on a schur complement approximation strategy robustness computational efficiency and scalability of the proposed approach are theoretically discussed and tested using challenging benchmark problems on massively parallel architectures

three dimensional finite element simulation of three phase flow in a deforming fissured reservoir the development of a capacity to predict the exploitation of structurally complicated and fractured oil reservoirs is essential for the rational use of investment capital a poor understanding of how the reservoir behaves during production may lead to inept costly and inefficient development schemes the mathematical formulation of a three phase three dimensional fluid flow and rock deformation in fractured reservoirs is hence presented the present formulation consisting of both the equilibrium and multiphase mass conservation equations accounts for the significant influence of coupling between the fluid flow and solid deformation an aspect usually ignored in the reservoir simulation literature a galerkin based finite element method is applied to discretise the governing equations in space and a finite difference scheme is used to march the solution in time the final set of equations which contain the additional cross coupling terms as compared to similar existing models are highly non linear and the elements of the coefficient matrices are updated implicitly during each iteration in terms of the independent variables a field scale example is employed as an alpha case to test the validity and robustness of the currently formulation and numerical scheme the results illustrate a significantly different behaviour for the case of a reservoir where the impact of coupling is also considered

minimal hypersurfaces of least area in this paper we study closed embedded minimal hypersurfaces in a riemannian n 1 manifold 2 n 6 that minimize area among such hypersurfaces we show they exist and arise either by minimization techniques or by min max methods they have index at most 1 we apply this to obtain a lower area bound for such minimal surfaces in some hyperbolic 3 manifolds

guaranteed and computable error bounds for approximations constructed by an iterative decoupling of the biot problem abstract the paper is concerned with guaranteed a posteriori error estimates for a class of evolutionary problems related to poroelastic media governed by the quasi static linear biot equations the system is decoupled by employing the fixed stress split scheme which leads to an iteratively solved semi discrete system the error bounds are derived by combining a posteriori estimates for contractive mappings with functional type error control for elliptic partial differential equations the estimates are applicable to any approximation in the admissible functional space and are independent of the discretization method they are fully computable do not contain mesh dependent constants and provide reliable global estimates of the error measured in the energy norm moreover they suggest efficient error indicators for the distribution of local errors and can be used in adaptive procedures

two way coupling in reservoir geomechanical models vertex centered galerkin geomechanical model cell centered and vertex centered finite volume reservoir models procedures to couple reservoir and geomechanical models are reviewed the focus is on immiscible compressible non compositional reservoir geomechanical models such models require the solution to coupled stress pressure saturation and temperature equations although the couplings between saturation and temperature with stress and fluid pressure are weak and can be adequately captured thru staggered fixed point iterations the couplings between stress and pressure equations are strong and require special procedures for accurate integration as shown and discussed in detail in prevost 2013 two way coupling i e simultaneous integration of pressure and stress equations is required if poromechanical effects are to be captured accurately in our previous work a galerkin implementation of both pressure and stress equations was used with equal order interpolant for both pressure and solid displacements

a mathematical framework for finite strain elastoplastic consolidation part 1 balance laws variational formulation and linearization a mathematical formulation for finite strain elasto plastic consolidation of fully saturated soil media is presented strong and weak forms of the boundary value problem are derived using both the material and spatial descriptions the algorithmic treatment of finite strain elastoplasticity for the solid phase is based on multiplicative decomposition and is coupled with the algorithm for fluid flow via the kirchhoff pore water pressure balance laws are written for the soil water mixture following the motion of the soil matrix alone it is shown that the motion of the fluid phase only affects the jacobian of the solid phase motion and therefore can be characterized completely by the motion of the soil matrix furthermore it is shown from energy balance consideration that the effective or intergranular stress is the appropriate measure of stress for describing the constitutive response of the soil skeleton since it absorbs all the strain energy generated in the saturated soil water mixture finally it is shown that the mathematical model is amenable to consistent linearization and that explicit expressions for the consistent tangent operators can be derived for use in numerical solutions such as those based on the finite element method

von neumann stability analysis of biot s general two dimensional theory of consolidation von neumann stability analysis is performed for a galerkin finite element formulation of biot s consolidation equations on two dimensional bilinear elements two dimensionless groups the time factor and void factor are identified and these quantities along with the time integration weighting are used to explore the stability implications for variations in physical property and discretization parameters the results show that the presence and persistence of stable spurious oscillations in the pore pressure are influenced by the ratio of time step size to the square of the space step for fixed time integration weightings and physical property selections in general increasing the time step or decreasing the mesh spacing has a smoothing effect on the discrete solution however special cases exist that violate this generality which can be readily identified through the von neumann approach the analysis also reveals that explicitly dominated schemes are not stable for saturated media and only become possible through a decoupling of the equilibrium and continuity equations in the case of unsaturated media a break down in the von neumann results has been shown to occur due to the influence of boundary conditions on stability 1998 john wiley sons ltd

formulation and finite element implementation of a multiplicative model of coupled poro plasticity at finite strains under fully saturated conditions in this paper we present the formulation and numerical analysis of a finite deformation model of coupled poro plasticity fully saturated conditions are assumed the model is based on a multiplicative decomposition of the deformation gradient in elastic and plastic parts together with an additive elastoplastic decomposition of the fluid content the final relations describing the evolution of the different fields are then obtained in the thermodynamic framework furnished by the principle of maximum plastic dissipation the particular case of an associated drucker prager model in effective stresses is considered as a model example a complete characterization of the dissipative structure of the problem of evolution is described through the derivation of an a priori stability estimate we describe in detail the finite element implementation of the model in the context of staggered algorithms for the solution of the physically coupled problem finally we present representative numerical simulations to illustrate the features of the proposed models in particular the case of strain localization in globally undrained conditions is studied in the proposed framework

dual algorithms for pure network problems this paper reports the development of a new algorithmic implementation of the dual algorithm for the capacitated minimum cost network flow problem furthermore it introduces dual reoptimization procedures and compares primal and dual algorithms for the optimization and reoptimization of network problems the implementation makes use of cut sets for the efficient execution of the entering variable selection and the selection of the leaving variable is tied to the basis structure at an iteration empirical testing of the dual algorithm for optimization shows that it dominates the primal procedure for assignment problems with 400 nodes transportation problems with more than 600 nodes and transshipment problems with more than 1500 nodes computational testing with typical changes found in decomposition and relaxation procedures indicates the superiority of dual reoptimization over primal reoptimization for a sequence of parametric changes as would be typical in large scale programming techniques on the

fractional step methods for index 1 differential algebraic equations in the numerical solution of ordinary differential systems the method of fractional steps also known as operator splitting yields high order accurate schemes based on separate computationally convenient treatments of distinct physical effects such schemes are equally desirable but much less accurate for semi explicit index 1 differential algebraic equations daes in the first half of this note it is shown that email protected ve application to daes of standard splitting schemes suffers from order reduction both first and second order schemes are only first order accurate for daes in the second half of this note a new family of higher order splitting schemes for semi explicit index 1 daes is developed the new schemes are based on a deferred correction paradigm in which an error equation is solved numerically and therefore inherit a simple computationally convenient structure higher order convergence of the new schemes is proved and numerical results confirm the expected order of accuracy in addition to establishing efficiency

stability of multirate explicit coupling of geomechanics with flow in a poroelastic medium abstract we consider single rate and multirate explicit schemes for the biot system modeling coupled flow and geomechanics in a poro elastic medium these schemes are widely used in practice that follows a sequential procedure in which the flow and mechanics problems are fully decoupled in such a scheme the flow problem is solved first with time lagging the displacement term followed by the mechanics solve the multirate explicit coupling scheme exploits the different time scales for the mechanics and flow problems by taking multiple finer time steps for flow within one coarse mechanics time step we provide fully discrete schemes for both the single and multirate approaches that use backward euler time discretization and mixed spaces for flow and conformal galerkin for mechanics we perform a rigorous stability analysis and derive the conditions on reservoir parameters and the number of finer flow solves to ensure stability for both schemes furthermore we investigate the computational time savings for explicit coupling schemes against iterative coupling schemes

iterative coupling of flow geomechanics and adaptive phase field fracture including level set crack width approaches in this work we present numerical studies of fixed stress iterative coupling for solving flow and geomechanics with propagating fractures in a porous medium specifically fracture propagations are described by employing a phase field approach the extension to fixed stress splitting to propagating phase field fractures and systematic investigation of its properties are important enhancements to existing studies moreover we provide an accurate computation of the fracture opening using level set approaches and a subsequent finite element interpolation of the width the latter enters as fracture permeability into the pressure diffraction problem which is crucial for fluid filled fractures our developments are substantiated with several numerical tests that include comparisons of computational cost for iterative coupling and nonlinear and linear iterations as well as convergence studies in space and time fixed stress formulations for fluid filled phase field fractures accurate fracture width finite element computations using level sets numerical examples in 2d and 3d with spatial and temporal convergence studies

on convergence conditions of partitioned solution procedures for consolidation problems abstract a systematic approach to determine conditions for convergence of partitioned staggered procedures for consolidation problems is presented consistency convergence and stability is investigated for linear and nonlinear problems and a comparison of the partitioned and direct method is made conclusions based also on numerical evidence are drawn the significance of the ratio of the time step length to the square of the finite element mesh length is shown

a stabilized iterative scheme for coupled hydro mechanical systems using reproducing kernel particle method summary in this paper a novel iterative coupling scheme is developed for solving coupled hydro mechanical problems using reproducing kernel particle method the numerical scheme calls the fluid and the solid solvers sequentially and iteratively until convergent solutions are obtained to overcome the numerical instability problem a simple stabilization technique is developed and proved to be unconditionally stable through stability analysis the accuracy and convergence of the proposed numerical scheme are demonstrated through extensive parametric studies of one dimensional and two dimensional consolidation simulations using fully saturated elastic medium as well as biaxial test using a fully nonlinear soil model copyright 2014 john wiley sons ltd

minimal hypersurfaces with bounded index we prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index and volume if the ambient dimension is greater than three in a riemannian manifold m n g 3 le n le 7 can degenerate loosely speaking our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces up to controlled errors several compactness finiteness theorems follow from our local picture

stabilized lowest order finite element approximation for linear three field poroelasticity a stabilized conforming mixed finite element method for the three field displacement fluid flux and pressure poroelasticity problem is developed and analyzed we use the lowest possible approximation order namely piecewise constant approximation for the pressure and piecewise linear continuous elements for the displacements and fluid flux by applying a local pressure jump stabilization term to the mass conservation equation we ensure stability and avoid pressure oscillations importantly the discretization leads to a symmetric linear system for the fully discretized problem we prove existence and uniqueness an energy estimate and an optimal a priori error estimate including an error estimate for the divergence of the fluid flux numerical experiments in two and three dimensions illustrate the convergence of the method show its effectiveness in overcoming spurious pressure oscillations and evaluate the added mass effect of the stabilization term

stability and convergence of sequential methods for coupled flow and geomechanics fixed stress and fixed strain splits abstract we analyze stability and convergence of sequential implicit methods for coupled flow and geomechanics in which the flow problem is solved first we employ the von neumann and energy methods for linear and nonlinear problems respectively we consider two sequential methods with the generalized midpoint rule for tn where is the parameter of time discretization namely the fixed strain and fixed stress splits the von neumann method indicates that the fixed strain split is only conditionally stable and that its stability limit is a coupling strength less than unity if 0 5 on the other hand the fixed stress split is unconditionally stable when 0 5 the amplification factors of the fixed stress split are different from those of the undrained split and are identical to the fully coupled method unconditional stability of the fixed stress split is also obtained from the energy method for poroelastoplasticity we show that the fixed stress split is contractive and b stable when 0 5 we also estimate the convergence behaviors for the two sequential methods by the matrix based and spectral analyses for the backward euler method in time from the estimates the fixed strain split may not be convergent with a fixed number of iterations particularly around the stability limit even though it is stable the fixed stress split however is convergent for a fixed number of iterations showing better accuracy than the undrained split even when we cannot obtain the exact local bulk modulus or exact rock compressibility at the flow step a priori due to complex boundary conditions or the nonlinearity of the materials the fixed stress split can still provide stability and convergence by an appropriate estimation of the local bulk modulus such as the dimension based estimation by which the employed local bulk modulus is less stiff than the exact local bulk modulus we provide numerical examples supporting all the estimates of stability and convergence for the fixed strain and fixed stress splits

on the mechanical energy and effective stress in saturated and unsaturated porous continua abstract this paper has two objectives a to formulate a mechanical theory of porous continua within the framework of strong discontinuity concept commonly employed for the analysis of strain localization in inelastic solids and b to introduce an effective stress tensor for three phase solid water air partially saturated porous continua emerging from principles of thermodynamics to achieve the first objective strong forms of the boundary value problems are compared between two formulations the first in which the velocity jump at the solid fluid interface is treated as a strong discontinuity problem and the second in which the strong discontinuity is smeared in the representative volume element as for the second objective an effective cauchy stress tensor of the form 1 k k s p 1 emerges from the formulation where is the total stress tensor k and ks are the bulk moduli of the solid matrix and solid phase respectively and p is the mean pore water and pore air pressures weighted according to the degree of saturation we show from the first and second laws of thermodynamics that this effective stress tensor is power conjugate to the solid rate of deformation tensor and that it includes the mechanical power required to compress the solid phase

a new numerically stable sequential algorithm for coupled finite strain elastoplastic geomechanics and flow abstract we propose a new numerically stable sequential implicit method for coupled flow and finite strain multiplicative elastoplastic geomechanics we find from stability analysis that the sequential method that solves the flow problem first by fixing the first piola kirchhoff total stress solving the geomechanics solid deformation problem at the next step is unconditionally stable contractive and b stable in this sense this method named the fixed first piola kirchhoff stress method is an extension of the fixed stress method in coupled flow and infinitesimal geomechanics we also study the fixed second piola kirchhoff stress method comparing it with the fixed first piola kirchhoff stress method because the constitutive relations are formulated by the second piola kirchhoff total stress although fixing the second piola kirchhoff total stress field does not provide theoretical unconditional stability in space discretization we use the finite element method for the geomechanics problem with the total lagrangian approach while employing the finite volume method for the flow problem geometrical nonlinearity from the total lagrangian approach results in full tensor permeability even if the initial permeability is isotropic to consider the full tensor permeability accurately we employ the multipoint flux approximation in solving the flow problem in time discretization the backward euler method is used then we perform numerical experiments of the two fixed stress sequential methods with various scenarios and find superiority of their numerical stability specifically the two fixed stress sequential methods provide almost identical results for elasticity for j 2 plasticity the two sequential methods yield slightly different numerical results although the trends of the results are similar for the drucker prager plasticity we find that the results from the two methods are almost identical for all the test cases the two sequential methods are numerically stable

flow through fissured porous media with deformable matrix an approach which is based on the double porosity concept and takes into account pore deformation has been presented to derive a set of coupled differential equations governing the behaviour of fissured porous media this approach has resulted in a set of non linear variable coefficient differential equations various coefficients involved in the formulation have been explicitly defined in terms of measurable physical parameters the finite element technique has been employed as the numerical tool for the solution of these equations the results based on the proposed non linear formulation have been compared with those of previously presented linear constant coefficient formulations it is found that in relatively rigid formations the linearity assumption is quite reasonable and can successfully model the behaviour of fissured porous media however in very deformable formations the linearity assumption could lead to a significant level of error in the numerical solutions

stability and convergence of sequential methods for coupled flow and geomechanics drained and undrained splits abstract we perform a stability and convergence analysis of sequential methods for coupled flow and geomechanics in which the mechanics sub problem is solved first we consider slow deformations so that inertia is negligible and the mechanical problem is governed by an elliptic equation we use biot s self consistent theory to obtain the classical parabolic type flow problem we use a generalized midpoint rule parameter between 0 and 1 time discretization and consider two classical sequential methods the drained and undrained splits the von neumann method provides sharp stability estimates for the linear poroelasticity problem the drained split with backward euler time discretization 1 is conditionally stable and its stability depends only on the coupling strength and it is independent of time step size the drained split with the midpoint rule 0 5 is unconditionally unstable the mixed time discretization with 1 0 for mechanics and 0 5 for flow has the same stability properties as the backward euler scheme the von neumann method indicates that the undrained split is unconditionally stable when 0 5 we extend the stability analysis to the nonlinear regime poro elastoplasticity via the energy method it is well known that the drained split does not inherit the contractivity property of the continuum problem thereby precluding unconditional stability for the undrained split we show that it is b stable therefore unconditionally stable at the algorithmic level when 0 5 we also analyze convergence of the drained and undrained splits and derive the a priori error estimates from matrix algebra and spectral analysis we show that the drained split with a fixed number of iterations is not convergent even when it is stable the undrained split with a fixed number of iterations is convergent for a compressible system i e finite biot modulus for a nearly incompressible system i e very large biot modulus the undrained split loses first order accuracy and becomes non convergent in time we also study the rate of convergence of both splits when they are used in a fully iterated sequential scheme when the medium permeability is high or the time step size is large which corresponds to a high diffusion of pressure the error amplification of the drained split is lower and therefore converges faster than the undrained split the situation is reversed in the case of low permeability and small time step size we provide numerical experiments supporting all the stability and convergence estimates of the drained and undrained splits in the linear and nonlinear regimes we also show that our spatial discretization finite volumes for flow and finite elements for mechanics removes the well documented spurious instability in consolidation problems at early times

on the optimization of the fixed stress splitting for biot s equations in this work we are interested in efficiently solving the quasi static linear biot model for poroelasticity we consider the fixed stress splitting scheme which is a popular method for iterative

on the fixed stress split scheme as smoother in multigrid methods for coupling flow and geomechanics the fixed stress split method has been widely used as solution method in the coupling of flow and geomechanics in this work we analyze the behavior of an inexact version of this algorithm as smoother within a geometric multigrid method in order to obtain an efficient monolithic solver for biot s problem this solver combines the advantages of being a fully coupled method with the benefit of decoupling the flow and the mechanics part in the smoothing algorithm moreover the fixed stress split smoother is based on the physics of the problem and therefore all parameters involved in the relaxation are based on the physical properties of the medium and are given a priori a local fourier analysis is applied to study the convergence of the multigrid method and to support the good convergence results obtained the proposed geometric multigrid algorithm is used to solve several tests in semi structured triangular grids in order to show the good behavior of the method and its practical utility

cell centered finite volume method for heterogeneous anisotropic poromechanics problem abstract the article introduces a finite volume method for the poromechanics problem for heterogeneous anisotropic media the media are characterized by the full tensors of elasticity permeability and biot coefficient the pressure and displacement unknowns are collocated at centers of cells poromechanics represents a multi physics problem of darcy flow and linear elasticity coupled via biot terms based on the flux and the displacement and pressure continuity conditions we derive the coupled block expression for the flux the flux expression is split into a block harmonic part a transversal part and an additional contribution due to gravity the block harmonic part is equivalent to the two point flux approximation method for the darcy problem for the approximation of the transversal part we derive an interpolation method for the coupled pressure and displacement fields valid over discontinuities of the tensors finally we derive the flux expression for generalized boundary conditions the flux expression does not require introduction of auxiliary unknowns at the boundary we evaluate the performance of the proposed method on a number of numerical tests and apply the method to a problem on a grid corresponding to a real oil field

obtaining a good primal solution to the uncapacitated transportation problem abstract sharma and sharma european journal of operational research 122 3 2000 37 have given a new procedure to solve the dual of the well known uncapacitated transportation problem this is expected to enhance the performance of dual based optimizing algorithms for solving the transportation problems in this paper we give a heuristic that obtains a very good starting solution with a duality gap of less than 2 for the primal transportation problem in o n 3 time and this is expected to enhance the performance of network simplex algorithm that obtains the optimal solution and runs in o n 3 log n time empirical investigation revealed that our heuristic gave significantly better solutions than the well known vogel s approximation method that runs in o n 2 time

a convergence study of monolithic simulations of flow and deformation in fractured poroelastic media a consistent linearisation has been carried out for a monolithic solution procedure of a poroelastic medium with fluid transporting fractures including a comprehensive assessment of the convergence behaviour the fracture has been modelled using a sub grid scale model with a continuous pressure across the fracture the contributions to the tangential stiffness matrix of the fracture have been investigated to assess their impact on convergence simulations have been carried out for different interpolation orders and for non uniform rational b splines as interpolants vs lagrangian polynomials to increase the generality of the results newtonian as well as non newtonian power law fluids have been considered unsurprisingly a consistent linearisation invariably yields a quadratic convergence but comes at the expense of a loss of symmetry and recalculation of the contribution of the interface to the stiffness matrix at each iteration when using a linear line search however the inclusion of only those terms of the interface stiffness which result in a symmetric and constant tangential stiffness matrix is sufficient to obtain a stable and convergent iterative process

a new dual based procedure for the transportation problem abstract we pose the transportation problem in a slightly different form to obtain a dual problem that has a special structure we exploit the structure of the dual problem to give a new heuristic that runs in o cn 2 time n is number of nodes in the network and c is a constant to improve the dual solution our heuristic obtained the optimal solution to several small sized transportation problems that we attempted and for large sized problems it obtained solutions that were within 82 of the optimal solution on average our approach gives good starting solutions for dual based approaches used for solving transportation problems and thus is expected to enhance their computational performance

non linear b stability and symmetry preserving return mapping algorithms for plasticity and viscoplasticity a class of second order accurate return mapping algorithms is presented which lead to symmetric algorithmic tangent moduli and contain the classical backward euler return maps as a particular case more importantly it is shown that this class of return maps is contractive relative to the natural norm defined by the complementary helmholz free energy function b stability since the equations of classical plasticity and viscoplasticity are shown to be contractive relative to this natural norm the requirement of b stability furnishes the appropriate notion of unconditionally stable algorithms for plasticity and viscoplasticity the analysis that follows depends critically on the assumption of convexity in particular the models of plasticity and viscoplasticity considered obey the principle of maximum plastic dissipation the proposed algorithms obey the discrete counterpart of this classical principle

a partially parallel in time fixed stress splitting method for biot s consolidation model in this work we study the parallel in time iterative solution of coupled flow and geomechanics in porous media modelled by a two field formulation of biot s equations in particular we propose a new version of the fixed stress splitting method which has been widely used as solution method of these problems this new approach forgets about the sequential nature of the temporal variable and considers the time direction as a further direction for parallelization the method is partially parallel in time we present a rigorous convergence analysis of the method and numerical experiments to demonstrate the robust behaviour of the algorithm

convergence analysis of multirate fixed stress split iterative schemes for coupling flow with geomechanics abstract we consider multirate iterative schemes for the biot system modeling coupled flow and geomechanics in a poro elastic medium the multirate iterative coupling scheme exploits the different time scales for the mechanics and flow problems by taking multiple finer time steps for flow within one coarse mechanics time step we adapt the fixed stress split algorithm that decouples the flow and mechanics equations for the multirate case and perform an iteration between the two problems until convergence we provide a fully discrete scheme that uses backward euler time discretization and mixed spaces for flow and conformal galerkin for mechanics our analysis is based on studying the equations satisfied by the difference of iterates and using banach contraction argument to prove that the corresponding scheme is a fixed point contraction the analysis provides the value of an adjustable coefficient used in the proposed iterative coupling algorithms furthermore we show that the converged quantities satisfy the variational weak form for the coupled discrete system

error propagation of staggered solution procedures for transient problems abstract closed formulas for the error of staggered and simultaneous procedures are derived and compared conditions of global convergence are determined it is shown that in some cases a complicated condition for stability of the staggered scheme can be replaced by a simpler one the simultaneous stability condition for one iteration step a counterexample for global convergence of the staggered method with a constant number of iterations is presented

a new unconditionally stable fractional step method for non linear coupled thermomechanical problems a new class of time stepping algorithms for strongly coupled thermomechanical problems is presented which retains the computational convenience of traditional staggered algorithms without upsetting the unconditional stability property characteristic of fully implicit monolithic schemes 11 the proposed schemes are fractional step methods associated with a two phase operator split of the full non linear system of thermoelasticity into an adiabatic elastodynamics phase followed by a heat conduction phase at fixed configuration this operator split is shown to inherit the contractivity property of the full problem of evolution thus leading to unconditionally b stable product formula algorithms in sharp contrast with this stability result traditional staggered algorithms based on an isothermal mechanical phase followed by a heat conduction phase with an effective heat source are shown to lead at best to conditionally stable schemes it is further shown that the actual implementation of these two classes of schemes is essentially identical the numerical simulations presented include both dynamic and quasi static problems and are shown to closely replicate the stability estimates and non linear stability results derived for these two classes of staggered schemes

a framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition part ii computational aspects computational aspects of the formulation of elastoplasticity proposed in part i of this work employing a multiplicative decomposition and based on the principle of maximum plastic dissipation are examined in detail in particular the implementation of the proposed hyperelastic j2 flow theory is shown to reduce to a trivial modification of the classical radial return algorithm which is amenable to exact linearization this results in a closed form expression for the consistent elastoplastic tangent moduli the algorithmic treatment of the general case is considered subsequently it is emphasized that objective rates and incrementally objective algorithms play no role in the present approach a number of numerical experiments are presented that illustrate the excellent performance of the proposed formulation

evaluating linear and nonlinear solvers for density driven flow abstract this study investigates properties of different solvers for density driven flow problems the focus is on both non linear and linear solvers for the non linear part we compare fully coupled method using a newton linearization and iteratively coupled versions of jacobi and gauss seidel type fully coupled methods require effective preconditioners for the jacobian to that end we present a transformation eliminating some couplings and present a strategy for employing algebraic multigrid to the transformed system as well the work covers theoretical aspects and provides numerical experiments although the primary focus is on density driven flow we believe that the analysis may well be extended beyond to similar equations with coupled phenomena such as geomechanics