Composition duality methods for quasistatic evolution elastoviscoplastic variational problems

Composition duality methods for dual quasistatic evolution elastoviscoplastic variational problems are studied. Dual evolution mixed analysis is performed, as well as corresponding primal static mixed analysis. For multi-constitutive modeling and parallel computing, macro-hybrid variational formulations are further considered at the continuous level.     

Optimal Control of Evolution Mixed Variational Inclusions

Optimal control problems of primal and dual evolution mixed variational inclusions, in reflexive Banach spaces, are studied. The solvability analysis of the mixed state systems is established via duality principles. The optimality analysis is performed in terms of perturbation conjugate duality methods, and proximation penalty-duality algorithms to mixed optimality conditions are further presented. Applications to nonlinear diffusion constrained problems as well as quasistatic elastoviscoplastic bilateral contact problems exemplify the theory.     

EVOLUTION FILTRATION PROBLEMS WITH SEAWATER INTRUSION: TWO-PHASE FLOW DUAL MIXED VARIATIONAL ANALYSIS

Tow-phase flow mixed variational formulations of evolution filtration problems with seawater intrusion are analyzed. A dual mixed fractional flow velocity-pressure model is considered with an air-fresh water and a fresh water-seawater characterization. For analysis and computational purposes, spatial decompositions based on nonoverlapping multidomains,above and below the sea level, are variationally introduced with internal boundary fluxes dualized as weak transmission constraints. Further, parallel augmented and exactly penalized duality algorithms, and proximation semi-implicit time marching schemes, are established and analyzed.     

Macro‐hybrid mixed variational two‐phase flow through fractured porous media

Macro‐hybrid mixed variational models of two‐phase flow, through fractured porous media, are analyzed at the mesoscopic and macroscopic levels. The mesoscopic models are treated in terms of nonoverlapping domain decompositions, in such a manner that the porous rock matrix system and the fracture network interact across rock–rock, rock–fracture, and fracture–fracture interfaces, with flux transmission conditions dualized. Alternatively, the models are scaled to a macroscopic level via an asymptotic process, where the width of the fractures tends to zero, and the fracture network turns out to be an interface system of one less spatial dimension, with variable high permeability. The two‐phase flow is characterized by a fractional flow dual mixed variational model. Augmented two‐field and three‐field variational reformulations are presented for regularization, internal approximations, and macro‐hybrid mixed finite element implementation. Also abstract proximal‐point penalty‐duality algorithms are derived and analyzed for parallel computing. Copyright © 2016 John Wiley & Sons, Ltd.     

Composition duality methods for evolution mixed variational inclusions

Composition duality methods are presented for the qualitative and discretization analysis of primal and dual evolution mixed variational inclusions in reflexive Banach spaces. Abstract applications to macro-hybrid variational formulations, semi-discrete internal approximations globally nonconforming and time marching schemes implementable as multidomain proximal-point algorithms are studied. Stationary fully discrete inclusions are considered as well as corresponding preconditioned penalty–duality algorithms. To illustrate the theory, a monotone distributed control diffusion problem is treated.     

Composition duality principles for evolution mixed variational inclusions

Composition duality principles for the qualitative and finite element analysis of primal and dual evolution mixed variational inclusions in reflexive Banach spaces are presented. A general distributed control diffusion problem illustrates the theory.     

Existence of solutions to various rock types odel model of two-phase flow in porous media

We study the flow of two immiscible and incompressible fluids through a porous media c,onsisting of different rock types: capillary pressure and relative permeablities curves are different in each type of porous media. This process can be formulated as a coupled system of partial differential equations which includes an elliptic pressurevelocity equation and a nonlinear degenerated parabolic saturation equation. Moreover the transmission conditions are nonlinear and the saturation is discontinuous at interfaces separating different media. A change of unknown leads to a new formulation of this problem. We derive a weak form for this new problem, which is a combination of a mixed formulation for the elliptic pressure-velocity equation and a standard variational formulation for the new parabolic equation. Under some realistic assumptions, we prove the existence of weak solutions to the implicit system given by time discretization.     

Numerical analysis of the equations of small strains quasistatic elastoviscoplasticity

Summary The present paper deals with the mathematical and the numerical analysis of small strains elastoviscoplasticity. By considering the problem as an evolution equation whose only unknown is the stress field, the quasistatic elastoviscoplastic evolution problem is proved to be well-posed, consistent mixed finite element approximations are introduced, and classical numerical algorithms are interpreted. In particular, augmented Lagrangian methods operating on the velocity appear as standard alternating-directions time-integrations of this stress evolution problem.     

Variational Phase Field Modeling of Fracture Caused by Fluid Transport in Poroelastic Solids

The numerical modeling of failure mechanisms due to fracture based on sharp crack discontinuities is extremely demanding and suffers in situations with complex crack topologies. This drawback can be overcome by recently developed diffusive crack modeling concepts, which are based on the introduction of a crack phase field. Such an approach is conceptually in line with gradient-extended continuum damage models which include internal length scales. In this paper, we extend our recently outlined mechanical framework [1–3] towards the phase field modeling of fracture in the coupled problem of fluid transport in deforming porous media. Here, extremely complex crack patterns may occur due to drying or hydraulic induced fracture, the so called fracking. We develop new variational potentials for Biot-type fluid transport in porous media at finite deformations coupled with phase field fracture. It is shown, that this complex coupled multi-field problem is related to an intrinsic mixed variational principle for the evolution problem. This principle determines the rates of deformation, fracture phase field and fluid content along with the fluid potential. We develop a robust computational implementation of the coupled problem based on the potentials mentioned above and demonstrate its performance by the numerical simulation of complex fracture patterns. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Simulation of Two-Phase Flow through Heterogeneous Porous Media

A mixed finite element method is combined to finite volume schemes on structured and unstructured grids for the approximation of the solution of incompressible flow in heterogeneous porous media. A series of numerical examples demonstrates the effectiveness of the methodology for a coupled system which includes an elliptic equation and a nonlinear degenerate diffusion–convection equation arising in modeling of flow and transport in porous media.     

Weak sharp solutions of mixed variational inequalities in Banach spaces

In this paper, using the approximate duality mapping, we introduce the definition of weak sharpness of the solution set to a mixed variational inequality in Banach spaces. In terms of the primal gap function associated to the mixed variational inequality, we give several characterizations of the weak sharpness.     

Analysis of augmented three-field macro-hybrid mixed finite element schemes

On the basis of composition duality principles, augmented three-field macro- hybrid mixed variational problems and finite element schemes are analyzed. The compatibility condition adopted here, for compositional dualization, is the coupling operator surjectivity, property that expresses in a general operator sense the Ladyenskaja-Babuka-Brezzi inf-sup condition. Variational macro-hybridization is performed under the assumption of decomposable primal and dual spaces relative to nonoverlapping domain decomp...     

Composition Duality Methods for Mixed Variational Inclusions

Composition duality methods for mixed variational inclusions are studied in a functional framework of reflexive Banach spaces. On the basis of duality principles, the solvability of maximal monotone and subdifferential mixed variational inclusions is established. For computational purposes, mass-preconditioned augmented formulations are introduced for regularization, as well as three-field and macro-hybrid variational versions. At a finite-dimensional level, corresponding discrete mixed and macro-hybrid internal approximations are discussed, as well as proximal-point iterative algorithms. Primal and dual mixed variational inclusions from contact mechanics illustrate the theory.     

Optimal Control of Nonlinear Transport-Flow Mixed Variational Problems

Optimal control of nonlinear transport-flow mixed variational problems are studied qualitatively, and the solvability analysis of the transport and flow mixed state systems is performed on the basis of primal and dual evolution duality principles. Corresponding primal and dual mixed optimality conditions are established by the application of some fundamental perturbation conjugate duality results recently proposed [8 G. Alduncin (2013). Optimal control of evolution mixed variational inclusions. Applied Mathematics and Optimization 68:445–473.[Crossref], [Web of Science ®] , [Google Scholar]]. Further, for computational purposes, two- and three-field proximation penalty-duality algorithms in the resolution of the mixed optimality conditions are finally presented and discussed.     

Variational formulations of nonlinear constrained boundary value problems

Variational formulations of nonlinear constrained boundary value problems in reflexive Banach spaces are discussed from a compositional duality approach. The mixed variational compatibility conditions of the theory correspond to the surjectivity of the primal coupling boundary and interior operators.     

A fully variational algorithmic formulation for wrinkling at finite strains

This contribution is concerned with an efficient novel algorithmic formulation for wrinkling at finite strains. In contrast to previously published numerical implementations, the advocated method is fully variational. More precisely, the parameters describing wrinkles or slacks, together with the unknown deformation mapping, are computed jointly by minimizing the potential energy of the considered mechanical system. Furthermore, the wrinkling criteria are naturally included within the presented variational framework. The presented approach allows to employ three–dimensional constitutive models directly, i.e., plane stress conditions characterizing membranes are variationally enforced by minimizing the potential energy with respect to the transversal strains. Since the proposed formulation for wrinkling in membranes is fully variational, it can be conveniently combined with other variational methods (based on energy minimization). As an example, a variationally consistent framework for finite strain plasticity theory is considered. More precisely, the minimization principle characterizing wrinkling in elastic membranes and that describing plasticity in inelastic solids are coupled leading to a novel variational approach for inelastic membranes. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

具广义V-不变凸多目标变分的混合对偶性

在函数广义V-不变凸性的条件下,建立了多目标变分关于有效解的混合对偶理论.     

Augmented Macro-Hybrid Mixed Finite Element Schemes for Elastic Contact Problems

On a setting of subdifferential models, variational augmented macro-hybrid mixed finite element schemes are formulated and analyzed for elastic unilateral contact problems with prescribed friction. Composition duality principles determine primal and dual mixed solvability, adopting coupling surjectivity for dualization. Macro-hybridization corresponds to nonoverlapping decompositions of elastic solid body systems, with displacement continuity and traction equilibrium transmission conditions dualized. In general, traction and displacement multipliers synchronize sub-bodies through nonmatching finite element interfaces. Three-field formulations give the basis for variational augmentation, in a sense of exact penalization, allowing speed-up of rates of convergence as well as proximation procedures of parallel numerical resolution algorithms.     

多目标分式变分问题的混合对偶性

在广义B-Ⅰ凸性条件下,建立了多目标分式变分问题的混合对偶模型,使得M ond-W e ir型对偶和W o lfe型成为其特殊情况,并建立了关于有效解的混合对偶理论.     

Fractional multi-commodity flow problem: Duality and optimality conditions

This paper deals with multi-commodity flow problem with fractional objective function. The optimality conditions and the duality concepts of this problem are given. For this aim, the fractional linear programming formulation of this problem is considered and the weak duality, the strong direct duality and the weak complementary slackness theorems are proved applying the traditional duality theory of linear programming problems which is different from same results in Chadha and Chadha (2007) [1]. In addition, a strong (strict) complementary slackness theorem is derived which is firstly presented based on the best of our knowledge. These theorems are transformed in order to find the new reduced costs for fractional multi-commodity flow problem. These parameters can be used to construct some algorithms for considered multi-commodity flow problem in a direct manner. Throughout the paper, the boundedness of the primal feasible set is reduced to a weaker assumption about solvability of primal problem which is another contribution of this paper. Finally, a real world application of the fractional multi-commodity flow problem is presented.