Two-phase Problem Research Articles

The two-phase problem for harmonic measure in VMO and the chord-arc condition

Let Ω + ⊂ R n + 1 \Omega ^+\subset \mathbb {R}^{n+1} be a bounded δ \delta -Reifenberg flat domain, with δ > 0 \delta >0 small enough, possibly with locally infinite surface measure. Assume also that Ω − = R n + 1 ∖ Ω + ¯ \Omega ^-= \mathbb {R}^{n+1}\setminus \overline {\Omega ^+} is an NTA (non-tangentially accessible) domain as well and denote by ω + \omega ^+ and ω − \omega ^- the respective harmonic measures of Ω + \Omega ^+ and Ω − \Omega ^- with poles p ± ∈ Ω ± p^\pm \in \Omega ^\pm . In this paper we show that the condition that log ⁡ d ω − d ω + ∈ VMO ⁡ ( ω + ) \log \dfrac {d\omega ^-}{d\omega ^+} \in \operatorname {VMO}(\omega ^+) is equivalent to Ω + \Omega ^+ being a chord-arc domain with inner unit normal belonging to VMO ⁡ ( H n | ∂ Ω + ) \operatorname {VMO}(\mathcal {H}^n|_{\partial \Omega ^+}) .

Two-phase heat conduction problems with Sturm-type boundary conditions

AbstractIn this work, the solution is justified using the method of separation of variables for initial-boundary value problems of the heat conduction equation with piecewise constant thermal conductivity coefficient under Sturm-type boundary conditions (separated boundary conditions). Various cases are considered. A system of eigenfunctions of the corresponding spectral problem is constructed and it is proved that this system of eigenfunctions forms a Riesz basis. The theorem of the existence of a unique classical solution to the problem is proved.

Solving the green reverse logistics problem in e-commerce using a reinforcement learning based genetic algorithm

This study explores the two-phase green reverse logistics problem with time windows and a focus on perishable items that pose a significant challenge in the management of returned goods in e-commerce. We proposed a mixed integer programming model that considers carbon emissions, fuel consumption costs, facility establishment and operating costs, among other factors.We incorporated reinforcement learning concepts to adjust parameters in traditional genetic algorithms, which often have inflexible parameter settings, thereby enhancing both the efficiency and quality of the solutions. The Q-learning algorithm was adopted as the learning method, and various action combinations of reinforcement learning were explored and compared. We further evaluated the performance of different genetic algorithm variations. The results indicate that the proposed algorithm provides high-quality solutions, and that effective parameter configuration significantly impacts the algorithm’s overall performance.

Elastic Localization With Particular Reference to Tape-Springs

Abstract Many elastic systems localize under applied displacement, precipitating into regions of lower and higher strain; further displacement is accommodated by growth of the high strain region at a constant load. Such systems can be studied as propagating instabilities, focusing on the work required to propagate the high strain region, or as two-phase energy minimization problems. It is shown that the Maxwell “equal-areas” construction, and the related common tangent construction, provide the solution to either approach. A new, graphical, proof of the Maxwell equal-areas construction using total strain energy diagrams is presented. Tape-springs are investigated as a case study, with localization presenting as the formation of elastic folds—developable regions with high curvature. One notable property of tape-spring folds is that the fold radius is approximately equal to the initial transverse radius. This result was first proven by Rimrott, and later improved by Calladine and Seffen. A further improvement is obtained here by application of the common tangent construction, and all solutions are shown to be approximations to the Maxwell equal-areas construction in the limit of zero thickness.

A mini immersed finite element method for two-phase Stokes problems on Cartesian meshes

Abstract This paper presents a mini immersed finite element (IFE) method for solving two- and three-dimensional two-phase Stokes problems on Cartesian meshes. The IFE space is constructed from the conventional mini element, with shape functions modified on interface elements according to interface jump conditions while keeping the degrees of freedom unchanged. Both discontinuous viscosity coefficients and surface forces are taken into account in the construction. The interface is approximated using discrete level set functions, and explicit formulas for IFE basis functions and correction functions are derived, facilitating ease of implementation.The inf-sup stability and the optimal a priori error estimate of the IFE method, along with the optimal approximation capabilities of the IFE space, are derived rigorously, with constants that are independent of the mesh size and the manner in which the interface intersects the mesh, but may depend on the discontinuous viscosity coefficients. Additionally, it is proved that the condition number has the usual bound independent of the interface. Numerical experiments are provided to confirm the theoretical results.

Multi-position two-phase person-position matching decision-making with intermediary participation

With the rapid development in society, intermediaries play an increasingly important role in person-position matching decision-making. To obtain reasonable enterprise-intermediary matching and enterprise-candidate matching schemes under multiple positions, preferences of enterprises, headhunting companies and candidates should be fully considered from a two-phase perspective. Thus, a two-sided matching decision-making method under the probabilistic linguistic environment is proposed to solve the multi-position two-phase person-position matching problem. First, related theories of probabilistic linguistic term sets and two-sided matching are given. On this basis, the problem of multi-position two-phase person-position matching under the probabilistic linguistic environment is described. The decision-making process is divided into the following two phases: In the first phase, satisfactions of enterprises and headhunting companies are calculated considering subject expectations based on prospect theory. Then, considering the maximization of satisfactions of two-sided subjects, the many-to-many two-sided matching model between enterprises and headhunting companies is established and solved to obtain the optimal many-to-many matching scheme. In the second phase, according to evaluation matrices and expectation matrices, satisfactions of enterprises and candidates under multiple positions are calculated considering intermediary evaluations. Then, considering the maximization of satisfactions of enterprises and candidates, one-to-many matching models and one-to-one stable matching models under multiple positions are developed and solved to obtain the optimal person-position matching schemes under multiple positions. Finally, an example of the person-position matching with intermediary participation is used to illustrate the feasibility and effectiveness of the proposed method.

Improved accuracy of continuum surface flux models for metal additive manufacturing melt pool simulations

Computational modeling of the melt pool dynamics in laser-based powder bed fusion metal additive manufacturing (PBF-LB/M) promises to shed light on fundamental mechanisms of defect generation. These processes are accompanied by rapid evaporation so that the evaporation-induced recoil pressure and cooling arise as major driving forces for fluid dynamics and temperature evolution. The magnitude of these interface fluxes depends exponentially on the melt pool surface temperature, which, therefore, has to be predicted with high accuracy. The present work utilizes a diffuse interface finite element model based on a continuum surface flux (CSF) description of interface fluxes to study dimensionally reduced thermal two-phase problems representative for PBF-LB/M in a finite element framework. It is demonstrated that the extreme temperature gradients combined with the high ratios of material properties between metal and ambient gas lead to significant errors in the interface temperatures and fluxes when classical CSF approaches, along with typical interface thicknesses and discretizations, are applied. It is expected that this finding is also relevant for other types of diffuse interface PBF-LB/M melt pool models. A novel parameter-scaled CSF approach is proposed, which is constructed to yield a smoother temperature field in the diffuse interface region, significantly increasing the solution accuracy. The interface thickness required to predict the temperature field with a given level of accuracy is less restrictive by at least one order of magnitude for the proposed parameter-scaled approach compared to classical CSF, drastically reducing computational costs. Finally, we showcase the general applicability of the parameter-scaled CSF to a 3D simulation of stationary laser melting of PBF-LB/M considering the fully coupled thermo-hydrodynamic multi-phase problem, including phase change.

Retraction Note: A heat polynomials method for the two-phase inverse Stefan problem

Retraction Note: A heat polynomials method for the two-phase inverse Stefan problem

Error analysis of a fully discrete projection method for Cahn–Hilliard Inductionless MHD problems

This article investigates the fully discrete finite element approximation and error analysis for a diffuse interface model of the two-phase incompressible inductionless magnetohydrodynamics problem. This model consists of Cahn–Hilliard equations, Navier–Stokes equations and Poisson equations, which are nonlinearly coupled through convection, stresses, and Lorentz forces. To address this highly nonlinear and multi-physics system, we propose a fully discrete energy stable scheme with the finite element projection method for spatial discretization, in which the velocity and pressure are decoupled. The temporal discretization is a combination of the first-order Euler semi-implicit scheme and a convex splitting energy strategy. We show that the proposed scheme is mass-conservative, charge-conservative and unconditional energy stable. The error estimates for the phase variable, chemical potential, velocity, pressure, current density and electric potential are rigorously established. Finally, several three-dimensional numerical experiments are performed to illustrate the features of the proposed numerical method and verify the theoretical conclusions.

Regularity in the two-phase Bernoulli problem for the p-Laplace operator

Regularity in the two-phase Bernoulli problem for the p-Laplace operator

A deep learning approach for solving the stationary compositional two-phase equilibrium problems

In this paper, we propose and investigate a deep neural network approach for solving the stationary compositional two-phase equilibrium problems in porous media. A recent approach is the unified formulation advocated by Lauser et al. (2011) which contains the complementarity conditions. The advantage of this formulation lies in its potential to handle the appearance and disappearance of phases automatically. To solve numerically the system of equations, a new strategy called NPIPM (NonParametric Interior-Point Method) is proposed by Vu et al. (2021). However, the method still has some disadvantages preventing the convergence to a solution. Taking inspiration from the work of Raissi et al. (2019) for the study of nonlinear partial differential equations using Physics-Informed Neural Networks (PINN), we realize that we can apply a deep neural network to obtain the solution of the stationary compositional model containing gas and liquid. We design a deep neural network structure and define a new loss function for the problem. The effectiveness of the proposed method is validated through numerical results obtained from a series of systems with increasing components. This machine learning approach provides a promising idea to solve the curse of dimensionality issue in the sense of the increasing number of components in stationary problems.

Regularization errors introduced by the one-fluid formulation in the solution of two-phase elliptic problems

This manuscript discusses the structure of the errors in the solution of elliptic problems introduced by the regularization of the fluid properties discontinuity in a small region of finite size. By using a multiscale approach, the problem of the error calculation is splitted into an outer problem, where the regularization region is replaced by a sharp interface, and an inner local one dimensional problem that eventually imposes effective jump conditions across the interface for the outer problem. Except in some particular cases, the use of regularization techniques introduces first order errors in the solution imposing an error flux jump that is proportional to the tangential Laplacian of the averaged solution and an error jump that is proportional to the normal flux, both multiplied by a prefactor that depends on the averaging rule used. In general, the optimal averaging procedure is shown to depend on the structure of the problem at hand. The errors introduced by standard arithmetic and harmonic averages are obtained for various analytical and numerical examples which are used to discuss the nature of the errors introduced and the importance of first order errors in the solution. The influence of the ratio between the regularization thickness and the grid size is also investigated in numerical implementations of the one-fluid model.

Existence of Weak Solutions for the Class of Singular Two-Phase Problems with a ψ-Hilfer Fractional Operator and Variable Exponents

In this paper, we prove the existence of at least two weak solutions to a class of singular two-phase problems with variable exponents involving a ψ-Hilfer fractional operator and Dirichlet-type boundary conditions when the term source is dependent on one parameter. Here, we use the fiber method and the Nehari manifold to prove our results.

Regularity of flat free boundaries for two-phase p(x)-Laplacian problems with right hand side

We consider viscosity solutions to two-phase free boundary problems for the p(x)-Laplacian with non-zero right hand side. We prove that flat free boundaries are C1,γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\gamma }$$\\end{document}. No assumption on the Lipschitz continuity of solutions is made. These regularity results are the first ones in literature for two-phase free boundary problems for the p(x)-Laplacian and also for two-phase problems for singular/degenerate operators with non-zero right hand side. They are new even when p(x)≡p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p(x)\\equiv p$$\\end{document}, i.e., for the p-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability.

(Quasi-)conformal methods in two-dimensional free boundary problems

In this paper, we relate the theory of quasi-conformal maps to the regularity of the solutions to nonlinear thin-obstacle problems; we prove that the contact set is locally a finite union of intervals and apply this result to the solutions of one-phase Bernoulli free boundary problems with geometric constraint. We also introduce a new conformal hodograph transform, which allows to obtain the precise expansion at branch points of both the solutions to the one-phase problem with geometric constraint and a class of symmetric solutions to the two-phase problem, as well as to construct examples of free boundaries with cusp-like singularities.

Portfolio-consumption choice problem with voluntary retirement and consumption constraints

We study the investment–consumption choice problem with voluntary retirement and upside/downside consumption constraints in infinite horizon, which can be formulated as a two-phase mixed stochastic control problem including stopping time coupled with controls. By verification theorem, we show the strong solution of a fully nonlinear differential variational inequality, satisfying some special properties, is the value function of this problem. Then we eliminate the nonlinearity through Legendre transformation, and obtain the closed-form formulas of the optimal investment–consumption feedback strategy, the optimal retirement strategy and the value function by solving the dual equation. Since the introduction of consumption constraints, these formulas are both segmented and semi-explicit, and bring certain difficulties to the analysis of the properties of the optimal strategies and the value function. The regularity of the feedback strategy and the value function are obtained, as well as their monotonicity or non-monotonicity with respect to various parameters are discussed by some stochastic analysis methods and differential equation techniques.

Unified framework for geotechnical cross-contact problems with interfacial fluid flow

In the hydro-mechanical coupling analysis of multi-body contact systems, challenges arise from the fluid flow at the interface and the over-constraint at the cross line, necessitating special treatments. In this paper, a novel algorithm is developed to address these challenges comprehensively. The proposed method introduces an intermediate interface, serving as a bridge for contact stress transfer and fluid exchange between different subdomains. The intermediate interface concurrently carries the degrees of freedom related to the displacement, pore pressure, and Lagrange multiplier. As a result, the contact constraint and interfacial fluid flow can be integrated into a unified formulation, resulting in a simplified algorithm implementation; the elimination of over-constraint behaviour can also be achieved in a simple manner. Additionally, the storage capacity of interfaces is adequately considered in the current framework, enabling the accurate computation of water suction generated by the opening of contact interfaces. Finally, several numerical experiments are conducted to validate the performance of our proposed method. Results show that our method can provide a reliable computational tool for the two-phase contact problems with interfacial fluid flow.

A hydro-mechanical coupled contact method for two-phase geotechnical large deformation problems within the SNS-PFEM framework

This paper proposes a novel numerical algorithm to solve frictional contact problems in two-phase porous media, accounting for the large deformation and interface fluid flow, within the framework of the Stable Node-based Smoothed Particle Finite Element Method (SNS-PFEM). The proposed approach builds on our previous work that was only designed for single-phase large deformation contact problems. Within this work, the hydro-mechanical coupling system is discretized using equal-order interpolation, with a polynomial pressure projection method to eliminate the pore pressure instability. Surface-to-surface discretization is then applied to formulate the contact constraint of the two-phase porous medium. Using the dual Lagrange multiplier approach, a coordinate transformation is introduced to enforce the contact constraint of the solid phase. This transformation is subsequently utilized to develop a new interface fluid element, which serves as a tool to model the fluid exchange behaviour between porous media and contact gaps. Different from existing interface fluid elements, the proposed element is constructed based on the contact algorithm, thereby enabling it to handle non-conforming interfaces and tackle challenges associated with large sliding problems. Furthermore, the proposed method can maintain a well-conditioned system matrix throughout the computation process. These distinctive attributes significantly enhance the flexibility of numerical modelling. Finally, the accuracy and performance of our method are verified through four numerical examples. The results show that the proposed method can effectively simulate various geotechnical problems, including multi-layer soil consolidation, fluid-soil-structure interaction, and geological interface-controlled slope failure.

Analytical solution of Stefan-type problems

Abstract In this paper, free surface problems of Stefan type for the parabolic heat equation are considered. The analytical solutions of the problems are based on the method of heat polynomials and integral error function in the form of series. Convergence of the series solution is considered and proved. Both one-and two-phase Stefan-type problems are investigated. Numerical results for one-phase inverse Stefan problem are presented and discussed.

Boundedness and higher integrability of minimizers to a class of two-phase free boundary problems under non-standard growth conditions

<abstract><p>In this paper, we are concerned with the existence, boundedness, and integrability of minimizers of heterogeneous, two-phase free boundary problems $ \mathcal {J}_{\gamma}(u) = \int_{\Omega}\left(f(x, \nabla u)+\lambda_{+}(u^{+})^{\gamma}+\lambda_{-}(u^{-})^{\gamma}+gu\right)\text{d}x \rightarrow \text{min} $ under non-standard growth conditions. Included in such problems are heterogeneous jets and cavities of Prandtl-Batchelor type with $ \gamma = 0 $, chemical reaction problems with $ 0 < \gamma < 1 $, and obstacle type problems with $ \gamma = 1 $, respectively.</p></abstract>