Multiphase Problems Research Articles

An extended finite element method for fluid flow in partially saturated porous media with weak discontinuities; the convergence analysis of local enrichment strategies

In this paper, a numerical model is developed for the fully coupled analysis of deforming porous media containing weak discontinuities which interact with the flow of two immiscible, compressible wetting and non-wetting pore fluids. The governing equations involving the coupled solid skeleton deformation and two-phase fluid flow in partially saturated porous media are derived within the framework of the generalized Biot theory. The solid phase displacement, the wetting phase pressure and the capillary pressure are taken as the primary variables of the three-phase formulation. The other variables are incorporated into the model via the experimentally determined functions that specify the relationship between the hydraulic properties of the porous medium, i.e. saturation, permeability and capillary pressure. The spatial discretization by making use of the extended finite element method (XFEM) and the time domain discretization by employing the generalized Newmark scheme yield the final system of fully coupled non-linear equations, which is solved using an iterative solution procedure. Numerical convergence analysis is carried out to study the approximation error and convergence rate of several enrichment strategies for bimaterial multiphase problems exhibiting a weak discontinuity in the displacement field across the material interface. It is confirmed that the problems which arise in the blending elements can have a significant effect on the accuracy and convergence rate of the solution.

Numerical solution of gas–solid flow in fluidised bed at sub-atmospheric pressures

Fluidised beds are characterised by excellent thermal and chemical uniformity and have a wide application range including heat and surface treatment, ore roasting and catalyst production. However, compared to other gas-based systems, to fluidise a particulate mass, a significant quantity of gas is required. To conserve gas there is potential to operate the fluid bed under low-pressure conditions. It is also observed that heat transfer remains constant with reduction in pressure. The present work has numerically studied the nature of hydrodynamics in fluidised bed at sub-atmospheric conditions and a new drag law is proposed to account for the increased mean free path of the fluid. A wide range of sub-atmospheric pressures were considered such that slip flow regime, which is characterised with Kn∼1, is applicable. An open source code (MFIX) is used to numerically solve the multiphase problem of a jet in the fluidised bed column with an immersed surface at vacuum pressure conditions. Bubbling fluidisation in shallow and deep beds are also solved. The new drag model takes into consideration the effect of slip flow to model drag force on the particles and the results of velocity distributions in the column and around the submerged surface is presented. The results of velocity distributions from the slip flow model are compared with the existing Gidaspow’s model. Significant differences were observed in the simulation results of velocity distributions and flow structure in the fluidised bed under vacuum conditions.

A Hybrid Differential Dynamic Programming Algorithm for Constrained Optimal Control Problems. Part 2: Application

In the first part of this paper series, a new solver, called HDDP, was presented for solving constrained, nonlinear optimal control problems. In the present paper, the algorithm is extended to include practical safeguards to enhance robustness, and four illustrative examples are used to evaluate the main algorithm and some variants. The experiments involve both academic and applied problems to show that HDDP is capable of solving a wide class of constrained, nonlinear optimization problems. First, the algorithm is verified to converge in a single iteration on a simple multi-phase quadratic problem with trivial dynamics. Successively, more complicated constrained optimal control problems are then solved demonstrating robust solutions to problems with as many as 7 states, 25 phases, 258 stages, 458 constraints, and 924 total control variables. The competitiveness of HDDP, with respect to general-purpose, state-of-the-art NLP solvers, is also demonstrated.

On multiphase negative flash for ideal solutions

There is a recent interest to solve multiphase negative flash problems where the phase amounts can be negative for normal positive feed composition. Solving such a negative flash problem using successive substitution needs an inner loop for phase distribution calculation at constant fugacity coefficients. It is shown that this inner loop, named here as multiphase negative flash for ideal solutions, can be solved either by Michelsen's algorithm for multiphase normal flash, or by its variation which uses F−1 phase amounts as independent variables. In either case, the resulting algorithm is actually simpler than the corresponding normal flash algorithm. Unlike normal flash, multiphase negative flash for ideal solutions can diverge if the feasible domain for phase amounts is not closed. This can be judged readily during the iteration process. The algorithm can also be extended to the partial negative flash situation where several phase amounts must be non-negative and at least one phase amount can be negative.

Thermodynamics of multiphase problems in viscoelasticity

AbstractThis paper deals with a three‐dimensional mixture model describing materials undergoing phase transition with thermal expansion. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the ßow rule with the heat transfer equation. A global solution for this thermodynamically consistent problem is obtained by using a Þxed‐point argument combined with global energy estimates (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Three-dimensional trajectory optimization of soft lunar landings from the parking orbit with considerations of the landing site

Minimum fuel, three-dimensional trajectory optimization from a parking orbit considering the desired landing site is addressed for soft lunar landings. The landing site is determined by the final longitude and latitude; therefore, a two-dimensional approach is limited and a three-dimensional approach is required. In addition, the landing site is not usually considered when performing lunar landing trajectory optimizations, but should be considered in order to design more accurate and realistic lunar landing trajectories. A Legendre pseudospectral (PS) method is used to discretize the trajectory optimization problem as a nonlinear programming (NLP) problem. Because the lunar landing consists of three phases including a de-orbit burn, a transfer orbit phase, and a powered descent phase, the lunar landing problem is regarded as a multiphase problem. Thus, a PS knotting method is also used to manage the multiphase problem, and C code for Feasible Sequential Quadratic Programming (CFSQP) using a sequential quadratic programming (SQP) algorithm is employed as a numerical solver after formulating the problem as an NLP problem. The optimal solutions obtained satisfy all constraints as well as the desired landing site, and the solutions are verified through a feasibility check.

A coupled computational method for multi-solver, multi-domain transient problems in elastodynamics

This work presents an efficient and stable methodology for the coupling of Finite Element Methods (FEM) and Boundary Element Methods (BEM) that is independent of the particular solver and allows for independent temporal discretizations among solvers. The approach satisfies explicitly compatibility conditions and equilibrium of forces at the contact interfaces. Although the proposed approach has been developed in view of the soil–rail–vehicle dynamic interaction problem in High Speed Rail applications, it is expressed in a general form applicable to any multi-domain, multi-phase transient problem. The method development and formulations are presented in detail. Verification and application studies demonstrate the accuracy, efficiency and versatility of the method for the direct time domain solution of dynamic problems including structure–structure interaction and soil–structure interaction. The proposed approach demonstrates high accuracy and efficiency to that of direct coupling solutions and more rigorous methods.

Numerical simulation of molten pool dynamics in high power disk laser welding

A single-phase problem is solved rather than a multiphase problem for numerical simplicity: and the solution is based on the assumption that the region of gas or plasma can be treated as a void because solid or liquid steel has a greater density level than gas or plasma. The volume-of-fluid method, which can calculate the free surface shape of the keyhole, is used in conjunction with a ray-tracing algorithm to estimate the multiple reflections. Fresnel's reflection model is simplified by the Hagen–Rubens relation for handling a laser beam interaction with materials. Factors considered in the simulations include buoyancy force, Marangoni force and recoil pressure; furthermore, pore generation is simulated by means of an adiabatic bubble model, which can also lead to the phenomenon of a keyhole collapse. Models of the shear stress on the keyhole surface and of the heat transfer to the molten pool via a plasma plume are introduced in simulations of the weld pool dynamics. Analysis of the temperature profile characteristics of the weld bead and molten pool flow in the molten pool is based on the results of the numerical simulations. The simulation results are used to estimate the weld fusion zone shape; and the results of the simulated fusion zone formation are compared with the results of the experimental fusion zone formation and found to be in good agreement. The effects of laser beam profile (Gaussian vs. measured), vapor shear stress, vapor heat source and sulfur content on the molten pool behavior and fusion zone shape are analyzed.

Fluid-fiber-interactions in rotational spinning process of glass wool production

AbstractThe optimal design of rotational production processes for glass wool manufacturing poses severe computational challenges to mathematicians, natural scientists and engineers. In this paper we focus exclusively on the spinning regime where thousands of viscous thermal glass jets are formed by fast air streams. Homogeneity and slenderness of the spun fibers are the quality features of the final fabric. Their prediction requires the computation of the fluid-fiber-interactions which involves the solving of a complex three-dimensional multiphase problem with appropriate interface conditions. But this is practically impossible due to the needed high resolution and adaptive grid refinement. Therefore, we propose an asymptotic coupling concept. Treating the glass jets as viscous thermal Cosserat rods, we tackle the multiscale problem by help of momentum (drag) and heat exchange models that are derived on basis of slender-body theory and homogenization. A weak iterative coupling algorithm that is based on the combination of commercial software and self-implemented code for flow and rod solvers, respectively, makes then the simulation of the industrial process possible. For the boundary value problem of the rod we particularly suggest an adapted collocation-continuation method. Consequently, this work establishes a promising basis for future optimization strategies.

On some multiscale algorithms for sector modeling in multiphase flow in porous media

A multiscale algorithm for a multiphase filtration problem is proposed. Filtration fluxes on a fine grid are determined from the solution of the pressure equation on a coarse grid. Further, the domain is decomposed into subdomains with an acceptable number of cells and the full second boundary condition filtration problem is solved using the fluxes. The support operator method has been improved for a complex structure cell for solution of the pressure equation on a coarse grid. This method is a high resolution one: a divergence operator has the approximation of the second order, while fluxes have the approximation of the first order. At the same time, the method allows revealing the solution’s properties related to a fine grid structure.

Image segmentation by iterative optimization of multiphase multiple piecewise constant model and Four-Color relabeling

In the paper an iteratively unsupervised image segmentation algorithm is developed, which is based on our proposed multiphase multiple piecewise constant (MMPC) model and its graph cuts optimization. The MMPC model use multiple constants to model each phase instead of one single constant used in Chan and Vese (CV) model and cartoon limit so that heterogeneous image object segmentation can be effectively dealt with. We show that the multiphase optimization problem based on our proposed model can be approximately solved by graph cuts methods. Four-Color theorem is used to relabel the regions of image after every iteration, which makes it possible to represent and segment an arbitrary number of regions in image with only four phases. Therefore, the computational cost and memory usage are greatly reduced. The comparison with some typical unsupervised image segmentation methods using a large number of images from the Berkeley Segmentation Dataset demonstrates the proposed algorithm can effectively segment natural images with a good performance and acceptable computational time.

Multi-level sequential circuit partitioning for test vector generation for low power test in VLSI

Sequential graph partitioning algorithms have been developed to fulfill the requirements of emerging multi-phase problems in circuit testing models. In this paper, we present a multi-level graph partitioning algorithm for circuit partitioning, which will minimize the number of test vectors during a low power test in VLSI circuits. By reducing the number of test vectors, we can reduce the energy consumption during the test. Our experimental results with ISCAS bench mark circuits have shown that the power can be reduced up to 55%.

Development of X-FEM methodology and study on mixed-mode crack propagation

The extended finite element method (X-FEM) is a novel numerical methodology with a great potential for using in multi-scale computation and multi-phase coupling problems. The algorithm is discussed and a program is developed based on X-FEM for simulating mixed-mode crack propagation. The maximum circumferential stress criterion and interaction integral are deduced. Some numerical results are compared with the experimental data to prove the capability and efficiency of the algorithm and the program. Numerical analyses of sub-interfacial crack growth in bi-materials give a clear description of the effect on fracture made by interface and loading condition.

A THEORETICAL AND NUMERICAL MULTISCALE FRAMEWORK FOR THE ANALYSIS OF PATTERN FORMATION IN PROTEIN CRYSTAL ENGINEERING

The relevance of self-organization, pattern formation, nonlinear phenomena, and nonequilibrium behavior in a wide range of problems related to macromolecular crystal engineering calls for a concerted approach using the tools of statistical physics, thermodynamics, fluid dynamics, nonlinear dynamics, mathematical modeling, and numerical simulation in synergy with experimentally oriented work. The reason behind such a need is that in many instances of relevance in this field one witnesses an interplay between molecular and macroscopic-level entities and processes. Along these lines, two models are defined here and discussed in detail, one dealing with issues of complex behavior at the microscopic level and the other referring to the strong nonlinear nature of macroscopic evolution. Such models share a common fundamental feature, a group of equations strictly related from a mathematical point of view to the kinetic conditions used to model mass transfer at the crystal surface. Model diversification then occurs on the basis of the desired scale length; i.e., according to the level of detail required by the analysis (local or global). If the local evolution of the crystal surface is the subject of the investigation (distribution of the local growth rate along the crystal face, shape instabilities, onset of surface depressions due to diffusive and/or convective effects, etc.; i.e., all those factors dealing with the local history of the shape) the model is conceived to provide microscopic and morphological details. For this specific case a kinetic-coefficient-based moving boundary numerical (computational fluid dynamics) strategy is carefully developed on the basis of the volume-of-fluid methods (also known as the volume tracking methods) and level-set techniques, which have become popular in the last years as numerical techniques capable of modeling complex multiphase problems as well as for their capability to undertake a fixed-grid solution without resorting to mathematical manipulations and transformations. On the contrary, if the size of the crystals is negligible with respect to the size of the reactor (i.e., if they are small and undergo only small dimensional changes with respect to the overall dimensions of the cell containing the feeding solution), the shape of the crystals is ignored and the proposed approach relies directly on an algebraic formulation of the nucleation events and on the application of an integral form of the mass balance kinetics for each protein crystal. The applicability and the suitability of the different submodels are discussed according to some worked examples of practical interest. Pattern formation in these processes is described here with respect to crystal shapes, nuclei spatial discrete arrangements, and the convective multicellular structures arising as a consequence of buoyancy forces, thus enriching the discussions with some interdisciplinary flavor.

Computation of the Coupled Dynamics of Fibers in a Spray

AbstractThe PUR‐fiber‐spray molding technology is a manufacturing process which produces polyurethane‐based (PUR) composites by spraying the matrix together with reinforcing fibers in a tool form or on a substrate. Thereby chopped fibers are laterally (sidewise) injected in the polyurethane‐air spray cone for wetting before the entire composite is spread on the substrate, where it starts curing. To investigate and compute the fiber orientation and density distribution in the final composites manufactured by this process, a new approach simplifying the multiply coupled interaction of the three phases is presented in this paper. Hereby it is presumed that the final position and orientation of a fiber on a substrate results from its dynamics and coupled interactions with air, PUR‐droplets and other fibers within the spray cone. Thus, a model of the process is built, that computes the transient behavior of the air‐liquid droplets mixture by the CFD code ANSYS Fluent and its influence on the dynamics of the fibers by an extra code called FIDYST. For this multiphase problem two approaches are presented for the droplet‐fiber coupling using a concept called “homogenization” of the liquid phase (droplets in the continuous phase). (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Multiphase mixing quantification by computational homology and imaging analysis

The purpose of this study is to introduce a new technique for quantifying the efficiency of multiphase mixing. This technique based on algebraic topology is illustrated by using the hydraulic modeling of gas agitated reactors stirred by top lance gas injection and image analysis. The zeroth Betti numbers are used to estimate the numbers of pieces in the patterns, leading to a useful parameter to characterize the mixture homogeneity. The first Betti numbers are introduced to characterize the nonhomogeneity of the mixture. The mixing efficiency can be characterized by the Betti numbers for binary images of the patterns. This novel method may be applied for studying a variety of multiphase mixing problems in which multiphase components or tracers are visually distinguishable.

Numerical Simulation of Rheological Behavior in Melting Metal Using Finite Volume Particle Method

In multiphase flow analyses, rheological behavior has a significant influence on not only the heat and mass transfer but also the dynamics of the solid and fluid during melting and solidification. Based on previous work, it is possible to consider rheological behavior by estimating the viscosity of the liquid phase with its compositional development. The present study investigates this rheological behavior through simulations of multiphase heat transfer problems using the moving finite volume particle (FVP) method, by introducing a viscosity model that takes into account viscosity changes due to phase changes. To validate the applicability of this viscosity model, a series of melting experiments using Wood's metal are conducted, and the observed melting characteristics form the basis for computer simulations and 3D numerical analysis using the FVP method. Good agreement between simulation and experiment indicates that the proposed viscosity model reproduces well the rheological behavior during melting.

A Comparison of Two-Phase Computational Fluid Dynamics Codes Applied to the ITER First Wall Hypervapotron

Enhanced radial transport in the plasma and the effect of ELMS may increase the ITER first wall heat loads to as much as 4 to 5 MW/m2 over localized areas. One proposed heatsink that can handle these higher loads is a CuCrZr hypervapotron. One concept for a first wall panel consists of 20 hypervapotron channels, each measuring 1400 mm long and 48.5 mm wide. The nominal cooling conditions anticipated for each channel are 400 g/s of water at 3 MPa and 100°C. This will result in boiling over a portion of the total length. A two-phase thermalhydraulic analysis is required to predict accurately the thermal performance. Existing heat transfer correlations used for nucleate boiling are not appropriate here because the flow does not reach fully developed conditions in the multi-segmented channels. Our design-by-analysis approach used two commercial codes, Fluent and Star-CCM+, to perform computational fluid dynamics analyses with conjugate heat transfer. Both codes use the Rensselear (RPI) model for wall heat flux partitioning to model nucleate boiling as implemented in user-defined functions. We present a comparison between the two codes for this Eulerian multiphase problem that relies on temperature dependent materials properties. The analyses optimized the hypervapotron geometry, including teeth height and pitch, as well as the depth of the back channel to permit highly effective boiling heat transfer in the grooves between the teeth while ensuring that no boiling could occur at the back channel exit. The analysis used a representative heat flux profile with the peak heat flux of 5 MW/m2 limited to a 50 mm length. The maximum surface temperature of the heatsink is 415°C. The baseline design uses 2 mm for the teeth height, a 3 mm width and 6 mm pitch, and a back channel depth of 8 mm. The teeth are detached from the sidewall by a 2-mm-wide slot on both sides that aids in sweep-out and quenching of the vapor bubbles.

Um tutorial sobre métodos pseudo-espectrais para controle ótimo computacional

Este artigo é um tutorial introdutório sobre controle ótimo pseudo-espectral. Em métodos pseudo-espectrais, uma função é aproximada como uma combinação linear de funções de base suaves, tipicamente escolhidas como polinômios de Legendre ou Chebyshev. A colocação de equações algébrico-diferenciais é realizada em pontos de colocação ortogonal, que são selecionados de modo a minimizar o erro de interpolação. Métodos pseudo-espectrais discretizam o problema de controle ótimo original de modo a convertê-lo em um problema de programação não-linear. Um otimizador numérico é então empregado para obter soluções localmente ótimas. Este artigo também descreve sucintamente a funcionalidade e a implementação de um pacote computacional de código aberto escrito em C++ chamado PSOPT. Tal pacote emprega métodos de discretização pseudo-spectrais para resolver problemas de controle ótimo com múltiplas fase. O PSOPT permite a utilização de métodos de Legendre ou Chebyshev, e possui características úteis tais como diferenciação automática, detecção de esparsidade e escalonamento automático. O uso de métodos pseudo-espectrais é ilustrado em dois problemas retirados da literatura de controle ótimo computacional.

Variational piecewise constant level set methods for shape optimization of a two-density drum

We apply the piecewise constant level set method to a class of eigenvalue related two-phase shape optimization problems. Based on the augmented Lagrangian method and the Lagrange multiplier approach, we propose three effective variational methods for the constrained optimization problem. The corresponding gradient-type algorithms are detailed. The first Uzawa-type algorithm having applied to shape optimization in the literature is proven to be effective for our model, but it lacks stability and accuracy in satisfying the geometry constraint during the iteration. The two other novel algorithms we propose can overcome this limitation and satisfy the geometry constraint very accurately at each iteration. Moreover, they are both highly initial independent and more robust than the first algorithm. Without penalty parameters, the last projection Lagrangian algorithm has less severe restriction on the time step than the first two algorithms. Numerical results for various instances are presented and compared with those obtained by level set methods. The comparisons show effectiveness, efficiency and robustness of our methods. We expect our promising algorithms to be applied to other shape optimization and multiphase problems.