Multiphase Problems Research Articles

Mathematical Modeling of Fast Biomass Pyrolysis and Bio-Oil Formation. Note I: Kinetic Mechanism of Biomass Pyrolysis

This paper discusses the research activities done at Politecnico di Milano in the field of the detailed kinetic modeling of pyrolysis and combustion of biomass and bio-oil formation. Different critical steps are involved in this multicomponent, multiphase and multiscale problem. The first complexity relies on biomass characterization with the selection of reference species: cellulose, hemicellulose, lignins, and extractives. Fast pyrolysis involves kinetic mechanisms, first in the solid phase for biomass pyrolysis, then in gas-phase for secondary reactions of released products. These mechanisms involve large number of species and reactions, which make computations expensive. They need to be simplified, while still maintaining their description capability. Lumping procedures are extensively applied to allow the development of the overall model. Multistep pyrolysis mechanisms of reference species are discussed in this Note, with several comparisons with experimental data. A peculiarity of the model is its a...

Numerical study of the droplet impact onto liquid film on the rough solid surface via lattice Boltzmann method

The process of the droplet impact onto the liquid film on the rough solid surface, as one of the basic multiphase problems, is very important in many fields of science and engineering. On the other hand, the problem is also very complicated since there are many parameters that may influence the process of the droplet impact on the rough solid surface with a liquid film. Up to now, there are still little research on this problem, and to gain a better understanding on the physical mechanics of the droplet impact onto the film on the rough solid surface, it is desirable to conduct a detailed study. To clearly understand the physical phenomena appearing in the process of droplet impact on the liquid film, a parametric study on this problem is also carried out based on a recently developed lattice Boltzmann method in which a MRT lattice Boltzmann model is used to solve the Navier-Stokes equations, and the other is adopted to solve the Cahn-Hilliard equation that is used to depict the interface between different phases. In this paper, the effects of the relative thickness of film ( h ), the relative width of cavity ( d *) and the relative depth of cavity ( L *) on the dynamic behavior of interface are investigated in detail, and the velocity and pressure fields are also presented. In order to reduce the influence of lattice, we fix the lattice to be 600×120 for gas, which is fine enough to give accurate results. In addition, in our simulations, We =500, Re =480, viscosity ratio and density ratio are set to be 2:1. The numerical results first show that, the phenomena of crown and entrainment can be observed obviously during the process of droplet impact onto the liquid film on the rough interface when We and Re are large. The radius of spray ( r ), which is formed by the droplet impact onto liquid film, is related to time through the relation r / 2 R ≈ α U t / 2 R when h is small, which is coincident with the result of droplet impact onto the liquid film on smooth surface, and additionally the coefficient α would decrease with the increase of h . However, this relation seems not accurate for the case with a large h , and simultaneously, the splashing phenomenon has not been observed. Secondly, the relative width of cavity d * plays an important role on the phenomena of splashing. When d *=1, there will be two small droplets through the splashing phenomenon (left half part), then with this parameter increase, the number of small droplet and the point where the splashing occur will also change, and there also are much difference in relation of spray radius and time. Actually, if d * is small, the coefficient α would first decrease and then increase with the increase of d *, while if d *>8, the cavity width would only have a little influence on the behavior of spray. Finally, it is also found that the pressure change near the cavity bottom is small at different L *, that is to say, the relative depth of cavity L * seems to has no apparent effect on the formation of spray, but it brings a great influence on the splashing of spray and the movement of the droplet which is produced in the process of splashing.

Pycalphad: CALPHAD-based Computational Thermodynamics in Python

The pycalphad software package is a free and open-source Python library for designing thermodynamic models, calculating phase diagrams and investigating phase equilibria using the CALPHAD method. It provides routines for reading thermodynamic databases and solving the multi-component, multi-phase Gibbs energy minimization problem. The pycalphad software project advances the state of thermodynamic modeling by providing a flexible yet powerful interface for manipulating CALPHAD data and models. The key feature of the software is that the thermodynamic models of individual phases and their associated databases can be programmatically manipulated and overridden at run-time without modifying any internal solver or calculation code. Because the models are internally decoupled from the equilibrium solver and the models themselves are represented symbolically, pycalphad is an ideal tool for CALPHAD database development and model prototyping.

Multiscale S-Fraction Reduced-Order Models for Massive Wavefield Simulations

We developed a novel reduced-order multiscale method for solving large time-domain wavefield simulation problems. Our algorithm consists of two main stages. During the first “off-line” stage the fine-grid operator (of the graph Laplacian type) is partitioned on coarse cells (subdomains). Then projection-type multiscale reduced-order models (ROMs) are computed for the coarse cell operators. The off-line stage is embarrassingly parallel as ROM computations for the subdomains are independent of each other. It also does not depend on the number of simulated sources (inputs) and it is performed just once before the entire time-domain simulation. At the second “on-line” stage the time-domain simulation is performed within the obtained multiscale ROM framework. The crucial feature of our formulation is the representation of the ROMs in terms of matrix Stieltjes continued fractions (S-fractions). The layered structure of the S-fraction introduces several hidden layers in the ROM representation that results in the block-tridiagonal dynamic system within each coarse cell. This allows us to sparsify the obtained multiscale subdomain operator ROMs and to reduce the communications between the adjacent subdomains, which is highly beneficial for a parallel implementation of the on-line stage. Our approach perfectly suits the high performance computing architectures; however, in this paper we present rather promising numerical results for a serial computing implementation only. These results include three-dimensional acoustic and multiphase anisotropic elastic problems.

On the numerical evaluation of local curvature for diffuse interface models of microstructure evolution

Within diffuse interface models for multiphase problems, the curvature of the phase boundary can be expressed as the difference of two terms, a Laplacian and a second, gradient, term of the diffuse interface variable, ø. In phase field simulations of microstructure evolution, the second term is often replaced by f’(ø) = df/dø, where f(ø) is the potential function in the free energy functional of the underlying physical model. We show here that this replacement systematically deteriorates the accuracy in local curvature evaluation as compared to a discretized evaluation of the second term. Analytic estimates reveal that the discretization errors in the Laplacian and in the second term have roughly the same spatial dependence across the interface, thus leading to a cancellation of errors in k. This is confirmed in a test case, where the discretization error can be determined via comparison to the exact solution. If, however, the second term is replaced by a quasi exact expression, the error in ∆ø enters k without being compensated and can obscure the behavior of the local curvature. Due to the antisymmetric variations of this error term, approaches using the average curvature, as obtained from an integral along the interface normal, are immune to this problem.

A numerical approach for a general class of the spatial segregation of reaction–diffusion systems arising in population dynamics

In the current work we consider the numerical solutions of equations of stationary states for a general class of the spatial segregation of reaction–diffusion systems with m≥2 population densities. We introduce a discrete multi-phase minimization problem related to the segregation problem, which allows to prove the existence and uniqueness of the corresponding finite difference scheme. Based on that scheme, we suggest an iterative algorithm and show its consistency and stability. For the special case m=2, we show that the problem gives rise to the generalized version of the so-called two-phase obstacle problem. In this particular case we introduce the notion of viscosity solutions and prove convergence of the difference scheme to the unique viscosity solution. At the end of the paper we present computational tests, for different internal dynamics, and discuss numerical results.

Stability criteria for multiphase partitioning problems with volume constraints

We study the stability of partitions involving two or more phases in convex domains under the assumption of at most two-phase contact, thus excluding in particular triple junctions. We present a detailed derivation of the second variation formula with particular attention to the boundary terms, and then study the sign of the principal eigenvalue of the Jacobi operator. We thus derive certain stability criteria, and in particular we recapture the Sternberg-Zumbrun result on the instability of the disconnected phases in the more general setting of several phases.

Computational investigations and grid refinement study of 3D transient flow in a cylindrical tank using OpenFOAM

The study of physic fluid for a liquid draining inside a tank is easily accessible using numerical simulation. However, numerical simulation is expensive when the liquid draining involves the multi-phase problem. Since an accurate numerical simulation can be obtained if a proper method for error estimation is accomplished, this paper provides systematic assessment of error estimation due to grid convergence error using OpenFOAM. OpenFOAM is an open source CFD-toolbox and it is well-known among the researchers and institutions because of its free applications and ready to use. In this study, three types of grid resolution are used: coarse, medium and fine grids. Grid Convergence Index (GCI) is applied to estimate the error due to the grid sensitivity. A monotonic convergence condition is obtained in this study that shows the grid convergence error has been progressively reduced. The fine grid has the GCI value below 1%. The extrapolated value from Richardson Extrapolation is in the range of the GCI obtained.

Coupled heat conduction and multiphase change problem accounting for thermal contact resistance

In this paper, heat conduction coupled with multiphase changes are considered in a cylindrical multilayer composite accounting for thermal contact resistance depending on contact pressures and roughness parameters. A numerical simulation is proposed using both analytical developments and numerical computations. The presented modeling strategy relies on an algorithm that alternates between heat conduction accounting for volumetric heat sources and a multiphase change model based on non-isothermal Avrami’s equation using the isokinetic assumption. Applications to coiling process (winding of a steel strip on itself) are considered. Indeed, phase changes determine the microstructure of the final material and are responsible for residual stresses that create flatness defects. A Finite Element modeling is used for validating the presented solution and numerical results are presented and discussed.

The hole-filling method and multiscale algorithm for the heat transfer performance of periodic porous materials

In this paper, we consider the transient heat transfer problem with rapidly oscillating coefficients in periodic porous materials. The hole-filling method through filling all holes with a very compliant material is investigated, and as the thermal conductivity parameters of the weak phase go to zero, the proof of the limiting process is derived in detail. Then, the multiscale analysis and computations based on the hole-filling method for the associated multiphase problem in a domain without holes are proposed. Finally, some numerical results are given, which show a good agreement and verify the feasibility of the hole-filling method.

A multiphase optimal control method for multi-train control and scheduling on railway lines

We consider a combined train control and scheduling problem involving multiple trains in a railway line with a predetermined departure/arrival sequence of the trains at stations and meeting points along the line. The problem is formulated as a multiphase optimal control problem while incorporating complex train running conditions (including undulating track, variable speed restrictions, running resistances, speed-dependent maximum tractive/braking forces) and practical train operation constraints on departure/arrival/running/dwell times. Two case studies are conducted. The first case illustrates the control and scheduling problem of two trains in a small artificial network with three nodes, where one train follows and overtakes the other. The second case optimises the control and timetable of a single train in a subway line. The case studies demonstrate that the proposed framework can provide an effective approach in solving the combined train scheduling and control problem for reducing energy consumption in railway operations.

A phase-field modeling approach of hydraulic fracture in saturated porous media

Continuum porous media theories, extended by a diffusive phase-field modeling (PFM) approach, introduce a convenient and efficient tool to the simulation of hydraulic fracture in fluid-saturated heterogeneous materials. In this, hydraulic- or tension-induced fracture occurs in the solid phase. This leads to permanent local changes in the permeability, the volume fractions of the constituents as well as the interstitial-fluid flow. In this work, the mechanical behaviors of the multi-field, multi-phase problem of saturated porous media, such as the pore-fluid flow and the solid-skeleton deformation, are described using the macroscopic Theory of Porous Media (TPM). To account for crack nucleation and propagation in the sense of brittle fracture, the energy-minimization-based PFM procedure is applied, which approximates the sharp edges of the crack by a diffusive transition zone using an auxiliary phase-field variable. Furthermore, the PFM can be implemented in usual continuum finite element packages, allowing for a robust solution of initial-boundary-value problems (IBVP). For the purpose of validation and comparison, simulations of a two-dimensional IBVP of hydraulic fracture are introduced at the end of this research paper.

NURBS enhanced HIFEM: A fully mesh-independent method with zero geometric discretization error

This manuscript introduces a NURBS-enhanced Hierarchical Interface-enriched Finite Element Method (NeHIFEM) for the mesh-independent treatment of multiphase problems with complex morphologies. The NeHIFEM implements a non-isoparametric mapping similar to that introduced in the NURBS Enhanced Finite Element Method (NEFEM) for the exact geometrical modeling of problems with curved materials interfaces. In this work, we propose a new generalized NURBS-enhanced mapping for triangular elements with three curvilinear edges, together with an isoparametric approach for evaluating basis functions in these elements to facilitate the construction of enrichment functions in the NeHIFEM. This unique capability allows the implementation of simple structured meshes for creating the computational model, while fully eliminating the geometric discretization error without introducing additional degrees of freedom. A new approach for the construction of higher-order enrichment functions in the NeHIFEM is also introduced to accurately capture gradient discontinuities along materials interfaces. Several example problems are presented to shed light on the accuracy and convergence rate of the NeHIFEM. We also show the application of this method for simulating the thermal and structural responses of heterogeneous materials with complex microstructures.

Non‐intrusive reduced‐order modeling for multiphase porous media flows using Smolyak sparse grids

SummaryIn this article, we describe a non‐intrusive reduction method for porous media multiphase flows using Smolyak sparse grids. This is the first attempt at applying such an non‐intrusive reduced‐order modelling (NIROM) based on Smolyak sparse grids to porous media multiphase flows. The advantage of this NIROM for porous media multiphase flows resides in that its non‐intrusiveness, which means it does not require modifications to the source code of full model. Another novelty is that it uses Smolyak sparse grids to construct a set of hypersurfaces representing the reduced‐porous media multiphase problem. This NIROM is implemented under the framework of an unstructured mesh control volume finite element multiphase model. Numerical examples show that the NIROM accuracy relative to the high‐fidelity model is maintained, whilst the computational cost is reduced by several orders of magnitude. Copyright © 2016 John Wiley & Sons, Ltd.

Maximum range trajectory optimization for a boost-glide vehicle using adaptive mesh refinement pseudospectral methods

The maximum down range trajectory optimization problem with multiple phases and multiple constraints corresponding to the flight of a boost-glide vehicle is considered. The longitudinal motion model was built as a multiphase optimization problem under constraints. Legendre–Gauss–Radau collocation points were used to transcribe the optimization problem into a finite-dimensional nonlinear programming problem, and the maximization down range trajectory was obtained based on adaptive mesh refinement pseudospectral methods. However, sometimes it is difficult to find interior points without position constraints. A novel optimization strategy based on dynamic programming theory was proposed to search the free interior points more accurately and quickly, which resulted in almost the same optimized trajectory while producing a small mesh. The results of numerical examples showed that the boost-glide vehicle trajectory optimization problem is solved using the adaptive mesh refinement pseudospectral methods.

Structural optimization of contact problems using Cahn–Hilliard model

The paper deals with an analysis and numerical solution of topology optimization problems for elastic bodies in unilateral contact using a phase field approach. The topology optimization problem consists in finding such distribution of the body material in a design domain to minimize the normal contact stress along the boundary of the contacting bodies. The optimization problem is formulated as multiphase problem in terms of material density function. The original cost functional is regularized using surface and bulk energy terms. These terms allow to control global perimeter constraint and the occurrence of the intermediate solution values. Using Lagrange multipliers approach the derivative of the cost functional with respect to density function is calculated. Necessary optimality condition is formulated as the steady state of the phase transition governed by this equation. The finite element and the finite difference methods are used as the approximation methods. Numerical examples are provided and discussed.

Smoothed Particle Hydrodynamics (SPH) simulation of a high-pressure homogenization process

High-pressure homogenization is a widely used process in the food, pharmaceutical, and cosmetic industry for producing emulsions. Because of small dimensions and high velocities, the experimental and numerical investigation of such a process is challenging. Hence, the development of products is mostly based on trial and error. In this paper, simulations of a generic high-pressure homogenization process using the Lagrangian, mesh-free smoothed particle hydrodynamics (SPH) method are presented and compared to experimental findings using Micro-Particle Image Velocimetry (μ-PIV). The SPH code has been developed and validated with the scope of simulating technical relevant multi-phase problems (Hofler et al. 2012). The present simulations cover the investigation of two different dynamic viscosities of the dispersed phase as well as different droplet trajectories. The comparison between the simulations and the experiments focusses on the velocity distribution of the continuous phase and the droplet deformation and breakup. In both cases a qualitatively good agreement is observed, demonstrating the ability of our SPH implementation for simulating technical relevant two-phase flows.

An Adaptive Finite Element Multi-Mesh Approach for Interacting Deformable Objects in Flow

Abstract We consider a hydrodynamic multi-phase field problem to model the interaction of deformable objects. The numerical approach considers one phase field variable for each object and allows for an independent adaptive mesh refinement for each variable. Using the special structure of various terms allows interpolating the solution on one mesh onto another without loss of information. Together with a general multi-mesh concept for the other terms speedup by a factor of two can be demonstrated which improves with the number of interacting objects. The general concept is demonstrated on an example describing the interaction of red blood cells in an idealized vessel.

A Multi-agent Approach to Computational Optimization of Metal Forming Processes

In the paper we present a new original approach to a complex multi-phase process optimization problem that relies on dividing the optimization task into partial tasks related to the implementation of individual phases. To implement this concept, we propose a multi-agent approach. Its practical realization is shown on the example of manufacturing process for auto body parts made of the Advanced High Strength Steels (AHSS). Although it is a rather simple process consisting of only two phases, we assumed, however, that the results obtained will allow to extend further research to more complex problems. We present the operating principles of a multi-agent model, the flow of the messages between agents, and the architecture of the system. To ensure the proper speed of the whole system a simple and flexible multi-agent framework called Eve was used to develop a prototype of the system. Research performed, as well as preliminary tests have shown that a multi-agent approach can be successfully applied to reduce complexity of the whole optimization processes. Due to splitting a single, complex optimization process into several, partially independent optimization processes, delegating them to autonomous agents, and application of a knowledge-based reasoning system, significant advantage could be observed over to the solutions described so far in the literature.

A Multiphase Shape Optimization Problem for Eigenvalues: Qualitative Study and Numerical Results

We consider the multiphase shape optimization problem $$\min\Big\{\sum_{i=1}^h\lambda_1(\Omega_i)+\alpha|\Omega_i|:\ \Omega_i\ \hbox{open},\ \Omega_i\subset D,\ \Omega_i\cap\Omega_j=\emptyset\Big\},$$ where $\alpha>0$ is a given constant and $ D\subset\Bbb{R}^2$ is a bounded open set with Lipschitz boundary. We give some new results concerning the qualitative properties of the optimal sets and the regularity of the corresponding eigenfunctions. We also provide numerical results for the optimal partitions.