Adjoint-Based Optimal Boundary Control of a Stefan Problem System Fully Coupled with Navier-Stokes Equations

Free boundary and moving boundary problems, that can be used to model crystal growth or the solidification and melting of pure materials, receive growing attention in science and technology. The optimal control of these problems appear even more interesting since certain desired shapes of the boundaries improve, e.g., the material quality in the case of crystal growth. We consider the so called two-phase Stefan problem that models a solid and a liquid phase separated by a moving interface. In the work presented, we take a sharp interface model approach and define a quadratic tracking-type cost functional that penalizes the deviation of the interface from the desired state at a final time as well as the control costs. Following the “optimize-then-discretize” paradigm, we formulate a first order optimality system using the formal Lagrange approach and derive the adjoint PDE system that provides the needed gradient of the cost functional. By means of an example setup of a container with an in- and outflow of water and a cooling unit at the bottom, we illustrate how the derived formulations can be used to achieve a desired interface between the solid and the fluid phase by controlling the flow at the inlet. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A comparison of numerical models for one-dimensional Stefan problems

In this paper, we present a critical comparison of the suitability of several numerical methods, level set, moving grid and phase field model, to address two well-known Stefan problems in phase transformation studies: melting of a pure phase and diffusional solid-state phase transformations in a binary system. Similarity solutions are applied to verify the numerical results. The comparison shows that the type of phase transformation considered determines the convenience of the numerical techniques. Finally, it is shown both numerically and analytically that the solid-solid phase transformation is a limiting case of the solid–liquid transformation.     

Towards Riccati-Feedback Control of Complex Flows with Moving Interfaces

Problems featuring moving interfaces appear in many applications. They can model solidification and melting of pure materials, crystal growth and other multi-phase problems. The control of the moving interface enables to, for example, influence production processes and, thus, the product material quality. We consider the two-phase Stefan problem that models a solid and a liquid phase separated by the moving interface. In the liquid phase, the heat distribution is characterized by a convection-diffusion equation. The fluid flow in the liquid phase is described by the Navier–Stokes equations which introduces a differential algebraic structure to the system. The interface movement is coupled with the temperature through the Stefan condition, which adds additional algebraic constraints. Our formulation uses a sharp interface representation and we define a quadratic tracking-type cost functional as a target of a control input. We compute an open loop optimal control for the Stefan problem using an adjoint system. For a feedback representation, we linearize the system about the trajectory defined by the open loop control. This results in a linear-quadratic regulator problem, for which we formulate the differential Riccati equation with time varying coefficients. This Riccati equation defines the corresponding feedback gain. Further, we present the feedback formulation that takes into account the structure and the differential algebraic components of the problem. Also, we discuss how the complexities that come, for example, with mesh movements, can be handled in a feedback setting. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A ternary phase bi-scale FE-model for diffusion-driven dendritic alloy solidification processes

It is of high interest to describe alloy solidification processes with numerical simulations. In order to predict the material behavior as precisely as possible, a ternary phase, bi-scale numerical model will be presented. This paper is based on a coupled thermo-mechanical, two-phase, two-scale finite element model developed by Moj et al. [2], where the theory of porous media (TPM) [1] has been used. Finite plasticity extended by secondary power-law creep is utilized to describe the solid phase and linear visco-elasticity with Darcy's law of permeability for the liquid phase, respectively. Here, the microscopic, temperature-driven phase transition approach is replaced by the diffusion-driven 0D model according to Wang and Beckermann [3]. The decisive material properties during solidification are captured by phenomenological formulations for dendritic growth and solute diffusion processes. A columnar as well as an equiaxial solidification example will be shown to demonstrate the principal performance of the presented model. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

AN EMBEDDED BOUNDARY METHOD FOR ELLIPTIC AND PARABOLIC PROBLEMS WITH INTERFACES AND APPLICATION TO MULTI-MATERIAL SYSTEMS WITH PHASE TRANSITIONS

The embedded boundary method for solving elliptic and parabolic problems in geometrically complex domains using Cartesian meshes by Johansen and Colella （1998, J. Comput. Phys. 147, 60） has been extended for elliptic and parabolic problems with interior boundaries or interfaces of discontinuities of material properties or solutions. Second order accuracy is achieved in space and time for both stationary and moving interface problems. The method is conservative for elliptic and parabolic problems with fixed interfaces. Based on this method, a front tracking algorithm for the Stefan problem has been developed. The accuracy of the method is measured through comparison with exact solution to a two-dimensional Stefan problem. The algorithm has been used for the study of melting and solidification problems.     

Numerical simulation two phase flows of casting filling process using SOLA particle level set method

Mould filling process is a typical gas–liquid metal two phase flow phenomenon. Numerical simulation of the two phase flows of mould filling process can be used to properly predicate the back pressure effect, the gas entrapment defects, and better understand the complex motions of the gas phase and the liquid phase. In this paper, a novel sharp interface incompressible two phase numerical model for mould filling process is presented. A simple ghost fluid method like discretization method and a density evaluation method at face centers of finite difference staggered grid are proposed to overcome the difficulties when solving two phase Navier–Stokes equations with large-density ratio and large-viscosity ratio. A new mass conservation particle level set method is developed to capture the gas–liquid metal phase interface. The classical pressure-correction based SOLA algorithm is modified to solve the two phase Navier–Stokes equations. Two numerical tests including the Zalesak disk problem and the broken dam problem are used to demonstrate the accuracy of the present method. The numerical method is then adopted to simulate three mould filling examples including two high speed CCD camera imaging water filling experiments and an in situ X-ray imaging experiment of pure aluminum filling. The simulation results are in good agreement with the experiments.     

A finite element method for steady-state conduction-advection phase change problems

A new finite element: technique is developed to solve steady-state conduction-advection problems with a phase change. The energy balance equation at the solid/liquid interface is employed to calculate the velocity of the solid/liquid interface in the Lagrangian frame. The position of the solid/liquid interface in the Eulerian frame is determined based on the composition of the velocity of the solid/liquid interface in the Lagrangian frame and the steady-state velocity of a rigid body. The interface position and the finite element mesh are continuously updated during an incremental process. No artificial diffusion is needed with this new finite element approach. An analytical solution for solidification of a pure material with a radiative boundary condition is also developed in this paper. Numerical experimentation is conducted and the corresponding results are compared with analytical solutions. The numerical results agree well with analytical solutions.     

A smoothed particle hydrodynamics-phase field method with radial basis functions and moving least squares for meshfree simulation of dendritic solidification

We present a robust and efficient approach to meshfree phase-field (PF) simulation of dendritic solidification on arbitrary domain geometries using smoothed particle hydrodynamics (SPH). We use radial basis functions (RBFs) and moving least squares (MLS) as alternative approaches for constructing kernel approximation functions exhibiting a higher order of consistency than traditional kernel functions used in SPH. In the proposed smoothed particle hydrodynamics-phase field method (SPH–PFM), proper discretization of the PF order parameter at the diffuse interface region can be easily accomplished independently from the particle spacing resolution used for computing the thermal field distribution. We use an implicit geometry construction approach to automatically generate virtual boundary particles to impose Neumann-type boundary conditions at the domain boundaries. We solve the Allen–Cahn equation locally at particles constructed at a narrow band around the interface region. Additionally, only first-order derivatives of the meshfree approximation functions are needed in our implementation to solve the governing equations. Mathematical formulation and detailed analysis will be presented and discussed where we investigate the effect of the meshfree approximation scheme on the final morphology of the grown dendrite.     

Modeling binary alloy solidification by a random projection method

This paper addresses the numerical modeling of the solidification of a binary alloy that obeys a liquidus–solidus phase diagram. In order to capture the moving melting front, we introduce a Lagrange projection scheme based on a random sampling projection. Using a finite volume formulation, we define accurate numerical fluxes for the temperature and concentration fields which guarantee the sharp treatment of the boundary conditions at the moving front, especially the jump of the concentration according to the liquidus–solidus diagram. We provide some numerical illustrations which assess the good behavior of the method: maximum principle, stability under CFL condition, numerical convergence toward self‐similar solutions, ability to handle two melting fronts.     

Stability of the mixed time partitioning methods in relation to the size of time step

In this paper we focus on stability of a mixed time partitioning methods in relation to time step size which is using in numerical modelling of two-component alloys solidification. We present the numerical integration methods to solve solidification problems in a fast and accurate way. Our approach exploits the fact that physical processes inside a mould are of different nature than those in a solidifying cast. As a result different time steps can be used to run computations within both sub-domains. Because processes that are modeled in the cast sub-domain are more dynamic they require very fine-grained time step. On the other hand a heat transfer within the mould sub-domain is less intense, and thus coarse-grained step is sufficient to guarantee desired precision of computations. We propose using a fixed time step in the cast and its integer multiple in other parts of mould. We use one-step explicit and implicit time integration Θ schemes. These time integration schemes are applied to equations obtained after spatial discretization. The implicit scheme is unconditionally stable, but stability of the explicit scheme depends on the size of time step. Critical time step size can be determined on the basis of eigenvalues of the amplification matrix that depend on the material properties, size and type of the finite element. In this work we present the manner of determining the critical time step and its affect on the course of numerical simulation of solidification. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Error analysis of the numerical solution of split differential equations

The operator splitting method is a widely used approach for solving partial differential equations describing physical processes. Its application usually requires the use of certain numerical methods in order to solve the different split sub-problems. The error analysis of such a numerical approach is a complex task. In the present paper we show that an interaction error appears in the numerical solution when an operator splitting procedure is applied together with a lower-order numerical method. The effect of the interaction error is investigated by an analytical study and by numerical experiments made for a test problem.     

The equimaterial approach for the numerical solution of the one dimensional transient diffusion equation with zero flux boundary conditions

A new approach is proposed for the grid motion for the numerical solution of a general transient diffusion equation in one spatial dimension with zero flux boundary conditions. The new criterion for grid motion is that the solute amount contained in each discretization section should be a pre-described fraction of the total solute amount at each time step. This requirement is not explicitly enforced to the solution technique but it is implicitly included in the equation through the appropriate variable transformation. The results showed that although the technique leads to the required grid motion the numerical results are of pure quality due to the appearance of singularities during the variable transformation procedure. Nevertheless, it is shown that by appropriate numerical handling of the solution at the singularity region the technique can lead to accurate results and potentially can replace the existing moving grid algorithms at least for the particular problem at hand.     

Combined grid-characteristic method for the numerical solution of three-dimensional dynamical elastoplastic problems

A combined method blending the advantages of smoothed particles hydrodynamics (SPH) and the grid-characteristic method (GCM) is proposed for simulating elastoplastic bodies. Various grid methods, including the GCM, have long been used for the numerical simulation of elastoplastic media. This method applies to the simulation of wave processes in elastic media, including elastic impacts, in which case an advantage is the use of moving tetrahedral meshes. Additionally, fracture processes can be simulated by applying various fracture criteria. However, this is a technically complicated task with the accuracy of the results degrading due to the continual updating of the grid. A more suitable approach to the simulation of processes involving substantial fractures and deformations is based on SPH, which is a meshless method. However, this method also has shortcomings: it produces spurious modes, and the simulation of oscillations requires particle refinement. Thus, two families of methods are available that are optimal as applied to two different groups of problems. However, a realworld problem can frequently be a mixed one, which requires a substantial tradeoff in the numerical methods applied. Aimed at solving such problems, a combined GCM-SPH method is developed that blends the advantages of two constituting techniques and partially eliminates their shortcomings.     

The discrete nature of grain attachment models in simulations of equiaxed solidification

Grain refiner is often added to aluminum and magnesium alloys during solidification processing to encourage the development of a fine equiaxed grain structure. Numerical modeling of such processes face the challenge of considering the effect of free floating grains that nucleate on grain refiner particles, are advected in the bulk fluid flow, and eventually coalesce to form a permeable, rigid solid structure. While several models have been developed to consider the advection of solid grains, the attachment of these grains is uniformly treated on a discrete, cell-by-cell basis. In a previous study, channel segregates were observed in the predicted composition field of equiaxed solidification simulations and were found to exhibit an extreme grid dependence. These channels were examined in the present study in detail for two different grain attachment models, one that assumed coalescence occurs at a constant and uniform volume fraction solid, and one that considered the effects of the local solid velocity field. The mechanism of the initiation and propagation of these channels was explored, and their physical relevance considered. It was concluded that these defects were primarily numerical artifacts arising from the discrete nature of the grain attachment models, and therefore, necessarily occurred on the length scale of the grid spacing. Development of an alternative attachment model that avoid this numerical problem is the subject of ongoing research.     

Two-grid finite element method for the dual-permeability-Stokes fluid flow model

In this paper, two-grid finite element method for the steady dual-permeability-Stokes fluid flow model is proposed and analyzed. Dual-permeability-Stokes interface system has vast applications in many areas such as hydrocarbon recovery process, especially in hydraulically fractured tight/shale oil/gas reservoirs. Two-grid method is popular and convenient to solve a large multiphysics interface system by decoupling the coupled problem into several subproblems. Herein, the two-grid approach is used to reduce the coding task substantially, which provides computational flexibility without losing the approximate accuracy. Firstly, we solve a global problem through standard P_k ? P_k??1 ? P_k ? P_k finite elements on the coarse grid. After that, a coarse grid solution is applied for the decoupling between the interface terms and the mass exchange terms to solve three independent subproblems on the fine grid. The three independent parallel subproblems are the Stokes equations, the microfracture equations, and the matrix equations, respectively. Four numerical tests are presented to validate the numerical methods and illustrate the features of the dual-permeability-Stokes model.

     

A numerical solution of a one-dimensional blood flow model—moving grid approach

We consider a one-dimensional blood flow model suitable for larger arteries. It consists of a hyperbolic system of two coupled nonlinear equations. The model has already been successfully used in practice. Its numerical solution is usually achieved by means of an explicit Taylor–Galerkin scheme. We have proposed a different approach. The system can be transformed to characteristic directions emphasizing the physical nature of the problem. We solved this system by using an operator splitting on a moving grid.     

The method of lines for the computation of incompressible viscous flows

A common approach to finding numerical solutions of the time-dependent incompressible Navier-Stokes equations is considered within the method of lines framework [1]. For the determination of viscous incompressible flows the stream-function formulation for the fourth-order equation [2, 3], an artificial compressibility method [4], and a modified velocity correction method [5] are designed. Some improved and extended results of numerical simulations obtained by the author in the previous works are presented. Test calculations have been done for various flows inside square, triangular and semicircular cavities with one moving wall, the backward-facing step, double bent channels and for the flow around an aerofoil at large angle of attack. An alternative and practical methodology for resolving the Navier-Stokes equations in arbitrarily complex geometries using Cartesian meshes is proposed. Some of complex geometrical configurations can be decomposed into a set of subdomains. The simplest approach for specifying boundary conditions near curved or irregular boundaries is to transfer all the variables from the boundaries to the nearest grid knots. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of the spherical metric tensor to grid adaptation and the solution of applied problems

New results concerning the construction and application of adaptive numerical grids for solving applied problems are presented. The grid generation technique is based on the numerical solution of inverted Beltrami and diffusion equations for a monitor metric. The capabilities of the spherical metric tensor as applied to adaptive grid generation are examined in detail. Adaptive hexahedral grids are used to numerically solve a boundary value problem for the three-dimensional heat equation with a moving boundary in a continuous medium with discontinuous thermophysical parameters; this problem models the interaction of a thermal wave with a thermocouple embedded in the solid.     

The development of a cellular automaton-finite volume model for dendritic growth

A cellular automaton to track the solid–liquid interface movement is linked to finite volume computations of solute diffusion to simulate the behavior of dendritic structures in binary alloys during solidification. A significant problem encountered in the CA formulation has been the presence of artificial anisotropy in growth kinetics introduced by a Cartesian CA grid. A new technique to track the interface movement is proposed to model dendritic growth in different crystallographic orientations while reducing the anisotropy due to grid orientation. The model stability with respect to the numerical parameters (cell size and time step) for various operating conditions is examined. A method for generating an operating window in Δt and Δx has been identified, in which the model gives a grid-independent set of results for calculated dendrite tip radius and tip undercooling. Finally, the model is compared to published experimental and analytical results for both directional and equiaxed growth conditions.