Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition

This paper describes a phase field method for the optimization of multimaterial structural topology with a generalized Cahn–Hilliard model. Similar to the well-known simple isotropic material with penalization method, the mass concentration of each material phase is considered as design variable. However, a variational approach is taken with the Cahn–Hilliard theory to define a thermodynamic model, taking into account of the bulk energy and interface energy of the phases and the elastic strain energy of the structure. As a result, the structural optimization problem is transformed into a phase transition problem defined by a set of nonlinear parabolic partial differential equations. The generalized Cahn–Hilliard model regularizes the original ill-posed topology optimization problem and provides flexibility of topology changes with interface coalescence and break-up due to phase separation and coarsening. We employ a powerful multigrid algorithm and extend it to include four material phases for numerical solution of the Cahn–Hilliard equations. We demonstrate our approach through several 2-D and 3-D examples to minimize mean compliance of the multimaterial structures.     

Phase field computations for surface diffusion and void electromigration in {\mathbb{R}^3}

We consider a finite element approximation of a phase field model for the evolution of voids by surface diffusion in an electrically conducting solid. The phase field equations are given by a degenerate Cahn–Hilliard equation with an external forcing induced by the electric field. We describe the iterative scheme used to solve the resulting nonlinear discrete equations and present some numerical experiments in three space dimensions. The first author was supported by the EPSRC grant EP/C548973/1.     

A generalized continuous surface tension force formulation for phase-field models for multi-component immiscible fluid flows

We present a new phase-field method for modeling surface tension effects on multi-component immiscible fluid flows. Interfaces between fluids having different properties are represented as transition regions of finite thickness across which the phase-field varies continuously. At each point in the transition region, we define a force density which is proportional to the curvature of the interface times a smoothed Dirac delta function. We consider a vector valued phase-field, the velocity, and pressure fields which are governed by multi-component advective Cahn–Hilliard and modified Navier–Stokes equations. The new formulation makes it possible to model any combination of interfaces without any additional decision criteria. It is general, therefore it can be applied to any number of fluid components. We give computational results for the four component fluid flows to illustrate the properties of the method. The capabilities of the method are computationally demonstrated with phase separations via a spinodal decomposition in a four-component mixture, pressure field distribution for three stationary drops, and the dynamics of two droplets inside another drop embedded in the ambient liquid.     

Solutions of the Cahn–Hilliard equation

In this paper, we found some exact solutions of the Cahn–Hilliard equation and the system of the equations by considering a modified extended tanh function method. A numerical solution to a Cahn–Hilliard equation is obtained using a homotopy perturbation method (HPM) combined with the Adomian decomposition method (ADM). The comparisons are given in the tables.     

Application of mortar elements to diffuse-interface methods

An adaptive 2D mesh refinement technique based on mortar spectral elements applied to diffuse-interface methods is presented. The refinement algorithm tracks the movement of the 2D diffuse-interface and subsequently refines the mesh locally at that interface, while coarsening the mesh in the rest of the computational domain, based on error estimators. Convergence of the method is validated using a Gaussian distribution problem and results are presented for a Cahn–Hilliard diffuse-interface model applied to capture the transient dynamics of polymer blends.     

Continuous finite element schemes for a phase field model in two-layer fluid Bénard–Marangoni convection computations

In this article, we study a phase field model for a two-layer fluid where the temperature dependence of both the density (buoyancy forces) and the surface tension (Marangoni effects) is considered. The phase field model consisting of a modified Navier–Stokes equation, a Cahn–Hilliard phase field equation and an energy transport equation is derived through an energetic variational procedure. An appropriate variational form and a continuous finite element method are adopted to maintain the underlying energy law to its greatest extent. A few examples for Bénard–Marangoni convection in an Acetonitrile and n-Hexane two-layer fluid system heated from above will be computed to justify our phase field model and further show the good performance of our methods. In addition, an interesting experiment will be performed to show the competition between the Marangoni effects and the buoyancy forces.     

Accurate contact angle boundary conditions for the Cahn–Hilliard equations

The contact angle dynamics between a two-phase interface and a solid surface is important in physical interpretations, mathematical modeling, and numerical treatments. We present a novel formulation based on a characteristic interpolation for the contact angle boundary conditions for the Cahn–Hilliard equation. The new scheme inherits characteristic properties, such as the mass conservation, the total energy decrease, and the unconditionally gradient stability. We demonstrate the accuracy and robustness of the proposed contact angle boundary formulation with various numerical experiments. The numerical results indicate a potential usefulness of the proposed method for accurately calculating contact angle problems.     

Optimal topologies derived from a phase-field method

A topology optimization method allowing for perimeter control is presented. The approach is based on a functional that takes the material density and the strain field as arguments. The cost for surfaces is included in the functional that is minimized. Diffuse designs are avoided by introducing a penalty term in the functional that is minimized. Equilibrium and a volume constraint are enforced via a Lagrange multiplier technique. The extremum to the functional is found by use of the Cahn–Hilliard phase-field method. It is shown that the optimization problem is suitable for finite element implementation and the FE-formulation is discussed in detail. In the numerical examples provided, the influence of surface penalization is investigated. It is shown that the perimeter of the structure can be controlled using the proposed scheme.     

Numerical modeling of multiphase flows in microfluidics and micro process engineering: a review of methods and applications

This article presents a comprehensive review of numerical methods and models for interface resolving simulations of multiphase flows in microfluidics and micro process engineering. The focus of the paper is on continuum methods where it covers the three common approaches in the sharp interface limit, namely the volume-of-fluid method with interface reconstruction, the level set method and the front tracking method, as well as methods with finite interface thickness such as color-function based methods and the phase-field method. Variants of the mesoscopic lattice Boltzmann method for two-fluid flows are also discussed, as well as various hybrid approaches. The mathematical foundation of each method is given and its specific advantages and limitations are highlighted. For continuum methods, the coupling of the interface evolution equation with the single-field Navier–Stokes equations and related issues are discussed. Methods and models for surface tension forces, contact lines, heat and mass transfer and phase change are presented. In the second part of this article applications of the methods in microfluidics and micro process engineering are reviewed, including flow hydrodynamics (separated and segmented flow, bubble and drop formation, breakup and coalescence), heat and mass transfer (with and without chemical reactions), mixing and dispersion, Marangoni flows and surfactants, and boiling.     

Simulation of the Paraxial Laser Propagation Coupled with Hydrodynamics in 3D Geometry

We address here numerical simulation problems for modeling some phenomena arising in plasmas produced in experimental devices for Inertial Confinement Fusion. The model consists of a compressible fluid dynamics system coupled with a paraxial equation for modeling the laser propagation. For the fluid dynamics system, a numerical method of Lagrange–Euler type is used. For the paraxial equation, a time implicit discretization is settled which preserves the laser energy balance; the method is based on a splitting of the propagation term and the diffraction terms according to the propagation spatial variable. We give some features on the 3D implementation of the method in the parallel platform HERA. Results showing the accuracy of the numerical scheme are presented and we give also numerical results related to cases corresponding to realistic simulations, with a mesh containing up to 500 millions of cells.     

Energy-stable linear schemes for polymer–solvent phase field models

We present new linear energy-stable numerical schemes for numerical simulation of complex polymer–solvent mixtures. The mathematical model proposed by Zhou et al. (2006) consists of the Cahn–Hilliard equation which describes dynamics of the interface that separates polymer and solvent and the Oldroyd-B equations for the hydrodynamics of polymeric mixtures. The model is thermodynamically consistent and dissipates free energy. Our main goal in this paper is to derive numerical schemes for the polymer–solvent mixture model that are energy dissipative and efficient in time. To this end we will propose several problem-suited time discretizations yielding linear schemes and discuss their properties.     

Energy stable and large time-stepping methods for the Cahn–Hilliard equation

We present the numerical methods for the Cahn–Hilliard equation, which describes phase separation phenomenon. The goal of this paper is to construct high-order, energy stable and large time-stepping methods by using Eyre's convex splitting technique. The equation is discretized by using a fourth-order compact difference scheme in space and first-order, second-order or third-order implicit–explicit Runge–Kutta schemes in time. The energy stability for the first-order scheme is proved. Numerical experiments are given to demonstrate the performance of the proposed methods.     

Mathematical modeling and numerical simulation of freezing processes of a supercooled melt under consideration of density changes

In this paper, we derive a model for the two-phase freezing process of supercooled fluids. Especially, we take density changes along the phase interfaces into account. Thus, besides heat diffusion and the interface phenomena, mass transport and convection in the fluid phase which is given by the full Navier–Stokes equations has to be considered. For the 2D case we implemented an algorithm for the numerical solution of the mathematical model using uniform volume cells for a finite difference discretization. Additionally, the phase boundaries are captured by a surface tracking method. We report on the mathematical model and its derivation, describe the numerical algorithm and present numerical experiments. Received: 11 September 1997 / Accepted: 19 June 1998     

Numerical modeling of electrowetting transport processes for digital microfluidics

Electrical actuation and control of liquid droplets in Hele-Shaw cells have significant importance for microfluidics and lab-on-chip devices. Numerical modeling of complex physical phenomena like contact line dynamics, dynamic contact angles or contact angle hysteresis involved in these processes do challenge in a significant manner classical numerical approaches based on macroscopic Navier–Stokes partial differential equations. In this paper, we analyze the efficiency of a numerical lattice Boltzmann model to simulate basic transport operations of sub-millimeter liquid droplets in electrowetting actuated Hele-Shaw cells. We use a two-phase three-dimensional D3Q19 lattice Boltzmann scheme driven by a Shan–Chen-type mesoscopic potential in order to simulate the gas–liquid equilibrium state of a liquid droplet confined between two solid plates. The contact angles at the liquid–solid–gas interface are simulated by taking into consideration the interaction between fluid particles and solid nodes. The electrodes are designed as regions of tunable wettability on the bottom plate and the contact angles adjusted by changing the interaction strength of the liquid with these regions. The transport velocities obtained with this approach are compared to predictions from analytical models and very good agreement is obtained.     

A Petrov Galerkin finite-element method for interface problems arising in sensitivity computations

Continuous sensitivity equation methods have been applied to a variety of applications ranging from optimal design, to fast algorithms in computational fluid dynamics to the quantification of uncertainty. In order to make use of these methods for interface problems, one needs fast and accurate numerical methods for computing sensitivities for problems defined by partial differential equations with solutions that have spatial discontinuities such as shocks and interfaces. In this paper we develop a discontinuous Petrov Galerkin finite-element scheme for solving the sensitivity equation resulting from a 1D interface problem. The 1D example is sufficient to motivate the theoretical and computational issues that arise when one derives the corresponding boundary value problem for the sensitivities. In particular, the sensitivity boundary value problem must be formulated in a very weak sense, and the resulting variational problem provides a natural framework for developing and analyzing numerical schemes. Numerical examples are presented to illustrate the benefits of this approach.     

Generalized mathematical model of the dynamics of consolidation processes with relaxation

Filtration consolidation of salt-saturated porous media is modeled taking into account time-space nonlocality effects. Asymptotic approximations of problem solutions for excess head are found, an algorithm of numerical modeling of the process dynamics within the framework of the model is proposed, and the results of numerical experiments are presented. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 25–34, September–October 2008.     

A finite element based level set method for two-phase incompressible flows

We present a method that has been developed for the efficient numerical simulation of two-phase incompressible flows. For capturing the interface between the phases the level set technique is applied. The continuous model consists of the incompressible Navier–Stokes equations coupled with an advection equation for the level set function. The effect of surface tension is modeled by a localized force term at the interface (so-called continuum surface force approach). For spatial discretization of velocity, pressure and the level set function conforming finite elements on a hierarchy of nested tetrahedral grids are used. In the finite element setting we can apply a special technique to the localized force term, which is based on a partial integration rule for the Laplace–Beltrami operator. Due to this approach the second order derivatives coming from the curvature can be eliminated. For the time discretization we apply a variant of the fractional step θ-scheme. The discrete saddle point problems that occur in each time step are solved using an inexact Uzawa method combined with multigrid techniques. For reparametrization of the level set function a new variant of the fast marching method is introduced. A special feature of the solver is that it combines the level set method with finite element discretization, Laplace–Beltrami partial integration, multilevel local refinement and multigrid solution techniques. All these components of the solver are described. Results of numerical experiments are presented.     

Analysis and formation of acoustic fields in inhomogeneous waveguides

The problem of numerical modeling and formation of acoustic fields with definite properties in an axisymmetric inhomogeneous underwater waveguide is considered. A numerical method to solve a boundary-value and extremal problems for a parabolic Schr?dinger-type wave equation with a complex nonself-adjoint operator is proposed and investigated. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 62–71, March–April 2009.