Operator Splitting Algorithm Research Articles

MR images reconstruction based on TVWL2–L1 model

Compressive sensing (CS) theory, which has been widely used in magnetic resonance (MR) image processing, indicates that a sparse signal can be reconstructed by the optimization programming process from non-adaptive linear projections. Since MR Images commonly possess a blocky structure and have sparse representations under certain wavelet bases, total variation (TV) and wavelet domain ℓ1 norm regularization are enforced together (TV-wavelet L1 method) to improve the recovery accuracy. However, the components of wavelet coefficients are different: low-frequency components of an image, that carry the main energy of the MR image, perform a decisive impact for reconstruction quality. In this paper, we propose a TV and wavelet L2–L1 model (TVWL2–L1) to measure the low frequency wavelet coefficients with ℓ2 norm and high frequency wavelet coefficients with ℓ1 norm. We present two methods to approach this problem by operator splitting algorithm and proximal gradient algorithm. Experimental results demonstrate that our method can obviously improve the quality of MR image recovery comparing with the original TV-wavelet method.

Structured populations, cell growth and measure valued balance laws

A well-posedness theory of measure valued solutions to balance laws is presented. Nonlinear semigroups are constructed by means of the operator splitting algorithm. This approach allows to separate the differential terms from the integral ones, leading to a significant simplification of the proofs. Continuous dependence with respect to parameters is also shown. The whole framework allows a unified approach to a variety of structured population models, providing to each of them the basic well posedness and stability results.

Design of Compliant Mechanisms of Distributed Compliance Using a Level-Set Based Topology Optimization Method

This paper presents a level set-based structural shape and topology optimization for the design of compliant mechanisms. The design boundary of the compliant mechanism is implicitly represented as the zero level-set of a higher-dimensional level set surface. A quadratic energy functional is introduced to augment the objective function in order to control the structural geometric size of the resulting mechanism. The optimization is thus changed to a numerical process that describes the design as a sequence of motions by updating the implicit boundaries until the optimized structure is achieved under specified constraints. A semi-implicit scheme with an additive operator splitting (AOS) algorithm is used to solve the Hamilton-Jacobi partial differential equation (PDE) in the level set method. In doing so, it is expected that numerical difficulties in most conventional level set methods can be eliminated. The final mechanism is characterized with strip-like members able to generate distributed compliance, and so that to resolve the hinge problem long sought-after in the design of compliant mechanisms. Typical numerical case is used to evidence the effectiveness of this method in the design of monolithic compliant mechanisms.

Copula density estimation by total variation penalized likelihood with linear equality constraints

A copula density is the joint probability density function (PDF) of a random vector with uniform marginals. An approach to bivariate copula density estimation is introduced that is based on maximum penalized likelihood estimation (MPLE) with a total variation (TV) penalty term. The marginal unity and symmetry constraints for copula density are enforced by linear equality constraints. The TV-MPLE subject to linear equality constraints is solved by an augmented Lagrangian and operator-splitting algorithm. It offers an order of magnitude improvement in computational efficiency over another TV-MPLE method without constraints solved by the log-barrier method for the second order cone program. A data-driven selection of the regularization parameter is through K-fold cross-validation (CV). Simulation and real data application show the effectiveness of the proposed approach. The MATLAB code implementing the methodology is available online.

Characteristic-based operator-splitting finite element method for Navier-Stokes equations

A new finite element method, which is the characteristic-based operator-splitting (CBOS) algorithm, is developed to solve Navier-Stokes (N-S) equations. In each time step, the equations are split into the diffusive part and the convective part by adopting the operator-splitting algorithm. For the diffusive part, the temporal discretization is performed by the backward difference method which yields an implicit scheme and the spatial discretization is performed by the standard Galerkin method. The convective part can be discretized using the characteristic Galerkin method and solved explicitly. The driven square flow and backward-facing step flow are conducted to validate the model. It is shown that the numerical results agree well with the standard solutions or existing experimental data, and the present model has high accuracy and good stability. It provides a prospective research method for solving N-S equations.

Coupled thermoviscoplasticity of glassy polymers in the logarithmic strain space based on the free volume theory

The paper outlines a constitutive model for finite thermo-visco-plastic behavior of amorphous glassy polymers and considers details of its numerical implementation. In contrast to existing kinematical approaches to finite plasticity of glassy polymers, the formulation applies a plastic metric theory based on an additive split of Lagrangian Hencky-type strains into elastic and plastic parts. The analogy between the proposed formulation in the logarithmic strain space and the geometrically linear theory of plasticity, makes this constitutive framework very transparent and attractive with regard to its numerical formulation. The characteristic strain hardening of the model is derived from a polymer network model. We consider the particularly simple eight chain model, but also comment on the recently developed microsphere model. The viscoplastic flow rule in the logarithmic strain space uses structures of the free volume flow theory, which provides a highly predictive modeling capacity at the onset of viscoplastic flow. The integration of this micromechanically motivated approach into a three-dimensional computational model is a key concern of this work. We outline details of the numerical implementation of this model, including elements such as geometric pre- and post-transformations to/from the logarithmic strain space, a thermomechanical operator split algorithm consisting of an isothermal mechanical predictor followed by a heat conduction corrector and finally, the consistent linearization of the local update algorithm for the dissipative variables as well as its relationship to the global tangent operator. The performance of the proposed formulation is demonstrated by means of a spectrum of numerical examples, which we compare with our experimental findings.

Rising of 3D catalyst particles in a natural convection dominated flow by a parallel DNS method

Abstract We investigate the problem of particulate flows with heat transfer by parallel direct numerical simulation (DNS). Among other heat transfer problems, we examine in detail the case of a 3D spherical catalyst rising in an enclosure due to natural convection though it is heavier than the suspending fluid. Natural convection is created by heat transferred from the warmer particle to the fluid. Heat is assumed to be produced at a constant rate in the particle bulk. As expected, there exists a critical production rate that leads to the rising of the catalyst. Compared to the 2D circular cylinder counterpart, momentum and heat transfers are slower in 3D and the spherical catalyst rises for a lower production rate. At the numerical level, we employ a Distributed Lagrange Multiplier/Fictitious Domain formulation together with an operator-splitting algorithm to solve the coupled problem. Two families of Lagrange multiplier are introduced to relax the velocity and temperature constraints respectively. As suggested in Wachs (2009), particle collisions are handled by an efficient Discrete Element Method granular solver. As it is, the model is restricted to the case of homogeneous temperature over the particles. From a computational viewpoint, this work might be regarded as an extension of the method proposed in our previous contributions (Dan & Wachs, 2010; Yu, Shao, et al., 2006) to distributed computing with our new parallel code PeliGRIFF. 1 1 ▪ Parallel Efficient LIbrary for GRains In Fluid Flow, see Wachs (2011) for details. This opens up new possibilities to study a broad range of applications in 3D and to get more insight in the comprehension of particulate flows with heat transfer. In particular, we examine how a bed of spherical catalysts can be self-fluidized as a result of the heat produced in the particles bulk, mimicking an exothermic catalyst reaction in a chemical engineering reactor.

Algorithm Research on Moving Vehicles Detection

Moving vehicles detection is an important part of Intelligence Transport System (ITS). Traditional method of moving vehicles detection from video is image subtraction which is apt to be affected by brightness changing, in this paper a kind of moving vehicles detection algorithm based on optical flow is purposed: estimate optical flow through two consecutive frames’ image pyramids and compute every optical flow image's threshold with which images are be segment into binaryzation images, after that, through morphological transformation operator and rectangular splitting algorithm on images, moving vehicles’ images will be extracted from background. Experiment shows that the algorithm framework is practicable.

Subgrid-Scale Modeling of Turbulent Flow Around Circular Cylinder by Mesh-Free Vortex Method

:This paper proposes a numerical model for the two-dimensional, incompressible, unsteady, turbulent flow applied to an impulsively started flow around a circular cylinder in the Reynolds number range from 1×104 to 6×105. A Lagrangian mesh-free vortex method blended with the Large Eddy Simulation theory is employed to simulate the large-scale motion, whereas the turbulent subgrid-scale motion is modeled with an eddy viscosity coefficient, expressed in terms of the Second-Order Velocity Structure Function. The filtered vorticity field is calculated by a superposition of Lamb vortices that are generated near the body surface such that circulation is conserved and the no-slip boundary condition is explicitly imposed at a finite number of points on the cylinder. The no-penetration boundary condition is satisfied exactly on the entire cylinder surface through the application of the circle theorem. The vorticity transport equation is solved using the convective-diffusive operator-splitting algorithm, where vorticity diffusion is simulated with the random walk method and the convective motion of the vortices is integrated in time using the second-order Adams-Bashforth scheme. Numerous simulations for high Reynolds numbers are carried out to determine the numerical parameters of the model. Results for the drag coefficient and the Strouhal number as a function of the Reynolds number present satisfactory agreement with other results from the literature.

Numerical simulation of profile control by clay particles after polymer flooding

A three-dimensional, two-phase, five-component mathematical model has been developed to describe flow characteristics of clay particles and flocs in the profile control process, in which the clay particle suspension is injected into the formation to react with residual polymer. This model considers the reaction of clay particles with residual polymer, apparent viscosity of the mixture, retention of clay particles and flocs, as well as the decline in porosity and permeability caused by the retention of clay particles and flocs. A finite difference method is used to discretize the equation for each component in the model. The Runge-Kutta method is used to solve the polymer flow equation, and operator splitting algorithms are used to split the flow equation for clay particles into a hyperbolic equation for convection and a parabolic equation for diffusion, which effectively ensures excellent precision, high speed and good stability. The numerical simulation had been applied successfully in the 4-P1920 unit of the Lamadian Oilfield to forecast the blocking capacity of clay particle suspension and to optimize the injection parameters.

A Coupled Chemo-Elastoplastic-Damage Constitutive Model for Plain Concrete Subjected to High Temperature

A coupled elastoplastic-damage constitutive model taking into account chemo-induced elastoplastic-damage effects for the modeling of coupled chemo-thermo-hygro-mechanical behavior of concretes at high temperature is proposed in this article. A three-step operator split algorithm for the integration of the rate coupled constitutive equations is developed. Consistent tangent modulus matrices for coupled chemo-thermo-hygro-mechanical analysis are derived to preserve the quadratic rate of convergence of the global Newton iterative procedure. Numerical results demonstrate the validity of the presented algorithm and illustrate the capability of the proposed constitutive model in reproducing coupled chemo-thermo-hygro-mechanical behavior in concretes subjected to fire and thermal radiation.

The modeling of realistic chemical vapor infiltration/deposition reactors

AbstractWe describe the low Mach number equations as well as a second‐order numerical integration procedure that are used to solve a realistic chemical vapor infiltration/chemical vapor deposition (CVI/CVD) problem. The simulation accounts for a homogeneous gas chemical reaction mechanism, a heterogeneous surface reaction mechanism, and an evolving pore structure model. The numerical solution of the model ultimately leads to the solution of a large system of stiff differential algebraic equations that are to be integrated over a long time. An operator splitting algorithm is employed to deal with the stiffness associated with chemical reactions, whereas a projection method is employed to overcome the difficulty arising from having to solve a large coupled system for velocity and pressure fields. Results show that the proposed integration procedure is very efficient for modeling long time CVI/CVD densification processes. Copyright © 2009 John Wiley & Sons, Ltd.

A One-Dimensional Theory of Solute Diffusion and Degradation in Elastic Solids

A theoretical framework for the description of the interaction between diffusion, mechanics, and degradation in elastic solids is developed. To avoid complications that obscure the essential features of these interactions, we work within a one-dimensional setting. A particular specialization of the general theory is selected and a numerical implementation based on the finite-element method, a backward Euler time-stepping scheming, and an operator-splitting algorithm is described. An application involving the time-independent end-loading of a notched cylindrical bar is used to illustrate the ability of the theory to describe some essential features of solute-assisted degradation.

A predictor–corrector algorithm for the coupling of stiff ODEs to a particle population balance

In this paper a novel predictor–corrector algorithm is presented for the solution of coupled gas-phase – particulate systems. The emphasis of this work is the study of soot formation, but the concepts can be applied to other systems. This algorithm couples a stiff ODE solver to a Monte Carlo population balance solver. Such coupling has been achieved previously for similar systems using a Strang operator splitting algorithm, however, that algorithm demonstrated several numerical issues which resulted in a high computational cost to acquire adequate precision. In particular a source-sink instability was identified whereby a large-magnitude source term present in the ODE system was competing with a similarly sized sink term in the population balance. This instability required that the splitting step size was very small in order to keep numerical error sufficiently low. A predictor–corrector algorithm has been formulated to negate this instability. An additional efficiency is gained with this algorithm as a principal computational cost of the Strang splitting algorithm is removed: the requirement to re-initialise the ODE solver every splitting step. The numerical convergence of the new algorithm is demonstrated, and its efficiency is compared to that of the Strang splitting algorithm. Substantial computation time savings are demonstrated, which allow a fixed error in three studied system functionals to be achieved with an order-of-magnitude reduction in computation time.

Design of piezoelectric actuators using a multiphase level set method of piecewise constants

This paper presents a multiphase level set method of piecewise constants for shape and topology optimization of multi-material piezoelectric actuators with in-plane motion. First, an indicator function which takes level sets of piecewise constants is used to implicitly represent structural boundaries of the multiple phases in the design domain. Compared with standard level set methods using n scalar functions to represent 2 n phases, each constant value in the present method denotes one material phase and 2 n phases can be represented by 2 n pre-defined constants. Thus, only one indicator function including different constant values is required to identify all structural boundaries between different material phases by making use of its discontinuities. In the context of designing smart actuators with in-plane motions, the optimization problem is defined mathematically as the minimization of a smooth energy functional under some specified constraints. Thus, the design optimization of the smart actuator is transferred into a numerical process by which the constant values of the indicator function are updated via a semi-implicit scheme with additive operator splitting (AOS) algorithm. In such a way, the different material phases are distributed simultaneously in the design domain until both the passive compliant host structure and embedded piezoelectric actuators are optimized. The compliant structure serves as a mechanical amplifier to enlarge the small strain stroke generated by piezoelectric actuators. The major advantage of the present method is to remove numerical difficulties associated with the solution of the Hamilton–Jacobi equations in most conventional level set methods, such as the CFL condition, the regularization procedure to retain a signed distance level set function and the non-differentiability related to the Heaviside and the Delta functions. Two widely studied examples are chosen to demonstrate the effectiveness of the present method.

A Numerical Method for a Non-Smooth Advection-Diffusion Problem Arising in Sand Mechanics

An operator-splitting algorithm is presented for the solution of a partial differential equation arising in the modeling of deposition processes in sand mechanics.Sand piles evolution is modeled by an advection-diffusion equation, with a non-smooth diffusion operator that contains a point-wise constraint on the gradient of the solution.Piecewise linear finite elements are used for the discretization in space.The advection operator is treated with a stabilized SUPG finite element method.An augmented Lagrangian method is proposed for the discretization of the fast/slow non-smooth diffusion operator.A penalization approach, together with a Newton method, is used for the treatment of inequality constraints.Numerical results are presented for the simulation of sand piles on flat and non-flat surfaces, and for extensions to water flows.

A new level set method for systematic design of hinge-free compliant mechanisms

This paper presents a new level set-based method to realize shape and topology optimization of hinge-free compliant mechanisms. A quadratic energy functional used in image processing applications is introduced in the level set method to control the geometric width of structural components in the created mechanism. A semi-implicit scheme with an additive operator splitting (AOS) algorithm is employed to solve the Hamilton–Jacobi partial differential equation (PDE) in the level set method. The design of compliant mechanisms is mathematically represented as a general non-linear programming with a new objective function augmented by the high-order energy term. The structural optimization is thus changed to a numerical process that describes the design as a sequence of motions by updating the implicit boundaries until the optimized structure is achieved under specified constraints. In doing so, it is expected that numerical difficulties such as the Courant–Friedrichs–Lewy (CFL) condition and periodically applied re-initialization procedures in most conventional level set methods can be eliminated. In addition, new holes can be created inside the design domain. The final mechanism configurations consist of strip-like members suitable for generating distributed compliance, and solving the de-facto hinge problem in the design of compliant mechanisms. Two widely studied numerical examples are studied to demonstrate the effectiveness of the proposed method in the context of designing distributed compliant mechanisms.

Parallel three‐dimensional simulation of the injection molding process

AbstractThis paper describes the development of a parallel three‐dimensional unstructured non‐isothermal flow solver for the simulation of the injection molding process. The numerical model accounts for multiphase flow in which the melt and air regions are considered to be a continuous incompressible fluid with distinct physical properties. This aspect avoids the complex reconstruction of the interface. A collocated finite volume method is employed, which can switch between first‐ and second‐order accuracy in both space and time. The pressure implicit with splitting of operators algorithm is used to compute the transient flow variables and couple velocity and pressure. The temperature equation is solved using a transport equation with convection and diffusion terms. An upwind differencing scheme is used for the discretization of the convection term to enforce a bounded solution. In order to capture the sharp interface, a bounded compressive high‐resolution scheme is employed. Parallelization of the code is achieved using the PETSc framework and a single program multiple data message passing model. Predicted numerical solutions for several example problems are considered. The first case validates the solution algorithm for moderate Reynolds number flows using a structured mesh. The second case employs an unstructured hybrid mesh showing the capability of the solver to describe highly viscous flows closer to realistic injection molding conditions. The final case presents the non‐isothermal filling of a thick cavity using three mesh sizes and up to 80 processors to assess parallel performance. The proposed algorithm is shown to have good accuracy and scalability. Copyright © 2008 John Wiley & Sons, Ltd.

On the rotating Navier-Stokes equations with mixed boundary conditions

The stationary and nonstationary rotating Navier-Stokes equations with mixed boundary conditions are investigated in this paper. The existence and uniqueness of the solutions are obtained by the Galerkin approximation method. Next, θ-scheme of operator splitting algorithm is applied to rotating Navier-Stokes equations and two subproblems are derived. Finally, the computational algorithms for these subproblems are provided.

Using adaptive proper orthogonal decomposition to solve the reaction–diffusion equation

We introduce an adaptive POD method to reduce the computational cost of reacting flow simulations. The scheme is coupled with an operator-splitting algorithm to solve the reaction–diffusion equation. For the reaction sub-steps, locally valid basis vectors, obtained via POD and the method of snapshots, are used to project the minor species mass fractions onto a reduced dimensional space thereby decreasing the number of equations that govern combustion chemistry. The method is applied to a one-dimensional, laminar premixed CH 4–air flame using GRImech 3.0; with errors less than 0.25 % , a speed-up factor of 3.5 is observed. The speed-up results from fewer source term evaluations required to compute the Jacobian matrices.