Multiscale Modeling for Food Drying: State of the Art

Plant‐based food materials are mostly porous in nature and heterogeneous in structure with huge diversity in cellular orientation. Different cellular environments of plant‐based food materials, such as intercellular, intracellular, and cell wall environments, hold different proportions of water with different characteristics. Due to this structural heterogeneity, it is very difficult to understand the drying process and associated morphological changes during drying. Transport processes and morphological changes that take place during drying are mainly governed by the characteristics of and the changes in the cells. Therefore, to predict the actual heat and mass transfer process that occurs in the drying process and associated morphological changes, development of multiscale modeling is crucial. Multiscale modeling is a powerful approach with the ability to incorporate this cellular structural heterogeneity with microscale heat and mass transfer during drying. However, due to the huge complexity involved in developing such a model for plant‐based food materials, the studies regarding this issue are very limited. Therefore, we aim in this article to develop a critical conceptual understanding of multiscale modeling frameworks for heterogeneous food materials through an extensive literature review. We present a critical review on the multiscale model formulation and solution techniques with their spatial and temporal coupling options. Food structure, scale definition, and the current status of multiscale modeling are also presented, along with other key factors that are critical to understanding and developing an accurate multiscale framework. We conclude by presenting the main challenges for developing an accurate multiscale modeling framework for food drying.     

A novel minimum weight formulation of topology optimization implemented with reanalysis approach

In this paper, we develop an efficient diagonal quadratic optimization formulation for minimum weight design problem subject to multiple constraints. A high-efficiency computational approach of topology optimization is implemented within the framework of approximate reanalysis. The key point of the formulation is the introduction of the reciprocal-type variables. The topology optimization seeking for minimum weight can be transformed as a sequence of quadratic program with separable and strictly positive definite Hessian matrix, thus can be solved by a sequential quadratic programming approach. A modified sensitivity filtering scheme is suggested to remove undesirable checkerboard patterns and mesh dependence. Several typical examples are provided to validate the presented approach. It is observed that the optimized structure can achieve lighter weight than those from the established method by the demonstrative numerical test. Considerable computational savings can be achieved without loss of accuracy of the final design for 3D structure. Moreover, the effects of multiple constraints and upper bound of the allowable compliance upon the optimized designs are investigated by numerical examples.     

Mesh independence of galerkin approach by preconditioning

Preconditioning the discrete Galerkin system for a bounded elliptic linear problem, by the Choleski factors of the Gramian, we obtain a mesh independent condition number. Moreover, the rate of convergence of the multigrid method is also mesh independent. Our example give a justification to the oscillatory movement of the discrete solution of a well known problem.     

The Navier–Stokes-αβ equations as a platform for a spectral multigrid method to solve the Navier–Stokes equations

This paper describes a spectral multigrid method for spatially periodic homogeneous and isotropic turbulent flows. The method uses the Navier–Stokes-αβ equations to accelerate convergence toward solutions of the Navier–Stokes equations. The Navier–Stokes-αβ equations are solved on coarse grids at various levels and the Navier–Stokes equations are solved on the “nest grid”. The method uses Crank–Nicolson time-stepping for the viscous terms, explicit time-stepping for the remaining terms, and Richardson iteration to solve linear systems encountered at each time step and on each grid level. To explore the computational efficiency of the method, comparisons are made with results obtained from an analogous spectral multigrid method for the Navier–Stokes equations. These comparisons are based on computing work units and residuals for multigrid cycles. Most importantly, we examine how choosing different values of the length scales α and β entering the Navier–Stokes-αβ equations influence the efficiency and accuracy of these multigrid schemes.     

Multigrid method with a new interpolation operator

We introduce new techniques to design interpolation in multigrid methods for elliptic problems with discontinuous coefficients. The new techniques employ the Nelder–Mead simplex algorithm and skills in space geometry. The Nelder–Mead algorithm was used to minimize a scalar-valued function, which is a sum of distances from a point to four planes. We derived interpolation scheme in space geometry. We observed that new interpolation is better than traditional bilinear interpolation and cubic interpolation, as prolongation operator in multigrid methods.     

Integrating load balancing and locality in the parallelization of irregular problems

An irregular problem models the evolution of a system where several elements are irregularly distributed in a domain. The evolution modifies this distribution in a way that cannot be foreseen and the behavior of each element depends upon the elements close to it according to a problem dependent relation. Starting from a hierarchical representation of the domain, we define a parallelization methodology that includes a load balancing strategy that preserves this locality property and a strategy to collect information distributed onto the processing nodes.     

A local multigrid X‐FEM strategy for 3‐D crack propagation

A multiscale method for 3‐D crack propagation simulation in large structures is proposed. The method is based on the extended finite element method (X‐FEM). The asymptotic behavior of the crack front is accurately modeled using enriched elements and no remeshing is required during crack propagation. However, the different scales involved in fracture mechanics problems can differ by several orders of magnitude and industrial meshes are usually not designed to account for small cracks. Enrichments are therefore useless if the crack is too small compared with the element size. To overcome this drawback, a project combining different numerical techniques was started. The first step was the implementation of a global multigrid algorithm within the X‐FEM framework and was presented in a previous paper (Eur. J. Comput. Mech. 2007; 16 :161–182). This work emphasized the high efficiency in cpu time but highlighted that mesh refinement is required on localized areas only (cracks, inclusions, steep gradient zones). This paper aims at linking the different scales by using a local multigrid approach. The coupling of this technique with the X‐FEM is described and computational aspects dealing with intergrid operators, optimal multiscale enrichment strategy and level sets are pointed out. Examples illustrating the accuracy and efficiency of the method are given. Copyright © 2008 John Wiley & Sons, Ltd.     

A FINITE-DIFFERENCE RELAXATION SCHEME BASED ON BOTH MULTIGRID TECHNIQUES AND HOMOTOPY METHOD FOR 2D STEADY-STATE NAVIER-STOKES EQUATIONS

In this paper, multigrid techniques together with homotopy method are applied to propose a kind of finite-difference relaxation scheme for 2D steady-state Navier-Stokes equations. The proposed numerical scheme can give convergent results for viscous flows with high Reynolds number. As an example, the results of shear-driven cavity flow with high Reynolds number up to 25000 on fine grid 257×257 are given.     

那比RCC重力坝6号溢流坝段坝基破坏模式及稳定性研究

文章采用三维非线性有限元法,对那比重力坝6号溢流坝段进行破坏模式及稳定性的研究,确定其在超载法下的破坏模式及稳定安全系数,并利用多重网格法对可能的滑移路径进行危险滑块搜索,从而确定出最不利的滑出路径。     

APPLICATION OF MULTIGRID METHOD IN 2-D MATHE-MATICAL MODEL IN OPEN CHANNELS

1.　 INTRODUCTIONIn 2 -D mathematical model of unsteady flow in open channel,one of the mostdifficulties is its slow convergence rate and too much time needed in computation.Themultigrid method is a new rapid iteration method developed in the resent 2 0 years. Now ithas been verified as an optimizing iteration method and its convergence rate has no relationswith mesh spacing.At present the multigrid method has been widely used in many fields(Liu Chaoqun,1 995;Peng Wenqi,1 994 ) ,especia…