A Fourth-Order Unstructured NURBS-Enhanced Finite Volume WENO Scheme for Steady Euler Equations in Curved Geometries

In Li and Ren(Int.J.Numer.Methods Fluids 70:742–763,2012),a high-order k-exact WENO finite volume scheme based on secondary reconstructions was proposed to solve the two-dimensional time-dependent Euler equations in a polygonal domain,in which the high-order numerical accuracy and the oscillations-free property can be achieved.In this paper,the method is extended to solve steady state problems imposed in a curved physical domain.The numerical framework consists of a Newton type finite volume method to linearize the nonlinear governing equations,and a geometrical multigrid method to solve the derived linear system.To achieve high-order non-oscillatory numerical solutions,the classical k-exact reconstruction with k=3 and the efficient secondary reconstructions are used to perform the WENO reconstruction for the conservative variables.The non-uniform rational B-splines(NURBS)curve is used to provide an exact or a high-order representation of the curved wall boundary.Furthermore,an enlarged reconstruction patch is constructed for every element of mesh to significantly improve the convergence to steady state.A variety of numerical examples are presented to show the effectiveness and robustness of the proposed method.

Genome-wide association study and genomic prediction of Fusarium ear rot resistance in tropical maize germplasm

Fusarium ear rot(FER)is a destructive maize fungal disease worldwide.In this study,three tropical maize populations consisting of 874 inbred lines were used to perform genomewide association study(GWAS)and genomic prediction(GP)analyses of FER resistance.Broad phenotypic variation and high heritability for FER were observed,although it was highly influenced by large genotype-by-environment interactions.In the 874 inbred lines,GWAS with general linear model(GLM)identified 3034 single-nucleotide polymorphisms(SNPs)significantly associated with FER resistance at the P-value threshold of 1×10^(-5),the average phenotypic variation explained(PVE)by these associations was 3%with a range from 2.33%to 6.92%,and 49 of these associations had PVE values greater than 5%.The GWAS analysis with mixed linear model(MLM)identified 19 significantly associated SNPs at the P-value threshold of 1×10^(-4),the average PVE of these associations was 1.60%with a range from 1.39%to 2.04%.Within each of the three populations,the number of significantly associated SNPs identified by GLM and MLM ranged from 25 to 41,and from 5 to 22,respectively.Overlapping SNP associations across populations were rare.A few stable genomic regions conferring FER resistance were identified,which located in bins 3.04/05,7.02/04,9.00/01,9.04,9.06/07,and 10.03/04.The genomic regions in bins 9.00/01 and 9.04 are new.GP produced moderate accuracies with genome-wide markers,and relatively high accuracies with SNP associations detected from GWAS.Moderate prediction accuracies were observed when the training and validation sets were closely related.These results implied that FER resistance in maize is controlled by minor QTL with small effects,and highly influenced by the genetic background of the populations studied.Genomic selection(GS)by incorporating SNP associations detected from GWAS is a promising tool for improving FER resistance in maize.

Study on Haploid Induction Rates in Different Maize Inducers

[Objective] The aim was to analyze the differences in haploid induction rates of different inducers. [Method] Six maize inducers with purple spot and purple color were selected as the male parents to pollinate six inbred lines. [Result] The mean haploid induction rates were significantly different among the inducers: KMS-3 >WY-1 >PR-2 >YP-13 >KMS-2 >KMS-1. The haploid induction rates of the different hybrid materials were significantly different: K410 >105A >103A >104A >107A >D271 >106A>L73>N21>KZ58. [Conclusion] The haploid inducer line PR-2, which had high haploid induction rate and low variation coefficient, was an elite haploid inducer.

An SAV Method for Imaginary Time Gradient Flow Model in Density Functional Theory

In this paper,based on the imaginary time gradient flow model in the density functional theory,a scalar auxiliary variable(SAV)method is developed for the ground state calculation of a given electronic structure system.To handle the orthonormality constraint on those wave functions,two kinds of penalty terms are introduced in designing the modified energy functional in SAV,i.e.,one for the norm preserving of each wave function,another for the orthogonality between each pair of different wave functions.A numerical method consisting of a designed scheme and a linear finite element method is used for the discretization.Theoretically,the desired unconditional decay of modified energy can be obtained from our method,while computationally,both the original energy and modified energy decay behaviors can be observed successfully from a number of numerical experiments.More importantly,numerical results show that the orthonormality among those wave functions can be automatically preserved,without explicitly preserving orthogonalization operations.This implies the potential of our method in large-scale simulations in density functional theory.

An Implicit Solver for the Time-Dependent Kohn-Sham Equation

The implicit numerical methods have the advantages on preserving the physical properties of the quantum system when solving the time-dependent Kohn-Sham equation.However,the efficiency issue prevents the practical applications of those implicit methods.In this paper,an implicit solver based on a class of Runge-Kutta methods and the finite element method is proposed for the time-dependent Kohn-Sham equation.The efficiency issue is partially resolved by three approaches,i.e.,an h-adaptive mesh method is proposed to effectively restrain the size of the discretized problem,a complex-valued algebraic multigrid solver is developed for efficiently solving the derived linear system from the implicit discretization,as well as the OpenMP based parallelization of the algorithm.The numerical convergence,the ability on preserving the physical properties,and the efficiency of the proposed numerical method are demonstrated by a number of numerical experiments.

ANURBS-Enhanced Finite Volume Method for Steady Euler Equations with Goal-Oriented h-Adaptivity

In[A NURBS-enhanced finite volume solver for steady Euler equations,X.C.Meng,G.H.Hu,J.Comput.Phys.,Vol.359,pp.77–92],aNURBS-enhanced finite volume method was developed to solve the steady Euler equations,in which the desired high order numerical accuracy was obtained for the equations imposed in the domain with a curved boundary.In this paper,the method is significantly improved in the following ways:(i)a simple and efficient point inversion technique is designed to compute the parameter values of points lying on a NURBS curve,(ii)with this new point inversion technique,the h-adaptive NURBS-enhanced finite volume method is introduced for the steady Euler equations in a complex domain,and(iii)a goal-oriented a posteriori error indicator is designed to further improve the efficiency of the algorithm towards accurately calculating a given quantity of interest.Numerical results obtained from a variety of numerical experiments with different flow configurations successfully show the effectiveness and robustness of the proposed method.

An h-Adaptive Finite Element Solution of the Relaxation Non-Equilibrium Model for Gravity-Driven Fingers

The study on the fingering phenomenon has been playing an important role in understanding the mechanism of the fluid flow through the porous media.In this paper,a numerical method consisting of the Crank-Nicolson scheme for the temporal discretization and the finite element method for the spatial discretization is proposed for the relaxation non-equilibrium Richards equation in simulating the fingering phenomenon.Towards the efficiency and accuracy of the numerical simulations,a predictor-corrector process is used for resolving the nonlinearity of the equation,and an h-adaptive mesh method is introduced for accurately resolving the solution around the wetting front region,in which a heuristic a posteriori error indicator is designed for the purpose.In numerical simulations,a traveling wave solution of the governing equation is derived for checking the numerical convergence of the proposed method.The effectiveness of the h-adaptive method is also successfully demonstrated by numerical experiments.Finally the mechanism on generating fingers is discussed by numerically studying several examples.

Moving Finite Element Simulations for Reaction-Diffusion Systems

This work is concerned with the numerical simulations for two reactiondiffusion systems,i.e.,the Brusselator model and the Gray-Scott model.The numerical algorithm is based upon a moving finite element method which helps to resolve large solution gradients.High quality meshes are obtained for both the spot replication and the moving wave along boundaries by using proper monitor functions.Unlike[33],this work finds out the importance of the boundary grid redistribution which is particularly important for a class of problems for the Brusselator model.Several ways for verifying the quality of the numerical solutions are also proposed,which may be of important use for comparisons.

An Optimization Method in Inverse Elastic Scattering for One-Dimensional Grating Profiles

Consider the inverse diffraction problem to determine a two-dimensional periodic structure from scattered elastic waves measured above the structure.We formulate the inverse problem as a least squares optimization problem,following the two-step algorithm by G.Bruckner and J.Elschner[Inverse Probl.,19(2003),315–329]for electromagnetic diffraction gratings.Such a method is based on the Kirsch-Kress optimization scheme and consists of two parts:a linear severely ill-posed problem and a nonlinear well-posed one.We apply this method to both smooth(C2)and piecewise linear gratings for the Dirichlet boundary value problem of the Navier equation.Numerical reconstructions from exact and noisy data illustrate the feasibility of the method.

A Robust WENO Type Finite Volume Solver for Steady Euler Equations on Unstructured Grids

A recent work of Li et al.[Numer.Math.Theor.Meth.Appl.,1(2008),pp.92-112]proposed a finite volume solver to solve 2D steady Euler equations.Although the Venkatakrishnan limiter is used to prevent the non-physical oscillations nearby the shock region,the overshoot or undershoot phenomenon can still be observed.Moreover,the numerical accuracy is degraded by using Venkatakrishnan limiter.To fix the problems,in this paper the WENO type reconstruction is employed to gain both the accurate approximations in smooth region and non-oscillatory sharp profiles near the shock discontinuity.The numerical experiments will demonstrate the efficiency and robustness of the proposed numerical strategy.

Towards Translational Invariance of Total Energywith Finite Element Methods for Kohn-Sham Equation

Numerical oscillation of the total energy can be observed when the Kohn-Sham equation is solved by real-space methods to simulate the translational move of an electronic system.Effectively remove or reduce the unphysical oscillation is crucial not only for the optimization of the geometry of the electronic structure,but also for the study of molecular dynamics.In this paper,we study such unphysical oscillation based on the numerical framework in[G.Bao,G.H.Hu,and D.Liu,An h-adaptive fi-nite element solver for the calculations of the electronic structures,Journal of Computational Physics,Volume 231,Issue 14,Pages 4967-4979,2012],and deliver some numerical methods to constrain such unphysical effect for both pseudopotential and all-electron calculations,including a stabilized cubature strategy for Hamiltonian operator,and an a posteriori error estimator of the finite element methods for Kohn-Sham equation.The numerical results demonstrate the effectiveness of our method on restraining unphysical oscillation of the total energies.