We consider a strongly heterogeneous medium saturated by an incompressible viscous fluid as it appears in geomechanical modeling.This poroelasticity problem suffers from rapidly oscillating material parameters,which c...We consider a strongly heterogeneous medium saturated by an incompressible viscous fluid as it appears in geomechanical modeling.This poroelasticity problem suffers from rapidly oscillating material parameters,which calls for a thorough numerical treatment.In this paper,we propose a method based on the local orthogonal decomposition technique and motivated by a similar approach used for linear thermoelasticity.Therein,local corrector problems are constructed in line with the static equations,whereas we propose to consider the full system.This allows to benefit from the given saddle point structure and results in two decoupled corrector problems for the displacement and the pressure.We prove the optimal first-order convergence of this method and verify the result by numerical experiments.展开更多
In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique ...In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.展开更多
In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al...In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al.in J Sci Comput 66:321–345,2016;Dong and Wang in J Comput Appl Math 380:1–11,2020)for a one-dimensional stationary Schrödinger equation.Previous work showed that penalty parameters were required to be positive in error analysis,but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes.In this work,by performing extensive numerical experiments,we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods,and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.展开更多
This paper develops fast multiscale collocation methods for a class of Fredholm integral equations of the second kind with singular kernels. A truncation strategy for the coefficient matrix of the corresponding discre...This paper develops fast multiscale collocation methods for a class of Fredholm integral equations of the second kind with singular kernels. A truncation strategy for the coefficient matrix of the corresponding discrete system is proposed, which forms a basis for fast algorithms. The convergence, stability and computational complexity of these algorithms are analyzed.展开更多
This paper examined how microstructure influences the homogenized thermal conductivity of cellular structures and revealed a surface-induced size-dependent effect.This effect is linked to the porous microstructural fe...This paper examined how microstructure influences the homogenized thermal conductivity of cellular structures and revealed a surface-induced size-dependent effect.This effect is linked to the porous microstructural features of cellular structures,which stems from the degree of porosity and the distri-bution of the pores.Unlike the phonon-driven surface effect at the nanoscale,the macro-scale surface mechanism in thermal cellular structures is found to be the microstructure-induced changes in the heat conduction path based on fully resolved 3D numerical simulations.The surface region is determined by the microstructure,characterized by the intrinsic length.With the coupling between extrinsic and intrinsic length scales under the surface mechanism,a surface-enriched multiscale method was devel-oped to accurately capture the complex size-dependent thermal conductivity.The principle of scale separation required by classical multiscale methods is not necessary to be satisfied by the proposed multiscale method.The significant potential of the surface-enriched multiscale method was demon-strated through simulations of the effective thermal conductivity of a thin-walled metamaterial struc-ture.The surface-enriched multiscale method offers higher accuracy compared with the classical multiscale method and superior efficiency over high-fidelity finite element methods.展开更多
Magneto-electro-elastic(MEE)materials are widely utilized across various fields due to their multi-field coupling effects.Consequently,investigating the coupling behavior of MEE composite materials is of significant i...Magneto-electro-elastic(MEE)materials are widely utilized across various fields due to their multi-field coupling effects.Consequently,investigating the coupling behavior of MEE composite materials is of significant importance.The traditional finite element method(FEM)remains one of the primary approaches for addressing such issues.However,the application of FEM typically necessitates the use of a fine finite element mesh to accurately capture the heterogeneous properties of the materials and meet the required computational precision,which inevitably leads to a reduction in computational efficiency.To enhance the computational accuracy and efficiency of the FEM for heterogeneous multi-field coupling problems,this study presents the coupling magneto-electro-elastic multiscale finite element method(CM-MsFEM)for heterogeneous MEE structures.Unlike the conventional multiscale FEM(MsFEM),the proposed algorithm simultaneously constructs displacement,electric,and magnetic potential multiscale basis functions to address the heterogeneity of the corresponding parameters.The macroscale formulation of CM-MsFEM was derived,and the macroscale/microscale responses of the problems were obtained through up/downscaling calculations.Evaluation using numerical examples analyzing the transient behavior of heterogeneous MEE structures demonstrated that the proposed method outperforms traditional FEM in terms of both accuracy and computational efficiency,making it an appropriate choice for numerically modeling the dynamics of heterogeneous MEE structures.展开更多
An extended multiscale finite element method （EMsFEM） is developed for solving the mechanical problems of heterogeneous materials in elasticity.The underlying idea of the method is to construct numerically the multi...An extended multiscale finite element method （EMsFEM） is developed for solving the mechanical problems of heterogeneous materials in elasticity.The underlying idea of the method is to construct numerically the multiscale base functions to capture the small-scale features of the coarse elements in the multiscale finite element analysis.On the basis of our existing work for periodic truss materials, the construction methods of the base functions for continuum heterogeneous materials are systematically introduced. Numerical experiments show that the choice of boundary conditions for the construction of the base functions has a big influence on the accuracy of the multiscale solutions, thus,different kinds of boundary conditions are proposed. The efficiency and accuracy of the developed method are validated and the results with different boundary conditions are verified through extensive numerical examples with both periodic and random heterogeneous micro-structures.Also, a consistency test of the method is performed numerically. The results show that the EMsFEM can effectively obtain the macro response of the heterogeneous structures as well as the response in micro-scale,especially under the periodic boundary conditions.展开更多
In this study,the transverse vibration of a traveling beam made of functionally graded material was analyzed.The material gradation was assumed to vary continuously along the thickness direction of the beam in the for...In this study,the transverse vibration of a traveling beam made of functionally graded material was analyzed.The material gradation was assumed to vary continuously along the thickness direction of the beam in the form of power law exponent.The effect of the longitudinally varying tension due to axial acceleration was highlighted,and the dependence of the tension on the finite support rigidity was also considered.A complex governing equation of the functionally graded beam was derived by the Hamilton principle,in which the geometric nonlinearity,material properties and axial load were incorporated.The direct multiscale method was applied to the analysis process of an axially moving functionally graded beam with timedependent velocity,and the natural frequency and solvability conditions were obtained.Based on the conditions,the stability boundaries of subharmonic resonance and combination resonance were obtained.It was found that the dynamic behavior of axial moving beams could be tuned by using the distribution law of the functional gradient parameters.展开更多
There are two separate traditional approaches to model contact problems: continuum and atomistic theory. Continuum theory is successfully used in many domains, but when the scale of the model comes to nanometer, conti...There are two separate traditional approaches to model contact problems: continuum and atomistic theory. Continuum theory is successfully used in many domains, but when the scale of the model comes to nanometer, continuum approximation meets challenges. Atomistic theory can catch the detailed behaviors of an individual atom by using molecular dynamics (MD) or quantum mechanics, although accurately, it is usually time-consuming. A multiscale method coupled MD and finite element (FE) is presented. To mesh the FE region automatically, an adaptive method based on the strain energy gradient is introduced to the multiscale method to constitute an adaptive multiscale method. Utilizing the proposed method, adhesive contacts between a rigid cylinder and an elastic substrate are studied, and the results are compared with full MD simulations. The process of FE meshes refinement shows that adaptive multiscale method can make FE mesh generation more flexible. Comparison of the displacements of boundary atoms in the overlap region with the results from full MD simulations indicates that adaptive multiscale method can transfer displacements effectively. Displacements of atoms and FE nodes on the center line of the multiscale model agree well with that of atoms in full MD simulations, which shows the continuity in the overlap region. Furthermore, the Von Mises stress contours and contact force distributions in the contact region are almost same as full MD simulations. The method presented combines multiscale method and adaptive technique, and can provide a more effective way to multiscale method and to the investigation on nanoscale contact problems.展开更多
Nanoscale adhesive contacts play a key role in micro/nano-electro-mechanical systems as the dimension of the components come to nanometer.Experimental studies on nanoscale adhesive contacts are limited by some uncerta...Nanoscale adhesive contacts play a key role in micro/nano-electro-mechanical systems as the dimension of the components come to nanometer.Experimental studies on nanoscale adhesive contacts are limited by some uncertain factors and the cost of experiments is too high.Besides,nanoscale textured surfaces are difficult to process and nanoscale adhesive contacts of textured surfaces are still lack of investigation.By using multiscale method,which couples molecular dynamics simulation and finite element method,two-dimensional nanoscale adhesive contacts between a rigid cylindrical tip and an elastic substrate are investigated.For the contacts between the rigid cylindrical tip and smooth surface,Von Mises stress distributions,the maximum Von Mises stresses,and contact forces are compared for different radii to show the size effects and adhesive effects.The phenomena of hysteresis are observed and more obvious as the radii of the tip increase.The influences of indentation depth and indentation speed are also discussed.Then two series of textured surfaces are employed,and the influences of the texture asperity shape,asperity height,and asperity spacing on contact forces are studied.The contact forces comparisons show that textured surfaces can reduce contact forces effectively in the range of the two series.Contact forces of textured surfaces increase as the asperity heights increase,and textured surfaces with smaller asperity spacing will get higher contact forces.Contact forces may be controlled through textured surfaces in the future.The obtained results will help to improve contact condition and provide theory basis for texture design.展开更多
In this paper,we first propose a new stabilized finite element method for the Stokes eigenvalue problem.This new method is based on multiscale enrichment,and is derived from the Stokes eigenvalue problem itself.The co...In this paper,we first propose a new stabilized finite element method for the Stokes eigenvalue problem.This new method is based on multiscale enrichment,and is derived from the Stokes eigenvalue problem itself.The convergence of this new stabilized method is proved and the optimal priori error estimates for the eigenfunctions and eigenvalues are also obtained.Moreover,we combine this new stabilized finite element method with the two-level method to give a new two-level stabilized finite element method for the Stokes eigenvalue problem.Furthermore,we have proved a priori error estimates for this new two-level stabilized method.Finally,numerical examples confirm our theoretical analysis and validate the high effectiveness of the new methods.展开更多
A concurrent multiscale method is developed for simulating quasi-static crack propagation in which the failure processes occur in only a small portion of the structure. For this purpose, a multiscale model is adopted ...A concurrent multiscale method is developed for simulating quasi-static crack propagation in which the failure processes occur in only a small portion of the structure. For this purpose, a multiscale model is adopted and both scales are discretized with finite-element meshes. The extended finite element method is employed to take into account the propagation of discontinuities on the fine-scale subregions. At the same time, for the other subregions, the coarse-scale mesh is employed and is resolved by using the extended multiscale finite element method. Several representative numerical examples are given to verify the validity of the method.展开更多
By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is propose...By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.展开更多
We proposed a practical way for mapping the results of coarse-grained molecular simulations to the observables in hydrogen change experiments.By combining an atomic-interaction based coarse-grained model with an all-a...We proposed a practical way for mapping the results of coarse-grained molecular simulations to the observables in hydrogen change experiments.By combining an atomic-interaction based coarse-grained model with an all-atom structure reconstruction algorithm,we reproduced the experimental hydrogen exchange data with reasonable accuracy using molecular dynamics simulations.We also showed that the coarse-grained model can be further improved by imposing experimental restraints from hydrogen exchange data via an iterative optimization strategy.These results suggest that it is feasible to develop an integrative molecular simulation scheme by incorporating the hydrogen exchange data into the coarse-grained molecular dynamics simulations and therefore help to overcome the accuracy bottleneck of coarse-grained models.展开更多
This paper is concerned with estimation of electrical conductivity in Maxwell equations. The primary difficulty lies in the presence of numerous local minima in the objective functional. A wavelet multiscale method is...This paper is concerned with estimation of electrical conductivity in Maxwell equations. The primary difficulty lies in the presence of numerous local minima in the objective functional. A wavelet multiscale method is introduced and applied to the inversion of Maxwell equations. The inverse problem is decomposed into multiple scales with wavelet transform, and hence the original problem is reformulated to a set of sub-inverse problems corresponding to different scales, which can be solved successively according to the size of scale from the shortest to the longest. The stable and fast regularized Gauss-Newton method is applied to each scale. Numerical results show that the proposed method is effective, especially in terms of wide convergence, computational efficiency and precision.展开更多
Viscoelastic flows play an important role in numerous engineering fields,and the multiscale algorithms for simulating viscoelastic flows have received significant attention in order to deepen our understanding of the ...Viscoelastic flows play an important role in numerous engineering fields,and the multiscale algorithms for simulating viscoelastic flows have received significant attention in order to deepen our understanding of the nonlinear dynamic behaviors of viscoelastic fluids.However,traditional grid-based multiscale methods are confined to simple viscoelastic flows with short relaxation time,and there is a lack of uniform multiscale scheme available for coupling different solvers in the simulations of viscoelastic fluids.In this paper,a universal multiscale method coupling an improved smoothed particle hydrodynamics(SPH)and multiscale universal interface(MUI)library is presented for viscoelastic flows.The proposed multiscale method builds on an improved SPH method and leverages the MUI library to facilitate the exchange of information among different solvers in the overlapping domain.We test the capability and flexibility of the presented multiscale method to deal with complex viscoelastic flows by solving different multiscale problems of viscoelastic flows.In the first example,the simulation of a viscoelastic Poiseuille flow is carried out by two coupled improved SPH methods with different spatial resolutions.The effects of exchanging different physical quantities on the numerical results in both the upper and lower domains are also investigated as well as the absolute errors in the overlapping domain.In the second example,the complex Wannier flow with different Weissenberg numbers is further simulated by two improved SPH methods and coupling the improved SPH method and the dissipative particle dynamics(DPD)method.The numerical results show that the physical quantities for viscoelastic flows obtained by the presented multiscale method are in consistence with those obtained by a single solver in the overlapping domain.Moreover,transferring different physical quantities has an important effect on the numerical results.展开更多
The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersi...The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersingular integral equation with the map x=cot（θ/2）. Second, we initiate the study of the multiscale Galerkin method for the 2π-periodic hypersingular integral equation. The trigonometric wavelets are used as trial functions. Consequently, the 2j＋1 × 2j＋1 stiffness matrix Kj can be partitioned j×j block matrices. Furthermore, these block matrices are zeros except main diagonal block matrices. These main diagonal block matrices are symmetrical and circulant matrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform and the inverse fast Fourier transform instead of the inverse matrix. Finally, we provide several numerical examples to demonstrate our method has good accuracy even though the exact solutions are multi-peak and almost singular.展开更多
A LES model is proposed to predict the dispersion of particles in the atmosphere in the context of Chemical,Biological,Radiological and Nuclear(CBRN)applications.The code relies on the Finite Element Method(FEM)for bo...A LES model is proposed to predict the dispersion of particles in the atmosphere in the context of Chemical,Biological,Radiological and Nuclear(CBRN)applications.The code relies on the Finite Element Method(FEM)for both the fluid and the dispersed solid phases.Starting from the Navier-Stokes equations and a general description of the FEM strategy,the Streamline Upwind Petrov-Galerkin(SUPG)method is formulated putting some emphasis on the related assembly matrix and stabilization coefficients.Then,the Variational Multiscale Method(VMS)is presented together with a detailed illustration of its algorithm and hierarchy of computational steps.It is demonstrated that the VMS can be considered as a more general version of the SUPG method.The final part of the work is used to assess the reliability of the implemented predictor/multicorrector solution strategy.展开更多
We provide a concise review of the exponentially convergent multiscale finite element method(ExpMsFEM)for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave pr...We provide a concise review of the exponentially convergent multiscale finite element method(ExpMsFEM)for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation.The ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions.Unlike most generalizations of the MsFEM in the literature,the ExpMsFEM does not rely on any partition of unity functions.In general,it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n-width barrier to achieve exponential convergence.Indeed,there are online and offline parts in the function representation provided by the ExpMsFEM.The online part depends on the right-hand side locally and can be computed in parallel efficiently.The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix;they are all independent of the right-hand side,so the stiffness matrix can be used repeatedly in multi-query scenarios.展开更多
We propose a multiscale multilevel Monte Carlo(MsMLMC)method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting.Our method consists of offline and online stages.In the offline stage,...We propose a multiscale multilevel Monte Carlo(MsMLMC)method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting.Our method consists of offline and online stages.In the offline stage,we construct a small number of reduced basis functions within each coarse grid block,which can then be used to approximate the multiscale finite element basis functions.In the online stage,we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis.The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid,without requiring the coarsest grid to resolve the smallestscale of the solution.We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients.Moreover,we provide convergence analysis of the proposed method.Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.展开更多
文摘We consider a strongly heterogeneous medium saturated by an incompressible viscous fluid as it appears in geomechanical modeling.This poroelasticity problem suffers from rapidly oscillating material parameters,which calls for a thorough numerical treatment.In this paper,we propose a method based on the local orthogonal decomposition technique and motivated by a similar approach used for linear thermoelasticity.Therein,local corrector problems are constructed in line with the static equations,whereas we propose to consider the full system.This allows to benefit from the given saddle point structure and results in two decoupled corrector problems for the displacement and the pressure.We prove the optimal first-order convergence of this method and verify the result by numerical experiments.
文摘In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.
基金supported by the National Science Foundation grant DMS-1818998.
文摘In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al.in J Sci Comput 66:321–345,2016;Dong and Wang in J Comput Appl Math 380:1–11,2020)for a one-dimensional stationary Schrödinger equation.Previous work showed that penalty parameters were required to be positive in error analysis,but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes.In this work,by performing extensive numerical experiments,we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods,and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.
基金The NSF (10371137 and 10201034) of China,the Foundation (20030558008) of Doctoral Program of National Higher Education,Guangdong Provincial Natural Science Foundation (1011170) of China and the Foundation of Zhongshan University Advanced Research Center.
文摘This paper develops fast multiscale collocation methods for a class of Fredholm integral equations of the second kind with singular kernels. A truncation strategy for the coefficient matrix of the corresponding discrete system is proposed, which forms a basis for fast algorithms. The convergence, stability and computational complexity of these algorithms are analyzed.
基金supported by the National Key Research and Development Program of China(Grant No.2021YFB1714600)the National Natural Science Foundation of China(Grant No.52175095)the Young Top-Notch Talent Cultivation Program of Hubei Province of China.
文摘This paper examined how microstructure influences the homogenized thermal conductivity of cellular structures and revealed a surface-induced size-dependent effect.This effect is linked to the porous microstructural features of cellular structures,which stems from the degree of porosity and the distri-bution of the pores.Unlike the phonon-driven surface effect at the nanoscale,the macro-scale surface mechanism in thermal cellular structures is found to be the microstructure-induced changes in the heat conduction path based on fully resolved 3D numerical simulations.The surface region is determined by the microstructure,characterized by the intrinsic length.With the coupling between extrinsic and intrinsic length scales under the surface mechanism,a surface-enriched multiscale method was devel-oped to accurately capture the complex size-dependent thermal conductivity.The principle of scale separation required by classical multiscale methods is not necessary to be satisfied by the proposed multiscale method.The significant potential of the surface-enriched multiscale method was demon-strated through simulations of the effective thermal conductivity of a thin-walled metamaterial struc-ture.The surface-enriched multiscale method offers higher accuracy compared with the classical multiscale method and superior efficiency over high-fidelity finite element methods.
基金supported by the National Natural Science Foundation of China(Grant Nos.42102346,42172301).
文摘Magneto-electro-elastic(MEE)materials are widely utilized across various fields due to their multi-field coupling effects.Consequently,investigating the coupling behavior of MEE composite materials is of significant importance.The traditional finite element method(FEM)remains one of the primary approaches for addressing such issues.However,the application of FEM typically necessitates the use of a fine finite element mesh to accurately capture the heterogeneous properties of the materials and meet the required computational precision,which inevitably leads to a reduction in computational efficiency.To enhance the computational accuracy and efficiency of the FEM for heterogeneous multi-field coupling problems,this study presents the coupling magneto-electro-elastic multiscale finite element method(CM-MsFEM)for heterogeneous MEE structures.Unlike the conventional multiscale FEM(MsFEM),the proposed algorithm simultaneously constructs displacement,electric,and magnetic potential multiscale basis functions to address the heterogeneity of the corresponding parameters.The macroscale formulation of CM-MsFEM was derived,and the macroscale/microscale responses of the problems were obtained through up/downscaling calculations.Evaluation using numerical examples analyzing the transient behavior of heterogeneous MEE structures demonstrated that the proposed method outperforms traditional FEM in terms of both accuracy and computational efficiency,making it an appropriate choice for numerically modeling the dynamics of heterogeneous MEE structures.
基金supported by the National Natural Science Foundation(10721062,11072051,90715037,10728205,91015003, 51021140004)the Program of Introducing Talents of Discipline to Universities(B08014)the National Key Basic Research Special Foundation of China(2010CB832704).
文摘An extended multiscale finite element method （EMsFEM） is developed for solving the mechanical problems of heterogeneous materials in elasticity.The underlying idea of the method is to construct numerically the multiscale base functions to capture the small-scale features of the coarse elements in the multiscale finite element analysis.On the basis of our existing work for periodic truss materials, the construction methods of the base functions for continuum heterogeneous materials are systematically introduced. Numerical experiments show that the choice of boundary conditions for the construction of the base functions has a big influence on the accuracy of the multiscale solutions, thus,different kinds of boundary conditions are proposed. The efficiency and accuracy of the developed method are validated and the results with different boundary conditions are verified through extensive numerical examples with both periodic and random heterogeneous micro-structures.Also, a consistency test of the method is performed numerically. The results show that the EMsFEM can effectively obtain the macro response of the heterogeneous structures as well as the response in micro-scale,especially under the periodic boundary conditions.
基金The authors acknowledge the support of National Natural Science Foundation of China(Nos.11672187,11572182)Natural Science Foundation of Liaoning Province(201602573)+2 种基金the Key Research Projects of Shanghai Science and Technology Commission(No.18010500100)Innovation Program of Shanghai Education Commission(No.2017-01-07-00-09-E00019)Beiyang Young Scholars of Tian jin University(2019XRX-0027).
文摘In this study,the transverse vibration of a traveling beam made of functionally graded material was analyzed.The material gradation was assumed to vary continuously along the thickness direction of the beam in the form of power law exponent.The effect of the longitudinally varying tension due to axial acceleration was highlighted,and the dependence of the tension on the finite support rigidity was also considered.A complex governing equation of the functionally graded beam was derived by the Hamilton principle,in which the geometric nonlinearity,material properties and axial load were incorporated.The direct multiscale method was applied to the analysis process of an axially moving functionally graded beam with timedependent velocity,and the natural frequency and solvability conditions were obtained.Based on the conditions,the stability boundaries of subharmonic resonance and combination resonance were obtained.It was found that the dynamic behavior of axial moving beams could be tuned by using the distribution law of the functional gradient parameters.
基金supported by National Natural Science Foundation of China (Grant Nos. 51205313, 50975232)Northwestern Polytechnical University Foundation for Fundamental Research of China (Grant No.JC20110249)
文摘There are two separate traditional approaches to model contact problems: continuum and atomistic theory. Continuum theory is successfully used in many domains, but when the scale of the model comes to nanometer, continuum approximation meets challenges. Atomistic theory can catch the detailed behaviors of an individual atom by using molecular dynamics (MD) or quantum mechanics, although accurately, it is usually time-consuming. A multiscale method coupled MD and finite element (FE) is presented. To mesh the FE region automatically, an adaptive method based on the strain energy gradient is introduced to the multiscale method to constitute an adaptive multiscale method. Utilizing the proposed method, adhesive contacts between a rigid cylinder and an elastic substrate are studied, and the results are compared with full MD simulations. The process of FE meshes refinement shows that adaptive multiscale method can make FE mesh generation more flexible. Comparison of the displacements of boundary atoms in the overlap region with the results from full MD simulations indicates that adaptive multiscale method can transfer displacements effectively. Displacements of atoms and FE nodes on the center line of the multiscale model agree well with that of atoms in full MD simulations, which shows the continuity in the overlap region. Furthermore, the Von Mises stress contours and contact force distributions in the contact region are almost same as full MD simulations. The method presented combines multiscale method and adaptive technique, and can provide a more effective way to multiscale method and to the investigation on nanoscale contact problems.
基金supported by National Natural Science Foundation of China (Grant No. 50975232)Fundamental Research Foundation of Northwestern Polytechnical University,China (Grant No. JC20110249)
文摘Nanoscale adhesive contacts play a key role in micro/nano-electro-mechanical systems as the dimension of the components come to nanometer.Experimental studies on nanoscale adhesive contacts are limited by some uncertain factors and the cost of experiments is too high.Besides,nanoscale textured surfaces are difficult to process and nanoscale adhesive contacts of textured surfaces are still lack of investigation.By using multiscale method,which couples molecular dynamics simulation and finite element method,two-dimensional nanoscale adhesive contacts between a rigid cylindrical tip and an elastic substrate are investigated.For the contacts between the rigid cylindrical tip and smooth surface,Von Mises stress distributions,the maximum Von Mises stresses,and contact forces are compared for different radii to show the size effects and adhesive effects.The phenomena of hysteresis are observed and more obvious as the radii of the tip increase.The influences of indentation depth and indentation speed are also discussed.Then two series of textured surfaces are employed,and the influences of the texture asperity shape,asperity height,and asperity spacing on contact forces are studied.The contact forces comparisons show that textured surfaces can reduce contact forces effectively in the range of the two series.Contact forces of textured surfaces increase as the asperity heights increase,and textured surfaces with smaller asperity spacing will get higher contact forces.Contact forces may be controlled through textured surfaces in the future.The obtained results will help to improve contact condition and provide theory basis for texture design.
基金supported by the National Key R&D Program of China(2018YFB1501001)the NSF of China(11771348)China Postdoctoral Science Foundation(2019M653579)。
文摘In this paper,we first propose a new stabilized finite element method for the Stokes eigenvalue problem.This new method is based on multiscale enrichment,and is derived from the Stokes eigenvalue problem itself.The convergence of this new stabilized method is proved and the optimal priori error estimates for the eigenfunctions and eigenvalues are also obtained.Moreover,we combine this new stabilized finite element method with the two-level method to give a new two-level stabilized finite element method for the Stokes eigenvalue problem.Furthermore,we have proved a priori error estimates for this new two-level stabilized method.Finally,numerical examples confirm our theoretical analysis and validate the high effectiveness of the new methods.
基金Project supported by the National Natural Science Foundation of China(Nos.11232003,11072051,11272003 and 91315302)the 111 Project(No.B08014)+2 种基金the National Basic Research Program of China(Nos.2010CB832704 and 2011CB013401)Program for New Century Excellent Talents in University(No.NCET-13-0088)Ph.D.Programs Foundation of Ministry of Education of China(No.20130041110050)
文摘A concurrent multiscale method is developed for simulating quasi-static crack propagation in which the failure processes occur in only a small portion of the structure. For this purpose, a multiscale model is adopted and both scales are discretized with finite-element meshes. The extended finite element method is employed to take into account the propagation of discontinuities on the fine-scale subregions. At the same time, for the other subregions, the coarse-scale mesh is employed and is resolved by using the extended multiscale finite element method. Several representative numerical examples are given to verify the validity of the method.
基金supported by the Natural Science Foundation of Zhejiang Province,China(Grant Nos.LY20A010021,LY19A010002,LY20G030025)the Natural Science Founda-tion of Ningbo City,China(Grant Nos.2021J147,2021J235).
文摘By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.
基金the National Natural Science Foundation of China(Grant Nos.11974173 and 11934008)the HPC Center of Nanjing University。
文摘We proposed a practical way for mapping the results of coarse-grained molecular simulations to the observables in hydrogen change experiments.By combining an atomic-interaction based coarse-grained model with an all-atom structure reconstruction algorithm,we reproduced the experimental hydrogen exchange data with reasonable accuracy using molecular dynamics simulations.We also showed that the coarse-grained model can be further improved by imposing experimental restraints from hydrogen exchange data via an iterative optimization strategy.These results suggest that it is feasible to develop an integrative molecular simulation scheme by incorporating the hydrogen exchange data into the coarse-grained molecular dynamics simulations and therefore help to overcome the accuracy bottleneck of coarse-grained models.
基金supported by the Program of Excellent Team of Harbin Institute of Technology
文摘This paper is concerned with estimation of electrical conductivity in Maxwell equations. The primary difficulty lies in the presence of numerous local minima in the objective functional. A wavelet multiscale method is introduced and applied to the inversion of Maxwell equations. The inverse problem is decomposed into multiple scales with wavelet transform, and hence the original problem is reformulated to a set of sub-inverse problems corresponding to different scales, which can be solved successively according to the size of scale from the shortest to the longest. The stable and fast regularized Gauss-Newton method is applied to each scale. Numerical results show that the proposed method is effective, especially in terms of wide convergence, computational efficiency and precision.
基金Project supported by the National Natural Science Foundation of China(No.52109068)the Water Conservancy Technology Project of Jiangsu Province of China(No.2022060)。
文摘Viscoelastic flows play an important role in numerous engineering fields,and the multiscale algorithms for simulating viscoelastic flows have received significant attention in order to deepen our understanding of the nonlinear dynamic behaviors of viscoelastic fluids.However,traditional grid-based multiscale methods are confined to simple viscoelastic flows with short relaxation time,and there is a lack of uniform multiscale scheme available for coupling different solvers in the simulations of viscoelastic fluids.In this paper,a universal multiscale method coupling an improved smoothed particle hydrodynamics(SPH)and multiscale universal interface(MUI)library is presented for viscoelastic flows.The proposed multiscale method builds on an improved SPH method and leverages the MUI library to facilitate the exchange of information among different solvers in the overlapping domain.We test the capability and flexibility of the presented multiscale method to deal with complex viscoelastic flows by solving different multiscale problems of viscoelastic flows.In the first example,the simulation of a viscoelastic Poiseuille flow is carried out by two coupled improved SPH methods with different spatial resolutions.The effects of exchanging different physical quantities on the numerical results in both the upper and lower domains are also investigated as well as the absolute errors in the overlapping domain.In the second example,the complex Wannier flow with different Weissenberg numbers is further simulated by two improved SPH methods and coupling the improved SPH method and the dissipative particle dynamics(DPD)method.The numerical results show that the physical quantities for viscoelastic flows obtained by the presented multiscale method are in consistence with those obtained by a single solver in the overlapping domain.Moreover,transferring different physical quantities has an important effect on the numerical results.
文摘The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersingular integral equation with the map x=cot（θ/2）. Second, we initiate the study of the multiscale Galerkin method for the 2π-periodic hypersingular integral equation. The trigonometric wavelets are used as trial functions. Consequently, the 2j＋1 × 2j＋1 stiffness matrix Kj can be partitioned j×j block matrices. Furthermore, these block matrices are zeros except main diagonal block matrices. These main diagonal block matrices are symmetrical and circulant matrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform and the inverse fast Fourier transform instead of the inverse matrix. Finally, we provide several numerical examples to demonstrate our method has good accuracy even though the exact solutions are multi-peak and almost singular.
基金The authors received the funding of the Royal Higher Institute for Defence(MSP16-06).
文摘A LES model is proposed to predict the dispersion of particles in the atmosphere in the context of Chemical,Biological,Radiological and Nuclear(CBRN)applications.The code relies on the Finite Element Method(FEM)for both the fluid and the dispersed solid phases.Starting from the Navier-Stokes equations and a general description of the FEM strategy,the Streamline Upwind Petrov-Galerkin(SUPG)method is formulated putting some emphasis on the related assembly matrix and stabilization coefficients.Then,the Variational Multiscale Method(VMS)is presented together with a detailed illustration of its algorithm and hierarchy of computational steps.It is demonstrated that the VMS can be considered as a more general version of the SUPG method.The final part of the work is used to assess the reliability of the implemented predictor/multicorrector solution strategy.
基金part supported by the NSF Grants DMS-1912654 and DMS 2205590。
文摘We provide a concise review of the exponentially convergent multiscale finite element method(ExpMsFEM)for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation.The ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions.Unlike most generalizations of the MsFEM in the literature,the ExpMsFEM does not rely on any partition of unity functions.In general,it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n-width barrier to achieve exponential convergence.Indeed,there are online and offline parts in the function representation provided by the ExpMsFEM.The online part depends on the right-hand side locally and can be computed in parallel efficiently.The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix;they are all independent of the right-hand side,so the stiffness matrix can be used repeatedly in multi-query scenarios.
基金partially supported by the Hong Kong Ph D Fellowship Schemesupported by the Hong Kong RGC General Research Funds(Projects 27300616,17300817,and 17300318)+2 种基金National Natural Science Foundation of China(Project 11601457)Seed Funding Programme for Basic Research(HKU)Basic Research Programme(JCYJ20180307151603959)of the Science,Technology and Innovation Commission of Shenzhen Municipality。
文摘We propose a multiscale multilevel Monte Carlo(MsMLMC)method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting.Our method consists of offline and online stages.In the offline stage,we construct a small number of reduced basis functions within each coarse grid block,which can then be used to approximate the multiscale finite element basis functions.In the online stage,we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis.The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid,without requiring the coarsest grid to resolve the smallestscale of the solution.We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients.Moreover,we provide convergence analysis of the proposed method.Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.
