Low order nonconforming mixed finite element method for nonstationary incompressible Navier-Stokes equations

This paper studies a low order mixed finite element method (FEM) for nonstationary incompressible Navier-Stokes equations. The velocity and pressure are approximated by the nonconforming constrained Q ₁ ^rot element and the piecewise constant, respectively. The superconvergent error estimates of the velocity in the broken H ¹-norm and the pressure in the L ²-norm are obtained respectively when the exact solutions are reasonably smooth. A numerical experiment is carried out to confirm the theoretical results.     

H 1 space-time discontinuous finite element method for convection-diffusion equations

An H~1 space-time discontinuous Galerkin (STDG) scheme for convectiondiffusion equations in one spatial dimension is constructed and analyzed. This method is formulated by combining the H~1 Galerkin method and the space-time discontinuous finite element method that is discontinuous in time and continuous in space. The existence and the uniqueness of the approximate solution are proved. The convergence of the scheme is analyzed by using the techniques in the finite difference and finite element methods. An optimal a-priori error estimate in the L~∞ (H~1 ) norm is derived. The numerical exper- iments are presented to verify the theoretical results.     

New nonconforming finite element method for solving transient Naiver-Stokes equations

For transient Naiver-Stokes problems, a stabilized nonconforming finite element method is presented, focusing on two pairs inf-sup unstable finite element spaces, i.e., pNC/pNC triangular and pNQ/pNQ quadrilateral finite element spaces. The semi- and full-discrete schemes of the stabilized method are studied based on the pressure projection and a variational multi-scale method. It has some attractive features： avoiding higher-order derivatives and edge-based data structures, adding a discrete velocity term only on the fine scale, being effective for high Reynolds number fluid flows, and avoiding increased computation cost. For the full-discrete scheme, it has second-order estimations of time and is unconditionally stable. The presented numerical results agree well with the theoretical results.     

A new weighted upwind finite volume element method based on non‐standard covolume for time‐dependent convection–diffusion problems

In modern numerical simulation of problems in energy resources and environmental science, it is important to develop efficient numerical methods for time‐dependent convection–diffusion problems. On the basis of nonstandard covolume grids, we propose a new kind of high‐order upwind finite volume element method for the problems. We first prove the stability and mass conservation in the discrete forms of the scheme. Optimal second‐order error estimate in L₂‐norm in spatial step is then proved strictly. The scheme is effective for avoiding numerical diffusion and nonphysical oscillations and has second‐order accuracy. Numerical experiments are given to verify the performance of the scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotic decomposition of a singular perturbation problem with unbounded energyDécomposition asymptotique d'une perturbation singulière d'énergie sans limite

We consider a singular perturbation with unbounded energy. We propose here an effective method of finite element computation, fit for accounting for the linear behavior of the solution. The Hilbert space of the variational formulation, H²₀(0,1), is replaced by a simpler subspace containing an asymptotic solution of the initial problem. Error estimates are derived by eliminating some degrees of freedom and a numerical experiment is developped. To cite this article: F. Fontvieille et al., C. R. Mecanique 330 (2002) 507–512.     

Positive‐definite q‐families of continuous subcell Darcy‐flux CVD(MPFA) finite‐volume schemes and the mixed finite element method

A new family of locally conservative cell‐centred flux‐continuous schemes is presented for solving the porous media general‐tensor pressure equation. A general geometry‐permeability tensor approximation is introduced that is piecewise constant over the subcells of the control volumes and ensures that the local discrete general tensor is elliptic. A family of control‐volume distributed subcell flux‐continuous schemes are defined in terms of the quadrature parametrization q (Multigrid Methods. Birkhauser: Basel, 1993; Proceedings of the 4th European Conference on the Mathematics of Oil Recovery, Norway, June 1994; Comput. Geosci. 1998; 2 :259–290), where the local position of flux continuity defines the quadrature point and each particular scheme. The subcell tensor approximation ensures that a symmetric positive‐definite (SPD) discretization matrix is obtained for the base member (q=1) of the formulation. The physical‐space schemes are shown to be non‐symmetric for general quadrilateral cells. Conditions for discrete ellipticity of the non‐symmetric schemes are derived with respect to the local symmetric part of the tensor. The relationship with the mixed finite element method is given for both the physical‐space and subcell‐space q‐families of schemes. M‐matrix monotonicity conditions for these schemes are summarized. A numerical convergence study of the schemes shows that while the physical‐space schemes are the most accurate, the subcell tensor approximation reduces solution errors when compared with earlier cell‐wise constant tensor schemes and that subcell tensor approximation using the control‐volume face geometry yields the best SPD scheme results. A particular quadrature point is found to improve numerical convergence of the subcell schemes for the cases tested. Copyright © 2007 John Wiley & Sons, Ltd.     

Analytical solutions for finite cylinders compressed between platens and the strain effect on the valence-band structure of Si1−xGex alloy

This paper presents an analytical solution for inhomogeneous strain and stress distributions within finite circular cylinders of Si_1−xGe_x alloy under compression test with end friction. The method follows Lekhnitskii’s stress function approach, but a new expression for the stress function is proposed so that all of the governing equations and boundary conditions are satisfied exactly. Numerical results show that the axial, radial, circumferential and shear strains are all inhomogeneous within finite cylinders, and local strain concentrations near two end surfaces were usually developed as long as friction exists between end surfaces and loading platens. Moreover, by using envelope-function method, the effect of strain on the valence-band structure of Si_1−xGe_x alloy is also studied. It was found that strain can induce band splitting, alteration of the shape of constant energy surfaces of the heavy-hole and the light-hole bands of Si_1−xGe_x alloy.     

A consistent numerical scheme for the von Mises mixed-hardening constitutive equations

Analytical solutions are derived for the von Mises mixed-hardening elastoplastic model under rectilinear strain paths, and the concept of response subspace is introduced such that the original five-dimensional problem in deviatoric stress space is reduced to a more economic two-dimensional problem, of which two coordinates (x,y) suffice to determine normalized active stress. Furthermore, in this subspace a Minkowski spacetime can be endowed, on which the group action is found to be a proper orthochronous Lorentz group SO_o(2,1). The existence of a fixed point attractor in the normalized active stress space is demonstrated by the long-term behavior deduced from the analytical solutions, which together with the response stability is further verified by Lyapunov's direct method. Two numerical schemes based on a nonlinear Volterra integral equation and on a group symmetry are derived, the latter of which exactly preserves the consistency condition for every time step. The consistent scheme is stable, accurate and efficient, because it updates the stress point automatically on the yield surface at each time step without any iteration. For the purpose of comparison and contrast, numerical results calculated by the above two schemes as well as by the radial return method were displayed for several loading examples.     

Higher-order accurate implicit time integration schemes for transport problems

The present paper is concerned with the numerical solution of transient transport problems by means of spatial and temporal discretization methods. The generalized initial boundary value problem of various nonlinear transport phenomena like heat transfer or mass transport is discretized in space by p-finite elements. After finite element discretization, the resulting first-order semidiscrete balance has to be solved with respect to time. Next to the classical generalized-α integration method predicated on the Newmark approach and the evaluation at a generalized midpoint also implicit Runge–Kutta time integration schemes, are presented. Both families of finite difference-based integration schemes are derived for general first-order problems. In contrast to the above-mentioned algorithms, temporal discontinuous and continuous Galerkin methods evaluate the balance equation not at a selected time instant within the timestep, but in an integral sense over the whole time step interval. Therefore, the underlying semidiscrete balance and the continuity of the primary variables are weakly formulated within time steps and between time steps, respectively. Continuous Galerkin methods are obtained by the strong enforcement of the continuity condition as special cases. The introduction of a natural time coordinate allows for the application of standard higher-order temporal shape functions of the p-Lagrange type and the well-known Gau?–Legendre quadrature of associated time integrals. It is shown that arbitrary order accurate integration schemes can be developed within the framework of the proposed temporal p-Galerkin methods. Selected benchmark analyses of calcium diffusion demonstrate the properties of all three methods with respect to non-smooth initial or boundary conditions. Furthermore, the robustness of the present time integration schemes is also demonstrated for the highly nonlinear reaction–diffusion problem of calcium leaching, including the pronounced changes of the reaction term and non-smooth changes of Dirichlet boundary conditions of calcium dissolution.     

Computing lower and upper bounds on stress intensity factors in bimaterials

This paper presents a finite element approach for finding complementary bounds of stress intensity factors (SIFs) in bimaterials. The SIF is formulated as an explicit computable linear function of displacements by means of the two-point extrapolation method. An appropriate and computable form of the SIF plays a crucial role in the dual problem involved in the computing procedure of both lower and upper bounds. In our discussions, computable forms of stress intensity factors, K₀ and K_r, are derived, which are related to the energy release rate, and the ratio of the open mode and shear mode SIFs, respectively. Based on a posteriori finite element error estimation, a bounding procedure is used to compute the bounds on the two stress intensity factors. Finally, bounds on the SIFs in a bimaterial interface crack problem are provided to verify the procedure.     

A finite element investigation of the effect of crack tip constraint on hole growth under mode i and mixed mode loading

In this work, the effect of constraint on hole growth near a notch tip in a ductile material under mode I and mixed mode loading (involving modes I and II) is investigated. To this end, a 2-D plane strain, modified boundary layer formulation is employed in which the mixed mode elastic K–T field is prescribed as remote boundary conditions. A finite element procedure that accounts for finite deformations and rotations is used along with an appropriate version of J₂ flow theory of plasticity with small elastic strains. Several analyses are carried out corresponding to different values of T-stress and remote elastic mode-mixity. The interaction between the notch and hole is studied by examining the distribution of hydrostatic stress and equivalent plastic strain in the ligament between the notch tip and the hole, as well as the growth of the hole. The implications of the above results on ductile fracture initiation due to micro-void coalescence are discussed.     

Managing false diffusion during second‐order upwind simulations of liquid micromixing

Liquid mixing is an important component of many microfluidic concepts and devices, and computational fluid dynamics (CFD) is playing a key role in their development and optimization. Because liquid mass diffusivities can be quite small, CFD simulation of liquid micromixing can over predict the degree of mixing unless numerical (or false) diffusion is properly controlled. Unfortunately, the false diffusion behavior of higher‐order finite volume schemes, which are often used for such simulations, is not well understood, especially on unstructured meshes. To examine and quantify the amount of false diffusion associated with the often recommended and versatile second‐order upwind method, a series of numerical simulations was conducted using a standardized two‐dimensional test problem on both structured and unstructured meshes. This enabled quantification of an ‘effective’ false diffusion coefficient (D_false) for the method as a function of mesh spacing. Based on the results of these simulations, expressions were developed for estimating the spacing required to reduce D_false to some desired (low) level. These expressions, together with additional insights from the standardized test problem and findings from other researchers, were then incorporated into a procedure for managing false diffusion when simulating steady, liquid micromixing. To demonstrate its utility, the procedure was applied to simulate flow and mixing within a representative micromixer geometry using both unstructured (triangular) and structured meshes. Copyright © 2016 John Wiley & Sons, Ltd.     

Towards variational constitutive updates for non-associative plasticity models at finite strain: Models based on a volumetric-deviatoric split

In this paper, an enhanced variational constitutive update suitable for a class of non-associative plasticity theories at finite strain is proposed. In line with classical numerical formulations for plasticity models, such as the by now established return-mapping algorithm, variational constitutive updates represent a numerical method for computing the unknown state variables. However, in contrast to conventional algorithms, variational constitutive updates are fully variational, i.e., all unknown variables follow jointly from minimizing a certain potential. In addition to the physical and mathematical elegance of these variational schemes, they show several practical advantages as well. For instance, numerically efficient and robust optimization schemes can be directly employed for solving the resulting minimization problem. Since mathematically, plasticity is a non-smooth problem and often, it leads to highly singular systems of equations as known from single crystal plasticity, a robust implementation is of utmost importance. So far, variational constitutive updates have been developed for different classes of standard dissipative solids, i.e., solids characterized by associative evolution equations and flow rules. In the present paper, this framework is extended to a certain class of non-associative plasticity models at finite strain. All models falling into this class show a volumetric-deviatoric split of the Helmholtz energy and the yield function. Typical prototypes are Drucker-Prager or Mohr-Coulomb models playing an important role in soil mechanics. The efficiency and robustness of the resulting algorithmic formulation is demonstrated by means of selected numerical examples.     

Some explicit triangular finite element schemes for the Euler equations

Several explicit schemes are presented for triangular P₀ and P₁ finite elements. A first-order accurate upwind P₀ scheme is compared to a FLIC type method. A second-order accurate Richtmyer scheme is constructed. Applications are given for the Euler system of conservation laws in the 2-dimensional case.     

Pore Space Microstructure Evolution of Regular Sphere Packings Undergoing Compaction and Cementation

We examine the pore space structure evolution of ordered uniform sphere packs: simple cubic (SC), body centered cubic (BCC), and face centered cubic (FCC), undergoing simple diagenetic processes that reduce their pore spaces. Focus is on the occurrence of pore space microstructure changes or transitions, which are followed through their characteristic or critical pore lengths (l _c). For almost all the cubic packings undergoing either compaction or cementation there are no singularities in l _c. This is a consequence of having a single pore shape controlling flow at all stages of the process. However, this is not so for the BCC packing under cementation, for which l _c is non-monotonic exhibiting a kink at ${\phi \approx 0.1452}$ , the porosity at which the pore shape controlling flow switches to a different form and position. These results for uniform compaction/cementation complement our previous works on pore networks under random shrinkage. Kinks in l _c as porosity decreases signal pore space microstructure transitions that anticipate sudden changes in the permeability?Cporosity relation as porosity decreases. The consequences are great; clearly l _c is not a constant unless the diagenetic process is mild. A l _c function of compaction/cementation advancement should be used above a transition and a different l _c function below. For the sphere packs here, once the diagenetic process has reduced the pore space substantially, a l _c function of compaction/cementation advancement is mandatory if we are to capture all significant flow features.     

The Space-Time Finite Element Method for Parabolic Problems

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｅｑｕａｔｉｏｎｓｗｅｃｏｎｓｉｄｅｒｅｄａｒｅａｓｆｏｌｌｏｗｓｕｔ-Δｕ =ｆ(ｕ) ,　　Ω× [0 ,Ｔ] ,ｕ｜ Ω =0 ,　　　　　 Ω× [0 ,Ｔ] ,ｕ( · ,0 ) =ｕ0 ,Ω ,( 1 )ｗｈｅｒｅΩ ∈Ｒ2 ,ｔｈｅｆｕｎｃｔｉｏｎｆ(ｕ)ｓａｔｉｓｆｉｅｓ｜ｆ(ｕ)｜≤ｃ｜ｕ｜ ,　　 ｕ∈Ｃ(Ω) . ( 2 )Ａｎｄｆ(ｕ)ｉｓＬｉｐｓｃｈｉｔｚｃｏｎｔｉｎｕｏｕｓ,ｉ.ｅ.ｉｔｓａｔｉｓｆｉｅｓ｜ｆ(ｕ) -ｆ(ｖ) ｜≤Ｌ｜ｕ-ｖ｜ ,　　 ｕ ,ｖ∈Ｃ(Ω) ,( 3 )ｗｈｅｒｅＬｉｓＬｉｐｓｃｈｉｔｚｃｏｎｓｔａ…     

Numerical methods for a fluid mixture model

In this paper, we firstly apply generalized difference methods to solve a fluid mixture model. The model is usually used to describe the tissue deformations and contains a nonlinear hyperbolic equation and an elliptic equation. Most people have used finite difference methods for solving the elliptic equation and other schemes for solving the hyperbolic equation. It is well known that the accuracy of traditional finite difference method is not high. This may be a serious disadvantage in the fluid mixture model, which describes cell movements and tissue deformations. The numerical methods we propose to improve accuracy are based on generalized Galerkin methods and dual decomposition. By choosing suitable trial function space and test function space, our generalized upwind difference schemes exhibit second‐order convergence in space for smooth problems and can eliminate numerical oscillations for discontinuous problems. Some numerical results are presented to demonstrate the advantages of our methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part III: Yield surface in tension–tension stress space (Al 6061–T 6511 and annealed 1100 Al)

Subsequent yield surfaces for aluminum alloys are determined for three proportional loading paths (i.e., axial, hoop, and combined hoop and axial stress) using 10 με deviation from linearity as the definition of yield. This paper is in continuation with Parts I and II of the author’s previous papers on subsequent yield surfaces under tension–torsion (σ11–√3σ₁₂) stress space using similar small offset definition of yield. In the current paper comprehensive experimental results on subsequent yield surfaces under tension–tension (σ₁₁–σ₂₂) stress space are presented. Comparison of subsequent yield surfaces under (σ₁₁–√3σ₁₂) stress space, investigated in the earlier papers, clearly indicated distinctive hardening behavior under various loading paths. However, subsequent yield surfaces for Al 6061–T 6511 (a low work hardening alloy) showed contraction and negative cross-effect with finite deformation as compared to the annealed 1100 Al (a high work hardening alloy) where expansion and positive cross-effect was observed.     

The inhomogeneous strain distributions within finite cylinders under the axial point load tests and the effects on the valence band of Si1−xGex alloy

An exact solution for inhomogeneous strain and stress distributions within a finite cubic isotropic cylinder of Si_1?xGe_x alloy under the axial Point Load Strength Test (PLST) is analytically derived. Lekhnitskii’s stress function is used to uncouple the equations of equilibrium, and a new expression for the stress function is proposed so that all of the governing equations and boundary conditions are satisfied exactly. The solution for isotropic cylinders under the axial PLST is covered as a special case. Numerical results show that the strain and stress distributions in the central region within half height and radius are relatively homogeneous, but strain and stress concentrations are usually developed near the point loads. The largest tensile strain and stress are always induced along the line joining the point loads, which gives theoretical explanation why most of the cylindrical specimens are split apart along the line joining the point loads under the axial PLST. In addition, by using envelope-function method, the effect of strain on the valence-band structure of Si_1?xGe_x alloy is analyzed. It is found that strain changes the band quantum gap and the shape of constant energy surfaces of the heavy-hole and the light-hole bands of Si_1?xGe_x alloy.     

A novel method in determination of dynamic fracture toughness under mixed mode I/II impact loading

The objective of this paper is to propose a novel methodology for determining dynamic fracture toughness (DFT) of materials under mixed mode I/II impact loading. Previous experimental investigations on mixed mode fracture have been largely limited to qusi-static conditions, due to difficulties in the generation of mixed mode dynamic loading and the precise control of mode mixity at crack tip, in absence of sophisticated experimental techniques. In this study, a hybrid experimental–numerical approach is employed to measure mixed mode DFT of 40Cr high strength steel, with the aid of the split Hopkinson tension bar (SHTB) apparatus and finite element analysis (FEA). A fixture device and a series of tensile specimens with an inclined center crack are designed for the tests to generate the components of mode I and mode II dynamic stress intensity factors (DSIF). Through the change of the crack inclination angle β (=90°, 60°, 45°, and 30°), the K_II/K_I ratio is successfully controlled in the range from 0 to 1.14. A mixed mode I/II dynamic fracture plane, which can also exhibit the information of crack inclination angle and loading rate at the same time, is obtained based on the experimental results. A safety zone is determined in this plane according to the characteristic line. Through observation of the fracture surfaces, different fracture mechanisms are found for pure mode I and mixed mode fractures.