PARTITION OF UNITY FINITE ELEMENT METHOD FOR SHORT WAVE PROPAGATION IN SOLIDS

A partition of unity finite element method for numerical simulation of short wave propagation in solids is presented. The finite element spaces were constructed by multiplying the standard isoparametric finite element shape functions, which form a partition of unity, with the local subspaces defined on the corresponding shape functions, which include a priori knowledge about the wave motion equation in trial spaces and approximately reproduce the highly oscillatory properties within a single element. Numerical examples demonstrate the performance of the proposed partition of unity finite element in both computational accuracy and efficiency.

A version of Hill's lemma for Cosserat continuum

On the basis of Hill＇s lemma for classical Cauchy continuum, a version of Hill＇s lemma for micro-macro homogenization modeling of heterogeneous Cosserat continuum is presented in the flame of average-field theory. The admissible boundary conditions required to prescribe on the representative volume element for the modeling are extracted and discussed to ensure the satisfaction of Hill-Mandel energy condition and the first-order average field theory.

PARTITION OF UNITY FINITE ELEMENT METHOD FOR SHORT WAVE PROPAGATION IN SOLIDS

A partition of unity finite element method for numerical simulation of short wave propagation in solids is presented. The finite element spaces were constructed by multiplying the standard isoparametric finite element shape functions, which form a partition of unity, with the local subspaces defined on the corresponding shape functions, which include a priori knowledge about the wave motion equation in trial spaces and approximately reproduce the highly oscillatory properties within a single element. Numerical examples demonstrate the performance of the proposed partition of unity finite element in both computational accuracy and efficiency.

PARAMETRIC VARIATIONAL PRINCIPLE BASED ELASTIC-PLASTIC ANALYSIS OF HETEROGENEOUS MATERIALS WITH VORONOI FINITE ELEMENT METHOD

The Voronoi cell finite element method （VCFEM） is adopted to overcome the limitations of the classic displacement based finite element method in the numerical simulation of heterogeneous materials. The parametric variational principle and quadratic programming method are developed for elastic-plastic Voronoi finite element analysis of two-dimensional problems. Finite element formulations are derived and a standard quadratic programming model is deduced from the elastic-plastic equations. Influence of microscopic heterogeneities on the overall mechanical response of heterogeneous materials is studied in detail. The overall properties of heterogeneous materials depend mostly on the size, shape and distribution of the material phases of the microstructure. Numerical examples are presented to demonstrate the validity and effectiveness of the method developed.

TIME DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR DYNAMIC ANALYSES IN SATURATED PORO-ELASTO-PLASTIC MEDIUM

A time-discontinuous Galerkin finite element method for dynamic analyses in saturated poro-elasto-plastic medium is proposed.As compared with the existing discontinuous Galerkin finite element methods,the distinct feature of the proposed method is that the continuity of the displacement vector at each discrete time instant is automatically ensured,whereas the discontinuity of the velocity vector at the discrete time levels still remains.The computational cost is then obviously reduced, particularly,for material non-linear problems.Both the implicit and explicit algorithms to solve the derived formulations for material non-linear problems are developed.Numerical results show a good performance of the present method in eliminating spurious numerical oscillations and providing with much more accurate solutions over the traditional Galerkin finite element method using the Newmark algorithm in the time domain.

Measurement of length-scale and solution of cantilever beam in couple stress elasto-plasticity

Owing to the absence of proper analytical solution of cantilever beams for couple stress/strain gradient elasto-plastic theory, experimental studies of the cantilever beam in the micro-scale are not suitable for the determination of material length-scale. Based on the couple stress elasto-plasticity, an analytical solution of thin cantilever beams is firstly presented, and the solution can be regarded as an extension of the elastic and rigid-plastic solutions of pure bending beam. A comparison with numerical results shows that the current analytical solution is reliable for the case of σ0 〈〈 H 〈〈 E, where σ0 is the initial yield strength, H is the hardening modulus and E is the elastic modulus. Fortunately, the above mentioned condition can be satisfied for many metal materials, and thus the solution can be used to determine the material length-scale of micro-structures in conjunction with the experiment of cantilever beams in the micro-scale.

STATIONARY DISCONTINUITY AND FLUTTER INSTABILITY OF WAVE PROPAGATION IN ELASTO-PLASTIC SATURATED POROUS MEDIA

In the present work a model based on the Biot theory for simulating coupled hydrodynamic behavior mi saturated porous media is utilized with integration of the inertial coupling effect between the solid-fluid phases of the media into the model. The non-associated Drucker-Prager criterion to describe nonlinear constitutive behavior of pressure dependent elasto-plasticity for the media is particularly considered. With no consideration of compressibility of solid grains and the pore fluid, the discontinuity and instability of the wave propagation in saturated porous media axe analyzed for the plane strain problems in detail. The critical conditions of stationary discontinuity and flutter instability in the wave propagation are given. The results and conclusions obtained by the present work will provide some bases or clues for overcoming the difficulties in numerical modeling of wave propagation in the media subjected to dynamic loading.

Computational multiscale methods for granular materials

The fine-scale heterogeneity of granular material is characterized by its polydisperse microstructure with randomness and no periodicity. To predict the mechanical response of the material as the microstructure evolves, it is demonstrated to develop computational multiscale methods using discrete particle assembly-Cosserat continuum modeling in micro- and macro- scales,respectively. The computational homogenization method and the bridge scale method along the concurrent scale linking approach are briefly introduced. Based on the weak form of the Hu-Washizu variational principle, the mixed finite element procedure of gradient Cosserat continuum in the frame of the second-order homogenization scheme is developed. The meso-mechanically informed anisotropic damage of effective Cosserat continuum is characterized and identified and the microscopic mechanisms of macroscopic damage phenomenon are revealed. c 2013 The Chinese Society of Theoretical and Applied Mechanics. [doi： 10.1063/2.1301101]

TIME PRECISE INTEGRATION METHOD FOR CONSTRAINED NONLINEAR CONTROL SYSTEM

For the constrained nonlinear optimal control problem, by taking the first term of Taylor series, the dynamic equation is linearized. Thus by, introducing into the dual variable (Lagrange multiplier vector), the dynamic equation can be transformed into Hamilton system from Lagrange system on the basis of the original variable. Under the whole state, the problem discussed can be described from a new view, and the equation can be precisely solved by, the time precise integration method established in linear dynamic system. A numerical example shows the effectiveness of the method.

Editorial: Computational mechanics of granular materials

Most of granular materials are highly heteroge- neous, composed of voids and particles with different sizes and shapes. Geological matter, soil and clay in nature, geo-structure, concrete, etc. are practical ex- amples among them. From the microscopic view, a lo- cal region in the medium is occupied by particles with small but finite sizes and granular material is naturally modeled as an assembly of discrete particles in contacts On the other hand, the local region is identified with a material point in the overall structure and this discon- tinuous medium can then be represented by an effective continuum on the macroscopic level

Prediction on dispersion in elastoplastic unsaturated granular media

This study focuses on the propagation of the plane wave in the elastoplastic unsaturated granular media,and the wave equations and dispersion equations are derived for the media under the framework of Cosserat theory.Due to symmetry,five different wave modes are considered and predicted for the elastoplastic unsaturated granular media based on the Cosserat theory,including two longitudinal waves,one rotational longitudinal wave and two coupled transverse–rotational transverse waves.The correspondence is discussed between these Cosserat wave modes and the classical wave modes.Based on the dispersion equations,the dispersion behaviors are obtained for the five Cosserat wave modes.The results indicated that the different stress-strain stages,including the elastic,hardening and softening stages,have obvious effect on the dispersion behaviors of the Cosserat wave modes.

A 3D pyramid spline element

In this paper,a 13-node pyramid spline element is derived by using the tetrahedron volume coordinates and the B-net method,which achieves the second order completeness in Cartesian coordinates.Some appropriate examples were employed to evaluate the performance of the proposed element.The numerical results show that the spline element has much better performance compared with the isoparametric serendipity element Q20 and its degenerate pyramid element P13 especially when mesh is distorted,and it is comparable to the Lagrange element Q27.It has been demonstrated that the spline finite element method is an efficient tool for developing high accuracy elements.

Functionally Graded Cellular Structure Design Using the Subdomain Level Set Method with Local Volume Constraints

Functional graded cellular structure(FGCS)usually shows superiormechanical behaviorwith lowdensity and high stiffness.With the development of additivemanufacturing,functional graded cellular structure gains its popularity in industries.In this paper,a novel approach for designing functionally graded cellular structure is proposed based on a subdomain parameterized level set method(PLSM)under local volume constraints(LVC).In this method,a subdomain level set function is defined,parameterized and updated on each subdomain independently making the proposed approach much faster and more cost-effective.Additionally,the microstructures on arbitrary two adjacent subdomains can be connected perfectly without any additional constraint.Furthermore,the local volume constraint for each subdomain is applied by virtue of the augmented Lagrange multiplier method.Finally,several numerical examples are given to verify the correctness and effectiveness of the proposed approach in designing the functionally graded cellular structure.From the optimized results,it is also found that the number of local volume constraints has little influence on the convergence speed of the developed approach.

Patch test function for axisymmetric element of conventional and couple stress theory

The enhanced patch test proposed by Chen W J(2006) can be used to assess the convergence of the problem with non-homogeneous differential equations.Based on this theory,we establish the patch test function for axisymmetric elements of conventional and couple stress theories,and reach an important conclusion that the patch test function for axisymmetric elements cannot contain non-zero constant shear.

A 17-node quadrilateral spline finite element using the triangular area coordinates

Isopaxametric quadrilateral elements are widely used in the finite element method. However, they have a disadvantage of accuracy loss when elements are distorted. Spline functions have properties of simpleness and conformality. A 17onode quadrilateral element has been developed using the bivaxiate quaxtic spline interpolation basis and the triangular area coordinates, which can exactly model the quartic displacement fields. Some appropriate examples are employed to illustrate that the element possesses high precision and is insensitive to mesh distortions.

Enhanced patch test of finite element methods

Theoretically, the constant stress patch test is not rigorous. Also, either the patch test of non-zero constant shear for Mindlin plate problem or non-zero strain gradient curvature of the microstructures cannot be performed. To improve the theory of the patch test, in this paper, based on the variational principle with relaxed continuity requirement of nonconforming element for homogeneous differential equations, the author proposed the individual element condition for passing the patch test and the convergence condition of the element: besides passing the patch test, the element function should include the rigid body modes and constant strain modes and satisfy the weak continuity condition, and no extra zero energy modes occur. Moreover, the author further established a variational principle with relaxed continuity requirement of nonconforming element for inhomogene-ous differential equations, the enhanced patch test condition and the individual element condition. To assure the convergence of the element that should pass the enhanced patch test, the element function should include the rigid body modes and non-zero strain modes which satisfied the equilibrium equations, and no spurious zero energy modes occur and should satisfy new weak continuity condition. The theory of the enhanced patch test pro-posed in this paper can be applied to both homogeneous and inhomogeneous differential equations. Based on this theory, the patch test of the non-zero constant shear stress for Mindlin plate and the C0-1 patch test of the non-zero constant curvature for the couple stress/strain gradient theory were established.