Finite boundary effects in problem of a crack perpendicular to and terminating at a bimaterial interface

In this paper, the problem of a crack perpendicular to and terminating at an interface in bimaterial structure with finite boundaries is investigated. The dislocation simulation method and boundary collocation approach are used to derive and solve the basic equations. Two kinds of loading form are considered when the crack lies in a softer or a stiffer material, one is an ideal loading and the other one fits to the practical experiment loading. Complete solutions of the stress field including the T stress are obtained as well as the stress intensity factors. Influences of T stress on the stress field ahead of the crack tip are studied. Finite boundary effects on the stress intensity factors are emphasized. Comparisons with the problem presented by Chen et al. (Int. J. Solids and Structure, 2003, 40, 2731–2755) are discussed also.

Scaled Boundary Finite Element Analysis of Wave Passing A Submerged Breakwater

The scaled boundary finite element method （SBFEM） is a novel semi-analytical technique combining the advantage of the finite element method （FEM） and the boundary element method （BEM） with its unique properties. In this paper, the SBFEM is used for computing wave passing submerged breakwaters, and the reflection coeffcient and transmission coefficient are given for the case of wave passing by a rectangular submerged breakwater, a rigid submerged barrier breakwater and a trapezium submerged breakwater in a constant water depth. The results are compared with the analytical solution and experimental results. Good agreement is obtained. Through comparison with the results using the dual boundary element method （DBEM）, it is found that the SBFEM can obtain higher accuracy with fewer elements. Many submerged breakwaters with different dimensions are computed by the SBFEM, and the changing character of the reflection coeffcient and the transmission coefficient are given in the current study.

Application of scaled boundary finite element method in static and dynamic fracture problems

The scaled boundary finite element method （SBFEM） is a recently developed numerical method combining advantages of both finite element methods （FEM） and boundary element methods （BEM） and with its own special features as well. One of the most prominent advantages is its capability of calculating stress intensity factors （SIFs） directly from the stress solutions whose singularities at crack tips are analytically represented. This advantage is taken in this study to model static and dynamic fracture problems. For static problems, a remeshing algorithm as simple as used in the BEM is developed while retaining the generality and flexibility of the FEM. Fully-automatic modelling of the mixed-mode crack propagation is then realised by combining the remeshing algorithm with a propagation criterion. For dynamic fracture problems, a newly developed series-increasing solution to the SBFEM governing equations in the frequency domain is applied to calculate dynamic SIFs. Three plane problems are modelled. The numerical results show that the SBFEM can accurately predict static and dynamic SIFs, cracking paths and load-displacement curves, using only a fraction of degrees of freedom generally needed by the traditional finite element methods.

Dynamic Crack Propagation Analysis Using Scaled Boundary Finite Element Method

The prediction of dynamic crack propagation in brittle materials is still an important issue in many engineering fields. The remeshing technique based on scaled boundary finite element method(SBFEM) is extended to predict the dynamic crack propagation in brittle materials. The structure is firstly divided into a number of superelements, only the boundaries of which need to be discretized with line elements. In the SBFEM formulation, the stiffness and mass matrices of the super-elements can be coupled seamlessly with standard finite elements, thus the advantages of versatility and flexibility of the FEM are well maintained. The transient response of the structure can be calculated directly in the time domain using a standard time-integration scheme. Then the dynamic stress intensity factor(DSIF) during crack propagation can be solved analytically due to the semi-analytical nature of SBFEM. Only the fine mesh discretization for the crack-tip super-element is needed to ensure the required accuracy for the determination of stress intensity factor(SIF). According to the predicted crack-tip position, a simple remeshing algorithm with the minimum mesh changes is suggested to simulate the dynamic crack propagation. Numerical examples indicate that the proposed method can be effectively used to deal with the dynamic crack propagation in a finite sized rectangular plate including a central crack. Comparison is made with the results available in the literature, which shows good agreement between each other.

THE COUPLING OF BOUNDARY ELEMENT AND FINITE ELEMENT METHODS FOR THE EXTERIOR NONSTATIONARY NAVIER-STOKES EQUATIONS

In this paper, we represent a new numerical method for solving the nonstationary Stokes equations in an unbounded domain. The technique consists in coupling the boundary integral and finite element methods. The variational formulation and well posedness of the coupling method are obtained. The convergence and optimal estimates for the approximation solution are provided.

Comparison of perfectly matched layer and multi-transmitting formula artificial boundary condition based on hybrid finite element formulation

The theory of perfectly matched layer （PML） artificial boundary condition （ABC）, which is characterized by absorption any wave motions with arbitrary frequency and arbitrarily incident angle, is introduced. The construction process of PML boundary based on elastodynamic partial differential equation （PDE） system is developed. Combining with velocity-stress hybrid finite element formulation, the applicability of PML boundary is investigated and the numerical reflection of PML boundary is estimated. The reflectivity of PML and multi-transmitting formula （MTF） boundary is then compared based on body wave and surface wave simulations. The results show that although PML boundary yields some reflection, its absorption performance is superior to MTF boundary in the numerical simulations of near-fault wave propagation, especially in comer and large angle grazing incidence situations. The PML boundary does not arise any unstable phenomenon and the stability of PML boundary is better than MTF boundary in hybrid finite element method. For a specified problem and analysis tolerance, the computational efficiency of PML boundary is only a little lower than MTF boundary.

A New Formulation of the Scaled Boundary Finite Element Method for Heterogeneous Media:Application to Heat Transfer Problems

The solution to heat transfer problems in two-dimensional heterogeneous media is attended based on the scaled boundary finite element method(SBFEM)coupled with equilibrated basis functions(EqBFs).The SBFEM reduces the model order by scaling the boundary solution onto the inner element.To this end,tri-lateral elements are emanated from a scaling center,followed by the development of a semi-analytical solution along the radial direction and a finite element solution along the circumferential/boundary direction.The discretization is thus limited to the boundaries of the model,and the semi-analytical radial solution is found through the solution of an eigenvalue problem,which restricts the methods’applicability to heterogeneous media.In this research,we first extracted the SBFEM formulation considering the heterogeneity of the media.Then,we replaced the semi-analytical radial solution with the EqBFs and removed the eigenvalue solution step from the SBFEM.The varying coefficients of the partial differential equation(PDE)resulting from the heterogeneity of the media are replaced by a finite series in the radial and circumferential directions of the element.A weighted residual approach is applied to the radial equation.The equilibrated radial solution series is used in the new formulation of the SBFEM.

A COUPLING METHOD OF DIFFERENCE WITH HIGH ORDER ACCURACY AND BOUNDARY INTEGRAL EQUATION FOR EVOLUTIONARY EQUATION AND ITS ERROR ESTIMATES

In the present paper, a new numerical method for solving initial-boundary value problems of evolutionary equations is proposed and studied, combining difference method with high accuracy with boundary integral equation method. The numerical approximate schemes for both problems on a bounded or unbounded domain in R3 are proposed and their prior error estimates are obtained.

Mathematical analysis of EEP method for one-dimensional finite element postprocessing

For a class of two-point boundary value problems, by virtue of onedimensional projection interpolation, it is proved that the nodal recovery derivative obtained by Yuan＇s element energy projection （EEP） method has the accuracy O（h^min{2k,k＋4}） The theoretical analysis coincides the reported numerical results.

Evaluation of Stress Intensity Factors for Multiple Cracked Circular Disks Under Crack Surface Tractions with SBFEM

Stress intensity factors （SIFs） for the cracked circular disks under different distributing surface tractions are evaluated with the scaled boundary finite element method （SBFEM）. In the SBFEM, the analytical advantage of the solution in the radial direction allows SIFs to be directly determined from its definition, therefore no special crack-tip treatment is necessary. Furthermore anisotropic material behavior can be treated easily. Different distributions of surface tractions are considered for the center and double-edge-cracked disks. The benchmark examples are modeled and an excellent agreement between the results in the present study and those in published literature is found. It shows that SBFEM is effective and possesses high accuracy. The SIFs of the cracked orthotropic material circular disks subjected to different surface tractions are also evaluated. The technique of substructure is applied to handle the multiple cracks problem.

Evaluation of Stress Intensity Factors Subjected to Arbitrarily Distributed Tractions on Crack Surfaces

The stress intensity factors （SIF） considering arbitrarily distributed surface tractions are evaluated based on the sealed boundary finite element method （SBFEM）. The semi-analytical solving process for the stress intensity factors including the effects of surface tractions is presented. Provided are the numerical examples for the evaluation of mode I and Ⅱ stress intensity factors with linear and non-linear distributing forces loaded on the crack surfaces. The crack problems of anisotropy and bimaterial interface are also studied and the stress intensity factors of single-edge-cracked orthotropic material and bi-material interface problems with surface tractions are calculated. Comparisons with the analytical solutions show that the proposed approach is effective and possesses high accuracy.

A Temporally Piecewise Adaptive Scaled Boundary Finite Element Method for Solving the Fuzzy Uncertain Viscoelastic Problems

The numerical solutions for uncertain viscoelastic problems have important theo- retical and practical significance. The paper develops a new approach by combining the scaled boundary finite element method （SBFEM） and fuzzy arithmetic. For the viscoelastic problems with zero uncertainty, the SBFEM and the temporally piecewise adaptive algorithm is employed in the space domain and the time domain, respectively, in order to provide an accurate semi- analytical boundary-based approach and to ensure the accuracy of discretization in the time domain with different sizes of time step at the same time. The fuzzy arithmetic is used to address the uncertainty analysis of viscoelastic material parameters, and the transformation method is used for computation with the advantages of effectively avoiding overestimation and reducing the computational costs. Numerical examples are provided to test the performance of the proposed method. By comparing with the analytical solutions and the Monte Carlo method, satisfactory results are achieved.

Transient heat conduction analysis using the NURBS-enhanced scaled boundary finite element method and modified precise integration method

The Non-uniform rational B-spline （NURBS） enhanced scaled boundary finite element method in combination with the modified precise integration method is proposed for the transient heat conduction problems in this paper. The scaled boundary finite element method is a semi-analytical technique, which weakens the governing differential equations along the circumferential direction and solves those analytically in the radial direction. In this method, only the boundary is discretized in the finite element sense leading to a re- duction of the spatial dimension by one with no fundamental solution required. Neverthe- less, in case of the complex geometry, a huge number of elements are generally required to properly approximate the exact shape of the domain and distorted meshes are often un- avoidable in the conventional finite element approach, which leads to huge computational efforts and loss of accuracy. NURBS are the most popular mathematical tool in CAD industry due to its flexibility to fit any free-form shape. In the proposed methodology, the arbitrary curved boundary of problem domain is exactly represented with NURBS basis functions, while the straight part of the boundary is discretized by the conventional Lagrange shape functions. Both the concepts of isogeometric analysis and scaled boundary finite element method are combined to form the governing equations of transient heat conduction analy- sis and the solution is obtained using the modified precise integration method. The stiffness matrix is obtained from a standard quadratic eigenvalue problem and the mass matrix is determined from the low-frequency expansion. Finally the governing equations become a system of first-order ordinary differential equations and the time domain response is solved numerically by the modified precise integration method. The accuracy and stability of the proposed method to deal with the transient heat conduction problems are demonstrated by numerical examples.

Short-Crested Waves Interaction with A Concentric Porous Cylinder System with Partially Porous Outer Cylinder

In this paper, based on the linear wave theory, the interaction of short-crested waves with a concentric dual cylindrical system with a partially porous outer cylinder is studied by using the scaled boundary finite element method （SBFEM）, which is a novel semi-analytical method with the advantages of combining the finite element method （FEM） with the boundary element method （BEM）. The whole solution domain is divided into one unbounded sub-domain and one bounded sub-domain by the exterior cylinder. By weakening the governing differential equation in the circumferential direction, the SBFEM equations for both domains can be solved analytically in the radial direction. Only the boundary on the circumference of the exterior porous cylinder is discretized with curved surface finite elements. Meanwhile, by introducing a variable porous-effect parameter G, non-homogeneous materials caused by the complex configuration of the exterior cylinder are modeled without additional efforts. Comparisons clearly demonstrate the excellent accuracy and computational efficiency associated with the present SBFEM. The effects of the wide range wave parameters and the structure configuration are examined. This parametric study will help determine the various hydrodynamic effects of the concentric porous cylindrical structure.

MODE III 2-D FRACTURE ANALYSIS BY THE SCALED BOUNDARY FINITE ELEMENT METHOD

The scaled boundary finite element method （SBFEM） is a novel semi-analytical technique that combines the advantages of the finite element method and the boundary element method with unique properties of its own. This method has proven very efficient and accurate for determining the stress intensity factors （SIFs） for mode I and mode II two-dimensional crack problems. One main reason is that the SBFEM has a unique capacity of analytically representing the stress singularities at the crack tip. In this paper the SBFEM is developed for mode III （out of plane deformation） two-dimensional fracture anMysis. In addition, cubic B-spline functions are employed in this paper for constructing the shape functions in the circumferential direction so that higher continuity between elements is obtained. Numerical examples are presented at the end to demonstrate the simplicity and accuracy of the present approach for mode Ⅲ two-dimensional fracture analysis.

THE COUPLING METHOD OF FINITE ELEMENTS AND BOUNDARY ELEMENTS FOR RADIATION PROBLEM

The paper presents the variational formulation and well posedness of the coupling method offinite elements and boundary elements for radiation problem. The convergence and optimal errorestimate for the approximate solution and numerical experiment are provided.

Interaction among fractures and stress field computation of fracture systems

The interaction problem among fractures under the action of compressional stress is studied in this paper by using the finite element method and boundary element method respectively.The mechanical criteria which differentiate between the independent fractures and fracture systems and their computation methods are presented in this paper.The proportional conditions between length and spacing of fractures that exist interaction for several kinds of fracture groups of different geometric arrangement are given.The effect of interaction among fractures on the displacement field,stress field and strain energy distribution are computed.The relations between the fracture system of conjugate array and conjugate earthquakes are also discussed in this paper.

VARIABLE MESH FINITE DIFFERENCE METHOD FOR SELF-ADJOINT SINGULARLY PERTURBED TWO-POINT BOUNDARY VALUE PROBLEMS

A numerical method based on finite difference method with variable mesh is given for self-adjoint singularly perturbed two-point boundary value problems. To obtain parameter- uniform convergence, a variable mesh is constructed, which is dense in the boundary layer region and coarse in the outer region. The uniform convergence analysis of the method is discussed. The original problem is reduced to its normal form and the reduced problem is solved by finite difference method taking variable mesh. To support the efficiency of the method, several numerical examples have been considered.

A FINITE ELEMENT/BOUNDARY ELEMENT——MODIFIED MODAL DECOMPOSITION METHOD FOR VIBRATION AND SOUND RADIATION FROM SUBMERGED SHELL OF REVOLUTION

A finite element / boundary element-modified modal decomposition method (FBMMD) is presented for predicting the vibration and sound radiation from submerged shell of revolution. Improvement has been made to accelerate the convergence to FBMD method by means of introducing the residual modes which take into accaunt the quasi -state contributiort of all neglected modes. As an example, the vibration and sound radiation of a submerged spherical shell excited by axisymmetric force are studied in cases of ka=l,2,3 and 4. From the calculated results we see that the FBMMD method shows a significant improvement to the accuracy of surface sound pressure, normal displacement and directivity patterns of radiating sound, especially to the directivity patterns.

ON THE FINITE DIFFERENCE METHODS FOR TWO DIMENSIONAL TURBULENCE BOUNDARY LAYER

In this paper,the author first establishes the general finite difference formula for the governing equations of the turbulent average velocities in a steady two dimensional incompressible fluid boundary layer-inner layer.Next, three key parameters of the difference scheme are determined respectively by several simple flow models with known analytical solutions.Finally a special five points difference system is given and its application value is showed by a numerical example for the vertical velocity distribution in an Ekman's layer.