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.

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.

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.

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.

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.

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.

Hydrodynamic performance of combined cylinders structure with dual arc-shaped porous outer walls

This study examines the hydrodynamic performance of short-crested wave interaction with a new porous cylindrical structure by using the scaled boundary finite element method （SBFEM）, which is a semi-analytical technique combining the advantages of the finite element method and the boundary element method and with its own special features as well. The cylindrical structure consists of dual arc-shaped porous outer cylinders circumscribing an impermeable inner cylinder. A central feature of the newly extended method is that two virtual outer cylinders extending the arc-shaped porous outer cylinders with the same centre are introduced and variable porous-effect parameters are also introduced for the two virtual cylinders, so that the final SBFEM quation still can be handled in a closed-form analytical manner in the radial direction and by a finite element approximation in the circumferential direction. The entire computational domain is divided into two bounded and one unbounded domains, and a variational principle formulation is used to derive the SBFEM equation in each sub-domain. The velocity potential in bounded and unbounded domains is formulated using sets of Bessel and Hankel functions respectively, and the unknown coefficients are determined from the matching conditions. The results of numerical verification show that the approach discretises only the outermost virtual cylinder with surface finite-elements and fewer elements are required to obtain very accurate results.Influences of the incident wave parameters and structural configurations on the hydrodynamics are examined.

An SBFEM-Based Model for Hydraulic Fracturing in Quasi-Brittle Materials

A numerical model based on the scaled boundary finite element method is devel- oped to simulate the hydraulic fracturing in concrete-like quasi-brittle materials using cohesive interface elements. The shadow domain method developed previously （Yang and Deeks in Eng Fract Mech 143（4）：333 354, 2007） is extended to consider crack-width-dependent hydraulic pres- sure and cohesive traction, so that the stress intensity factors caused by both crack-face forces are semi-analytically calculated separately in the same way. The crack propagation is determined by the criterion of KI ≥ 0, and the propagation direction by the linear elastic fracture mechanics criteria. Two examples of concrete structures are modeled, and the results are in good agreement with the experimental data and others numerical results.

The Static Solution for the Layered Piezoelectric Bounded Domain with Side Face Load by the Modified SBFEM

The static response of two-dimensional horizontal layered piezoelectric bounded domain with side face load was investigated.In this paper,the modified scaled boundary finite element method(SBFEM)is provided as an effective semi analytical methodology.The method is used to solve the static problem for the layered piezoelectric bounded domain.The scaling line definition extends the SBFEM to be more suitable for analyzing the multilayered piezoelectric bounded domain.It avoids the limitations of original SBFEM in modeling the horizontal layered bounded domain.The modified SBFEM governing equation with piezoelectric medium is derived by introducing Duality variable in the Hamilton system.This derivation technology makes the progress be concise.The novel displacement and electric governing equations of the modified SBFEM is given together by the first time.The node forces can be expressed as power exponent function with radial coordinate by introducing the auxiliary variable and using the eigenvalue decomposition.The novel modified SBFEM solution of layered bounded domain with piezoelectric medium is solved.The new power expansion function of layered piezoelectric medium with side face load is proposed.This technology significantly extends the application range of modified SBFEM.The novel treatment of side face load for the layered piezoelectric bounded domain is proposed.Numerical studies are conducted to demonstrate the accuracy of proposed technique in handling with the static problem of layered bounded domain with piezoelectric medium for side face load.The influence of the side face load type and depth are discussed in detail.