Semi-analytical Scaled Boundary Finite Element Method Research Articles

An efficient semi-analytical static and free vibration analysis of laminated and sandwich beams based on linear elasticity theory

Based on the two-dimensional (2D) elastic theory without enforcing any beam assumption, an efficient semi-analytical scaled boundary finite element method (SBFEM) is proposed to solve the bending and free vibration responses of composite laminated and sandwich beams under the mechanical load. The scaled center is placed at infinity, which produces the accurate result by discretizing only the longitudinal direction of the beam structure treated as a one-dimensional (1D) discretization problem. A new kind of 1D high-order spectral element shape functions with the advantages of high accuracy and superior convergence is introduced in SBFEM coordinate system to approximate the geometric model and corresponding variables. The principle of weighted residual in conjunction with the Green’s theorem are applied to obtain the SBFEM governing equation of each layer with respect to radial displacement fields. The solution of equation is indicated analytically by a matrix exponential function, which can be accurately solved by using the precise integration technique (PIT). Finally, an effective and simple stiffness matrix is obtained. By comparing two examples with the solutions based on the finite element method (FEM), the results show that the proposed method has good accuracy and rapid convergence with only a few meshes. The numerical examples are given to investigate the parametric effects of the stacking sequence, thickness ratio, boundary condition, and load form on the variation of the displacement, stress and natural frequency. The results validate that the present technique is also applicable to the complex beam structure with softcore layer inside.

A semi‐analytical approach for the efficient predicition of thermally induced interface crack initiation at bi‐material free edges

AbstractIn the present study, thermally induced interlaminar crack initiation emanating from the free edge of a bi‐material junction is investigated by the example of a uniformly cooled epoxy‐glass bi‐material specimen taken from literature. For this purpose, the coupled stress and energy criterion within the framework of finite fracture mechanics is applied. Interlaminar stresses and energy dissipation due to crack onset are determined efficiently using the semi‐analytical scaled boundary finite element method (SBFEM). Moreover, the effect of the layer thicknesses on crack initiation is studied. For this purpose, dimensional analyses yield simple scaling laws for interlaminar stresses and energy dissipation with respect to the layer thickness such that only few model evaluations are required. The obtained results are compared to a finite element reference solution. For validation purposes, a cohesive zone model is applied additionally. Finally, the obtained predictions are compared to experimental findings from literature.

Buckling analysis of three-dimensional functionally graded sandwich plates using two-dimensional scaled boundary finite element method

For the stability analysis of functionally graded material (FGM) sandwich plates, a semi-analytical scaled boundary finite element method (SBFEM) is established within the frame work of layerwise theory in this paper. Two different configurations of FGM sandwich plates are studied, one of which consists of ceramic-rich core and FGM face sheets, and another involves metal-rich and ceramic-rich material face sheets as well as a FGM core, and the modulus of elasticity for each FGM layer is assumed to be changed continuously along thickness according to a power-law distribution. The layerwise theory performed in present work is based on the three-dimensional theory of elasticity for each individual layer satisfying the displacement continuity boundary conditions at layer interfaces, and the two-dimensional high order spectral element with three degrees of freedom per node is employed to discrete the reference surface of each individual layer. And then, based on the principle of virtual work, the SBFEM governing equation for each individual layer of FGM sandwich plates is derived, and a dual variable including the displacement and internal nodal force is used to reduce the governing equations to a system of first-order ordinary differential equation, which is solved analytically by a general approach. Finally, the critical buckling load can be obtained by solving the eigenvalue equation once the bending stiffness matrix and geometric stiffness matrix of FGM sandwich plates are determined. Fast rate of convergence of the proposed formulations is confirmed and comparison studies are presented to construct its high accuracy and excellent predictive capability. Furthermore, effects of gradient index, side-to-thickness ratio, layer configuration, and boundary conditions on dimensionless critical buckling loads of FGM sandwich plates are also studied. The proposed semi-analytical approach is simple in program implementation, accurate and computationally efficient.

A study on full interaction of water waves with moored rectangular floating breakwater by applying 2DV scaled boundary finite element method

The problem of interaction between water wave and moored rectangular floating breakwater is studied using the semi-analytical Scaled Boundary Finite Element Method (SBFEM) in a two-dimensional vertical plane. The current study applies a generalized solution process by locating the scaling center within the bounded sub-domains instead of the body of structure, resulting in a homogeneous SBFEM equation. In spite of the merits of such choice for the scaling center that makes possible the analysis of non-rectangular objects, the method is changed to numerical-analytical one with its consequences. Such numerical solution in the problem of water wave radiation and diffraction is validated by comparing the results with previous reported studies. The generality and robustness of the current SBFEM-based model is also examined for non-rectangular cross-sections of the floating breakwater. Focusing on the proposed SBFEM-based model, the full wave-rigid body interaction is analyzed for a moored rectangular floating structure. Moreover, in order to make the linear numerical modeling accurate, taking into account the nonlinear aspects of the interaction problem, the artificial damping coefficients are quantified, and the relevant formula is derived.

Nonlocal damage modelling by the scaled boundary finite element method

Summary The progressive damage of structures is fruitfully simulated by using the semi-analytical scaled boundary finite element method (SBFEM). The integral-type nonlocal model combined with the isotropic damage model is extended to eliminate the mesh sensitivity concerning the strain localization. An automatic and efficient quadtree mesh generation algorithm is employed to refine the localized damage process zone (DPZ) and reduce the number of degrees of freedom (DOFs). Owing to the salient advantage of the SBFEM in using arbitrary polygonal subdomains, side-effects associated with hanging nodes can be eliminated. Furthermore, the computational effort of strain/stress field and damage variables can be considerably saved in the framework of the SBFEM. Four numerical benchmarks with regular-shaped domain and a porous plate with irregular holes are simulated to demonstrate the effectiveness and robustness of the proposed approach.

A Non-linear Scaled Boundary Finite-Element Analysis Applied to Geotechnical Problems

Soils as highly nonlinear materials with a non-homogeneous and anisotropic nature often display behavioral complexities under loads. When a soil mass is subjected to a loading, the soil displacements may either be linear or non-linear depending on the amount of load or strength of the soil. Several constitutive models have been so far presented for modeling geotechnical problems. These models have taken very large steps in analyzing and predicting the real behavior of soils. Two main approaches to analyze the stability problems in geotechnical engineering are limit and numerical methods. The limitations associated with the limit methods and inventions of numerical techniques have paved the way for extensive use of numerical methods in analysis of geotechnical problems. In this paper, the semi-analytical scaled boundary finite-element method is employed to perform the elasto-plastic analysis of geotechnical engineering problems. The well-known linear elastic-perfect plastic Mohr–Coulomb criterion is used as the constitutive model. The polygon elements are utilized to discretize the computational domain. Having presented the detailed formulation and solution procedures, three numerical examples are analyzed. The accuracy of the SBFEM in the elasto-plastic analysis of geo-materials is validated by comparing the results with the FEM ones. Excellent agreement is achieved.

Sloshing of liquid in partially liquid filled toroidal tank with various baffles under lateral excitation

A technique to study effect of various baffles on liquid oscillations in partially filled rigid toroidal tanks is first developed by extending the semi-analytical scaled boundary finite element method (SBFEM), which utilizes the advantages of both the boundary element (BEM) and finite element methods (FEM). As found in the BEM, only the boundary is discretized to reduce the space by one, and no fundamental solution is needed unlike the BEM. The calculated liquid domain is divided into several simple sub-domains so that the liquid velocity potential in each liquid sub-domain becomes the class C1 with continuity boundary conditions. Based on the linear potential theory and weighted residual method, the semi-analytical solutions of the liquid velocity potential corresponding to each sub-domain are obtained by means of SBFEM, where the geometry of each sub-domain is transformed into scaled boundary coordinates, including the radial and circumferential coordinates by using a scaling centre, and the finite-element approximation of the circumferential coordinate yields the analytical equation in the radial coordinate. By discretizing the flow boundaries, the integral equation governed on the boundary is formulated into a general matrix eigenvalue problem. Based on the eigenvalue problem and multimodal method, an efficient methodology is adopted to computer the sloshing masses and sloshing force. Accuracy, simple and fast numerical computations are observed by the convergence study, and excellent agreements have been achieved in the comparison of results obtained by the proposed approach with other methods. Meanwhile, several baffle configurations are considered including the horizontal bottom-mounted and surface-piercing ring baffles as well as their combination form, bottom-mounted and surface-piercing ring baffles as well as their combination, and free surface-touching baffles. The effects of baffled arrangement, the ratio b/a of elliptical cross section, liquid fill level, and baffles' length upon the sloshing frequencies, the associated sloshing mode shapes and sloshing forces are investigated in detail and some conclusions are outlined. The results show that the present method allows for the simulation of complex 3D sloshing phenomena using a relative small number of degrees of freedom while the mesh consists of two-dimensional elements only.

Transient Sloshing in Partially Filled Laterally Excited Horizontal Elliptical Vessels With T-Shaped Baffles

The transient sloshing in laterally excited horizontal elliptical containers with T-shaped baffles is first investigated by using a novel semi-analytical scaled boundary finite-element method (SBFEM). The proposed method combines the advantages of the finite-element and the boundary element methods (BEMs) with unique properties of its own, in which a new coordinate system including the circumferential local coordinate and the radial coordinate has been established. Only the boundary of the computational domain needs to be discretized in the circumferential direction as the same as the BEM and the solution in the radial direction is analytical. Assuming ideal, irrotational flow and small-amplitude free-surface elevation, the formulations (using a new variational principle formulation) and solutions of SBFEM equations for an eigenvalue problem under zero external excitation (free sloshing problem) are derived in detail. Subsequently, based on an appropriate decomposition of the container-fluid motion, and considering the eigenvalues and eigenmodes of the above eigenvalue problem, an efficient methodology is proposed for externally induced sloshing through the calculation of the corresponding sloshing masses and liquid motion. Several numerical examples are presented to demonstrate the simplicity, versatility, and applicability of the SBFEM during the simulation of sloshing problems of complex containers, and excellent agreement with the other methods is observed. Meanwhile, three T-shaped baffle configurations are considered including surface-piercing baffle, bottom-mounted baffle and their combination form, and Y-shaped configuration evolved from that of T-shaped baffle has been taken into consideration as well. The liquid fill level, arrangement and length of those baffles affecting the sloshing masses, and liquid motion are investigated in detail. The results also show that the present method can easily solve the singularity problems analytically by choosing the scaling center at the tip of the baffles and allows for the simulation of complex sloshing phenomena using far less number of degrees-of-freedom.

An efficient FE–SBFE coupled method for mesoscale cohesive fracture modelling of concrete

This study develops a method coupling the finite element method (FEM) and the scaled boundary finite element method (SBFEM) for efficient meso-scale fracture modelling of concrete for the first time. In this method, the aggregates are modelled by SBFE polygons with boundaries discretised only, while the mortar matrix is modelled by conventional finite elements. The semi-analytical SBFEM is implemented in ABAQUS by a user-defined element subroutine for the first time. Nonlinear cohesive interface elements with normal and shear traction-separation constitutive laws are pre-inserted within the mortar and on the aggregate-mortar interfaces to simulate potential cracks. Various meso-structures generated from both random aggregates and X-ray computed tomography images are modelled. The results demonstrate that the coupled method leads to considerable reductions in degrees of freedom and computational time against the conventional FEM, and these reductions become more significant when the aggregate volume fraction increases. The modelled crack paths and load-carrying capacities of a three-point bending beam and an L-shaped panel are in excellent agreement with the experimental data.

Modeling crack propagation with the extended scaled boundary finite element method based on the level set method

The extended scaled boundary finite element method (X-SBFEM) based on the level set method (LSM) is proposed in this paper to combine the advantages of the scaled boundary finite element method (SBFEM) and the extended finite element method (XFEM). The level set method (LSM) algorithm is applied to further develop the X-SBFEM, especially for the crack propagation problem. The Heaviside enrichment function is used to represent a jump across a discontinuity surface in a split element, and the non-smooth behavior around the crack tip is described using the semi-analytical SBFEM. The stiffness of the region containing the crack tip is computed directly, and the generalized stress intensity factors of many types of singularities are obtained directly from their definitions using consistent formulas. In the numerical simulations, a square plate with an edge crack under tension, a three-point bending beam, a four-point shear beam and a dam (the Koyna dam) with a single propagating crack are modeled. The results show that the proposed X-SBFEM is capable of calculating the stress intensity factors of cracks and predicting crack trajectories and load–displacement relations accurately. An analysis of the sensitivity of the parameters is employed to demonstrate that various mesh densities and crack propagation step lengths led to consistent results.

A numerical study of the effects of the T-shaped baffles on liquid sloshing in horizontal elliptical tanks

A semi-analytical scaled boundary finite-element method (SBFEM) is firstly extended to study the effects of the T-shaped baffle on liquid sloshing in horizontal elliptical tanks. The proposed method is based on the finite-element technology and gains the advantages of the boundary element method as well. Only the boundary is discretized reducing the spatial discretization by one, no fundamental solution is required and singularity problems can be modeled rigorously. Based on linear potential theory, the formulations (using new variational principle formulation) and the solutions of SBFEM equations for the sloshing problems are derived in details. As a key point, the solution for the inclusion of flow velocity along the baffles, which is called the side-faces of bounded domain in SBFEM, is also expressed in details. The accuracy, simplicity and convergence of the present method are demonstrated via numerical examples, and excellent agreement with the other method is observed. Meanwhile, three T-shaped baffled configurations are considered including surface-piercing baffle, bottom-mounted baffle and their combination form, and Y-shaped baffled configuration evolved from that of T-shaped baffle has been taken into consideration as well. The effects of liquid fill level, baffled arrangement and length of those baffles upon the sloshing frequencies, the associated sloshing mode shapes and sloshing wave height are investigated in details. The results also show that the present method has strong ability to resolve singularity problems analytically by choosing the scaling center at the tip of the baffles and allows for the simulation of complex sloshing phenomena using a very small number of degrees of freedom while the mesh consists of one-dimensional elements only.

AN ADAPTIVE TIME-STEPPING PROCEDURE BASED ON THE SCALED BOUNDARY FINITE ELEMENT METHOD FOR ELASTODYNAMICS

This study develops an adaptive time-stepping procedure of Newmark integration scheme for transient elastodynamic problems, based on the semi-analytical scaled boundary finite element method (SBFEM). In each time step, a posteriori local error estimator based on the linear distributed acceleration is employed to estimate the error caused by the time discretization. The total energy of the domain, consisting of the kinetic energy and the strain energy, is calculated semi-analytically. The time increment is automatically adjusted according to a simple criterion. Three examples with stress wave propagation were modeled. The numerical results show that the developed method is capable of limiting the local error estimator within specified targets by using an optimal time increment in each time step.

Short-crested waves interaction with a concentric cylindrical structure with double-layered perforated walls

This paper aims at extending the semi-analytical scaled boundary finite element method (SBFEM) to deal with the short-crested waves interaction with a surface-piercing concentric cylindrical structure, which consists of a solid inner cylinder and a coaxial double-layered perforated wall. The whole fluid domain is divided into three sub-domains including two bounded and one unbounded domains, and a variational principle formulation is used to derive the SBFEM equation in each sub-domain. Hankel functions and Bessel functions are chosen as the basis function for the solution of the unbounded and bounded domains, respectively. Although the structure adds a porous cylinder compared to that of Tao's work (2009), the SBFEM also needs to discretize only the outermost porous cylinder with curved surface finite-elements and keeps the radial differential equation solved completely analytically. However, the unknown coefficient vectors for solving the SBFEM equations increase compared with that of Tao's work. Thus we re-derive the solving process for the SBFEM equations. The results of numerical verification show that the present method yields excellent results with only a few nodes and quick convergence. The major factors including wave parameters and structure configuration that affect the wave forces, surface elevations and the diffracted wave contours are examined.

Calculation of transient dynamic stress intensity factors at bimaterial interface cracks using a SBFEM-based frequency-domain approach

A frequency-domain approach based on the semi-analytical scaled boundary finite element method (SBFEM) was developed to calculate dynamic stress intensity factors (DSIFs) at bimaterial interface cracks subjected to transient loading. Because the stress solutions of the SBFEM in the frequency domain are analytical in the radial direction, and the complex stress singularity at the bimaterial interface crack tip is explicitly represented in the stress solutions, the mixed-mode DSIFs were calculated directly by definition. The complex frequency-response functions of DSIFs were then used by the fast Fourier transform (FFT) and the inverse FFT to calculate time histories of DSIFs. A benchmark example was modelled. Good results were obtained by modelling the example with a small number of degrees of freedom due to the semi-analytical nature of the SBFEM.

A frequency-domain approach for modelling transient elastodynamics using scaled boundary finite element method

This study develops a frequency-domain method for modelling general transient linear-elastic dynamic problems using the semi-analytical scaled boundary finite element method (SBFEM). This approach first uses the newly-developed analytical Frobenius solution to the governing equilibrium equation system in the frequency domain to calculate complex frequency-response functions (CFRFs). This is followed by a fast Fourier transform (FFT) of the transient load and a subsequent inverse FFT of the CFRFs to obtain time histories of structural responses. A set of wave propagation and structural dynamics problems, subjected to various load forms such as Heaviside step load, triangular blast load and ramped wind load, are modelled using the new approach. Due to the semi-analytical nature of the SBFEM, each problem is successfully modelled using a very small number of degrees of freedom. The numerical results agree very well with the analytical solutions and the results from detailed finite element analyses.

Adaptive coupling of the finite-element and scaled boundary finite-element methods for non-linear analysis of unbounded media

A technique is developed for analysing elasto-plastic unbounded media by adaptively coupling the finite-element method with the scaled boundary finite-element method. The analysis begins with a finite-element mesh that tightly encloses the load–medium interface, capturing non-linearity in the very near field. The remainder of the problem is modelled accurately and efficiently using the semi-analytical scaled boundary finite-element method. Load increments are applied in the usual (finite-element) way and the plastic stress field grows outwards from the load–medium interface as the solution advances. If plasticity is detected at a Gauss point in the outer band of finite-elements, an additional band of finite-elements are added around the perimeter of the existing mesh and the scaled boundary finite-element domain is stepped out accordingly. This technique exploits the most attractive features of both the finite-element and scaled boundary finite-element methods. The technique is shown to be highly accurate and both user and computationally efficient.

Semi-analytical solution of Laplace’s equation in non-equilibrating unbounded problems

Some two-dimensional problems of elastostatics are governed by Laplace’s equation. Using the terminology of elastostatics, if the face loads and body loads are not self-equilibrating, even when the displacement at infinity is restricted to zero, displacements in the near field will be infinite. However, the stress field within the domain is well behaved, and is of practical interest. In this paper the semi-analytical scaled boundary finite-element method is extended to permit the analysis of such problems. The solutions in the primary variable so obtained include an infinite component, but the difference in value between any two points in the domain can be computed accurately. The method is also extended to solve the non-homogeneous form of Laplace’s equation.

Semi‐analytical elastostatic analysis of unbounded two‐dimensional domains

AbstractUnbounded plane stress and plane strain domains subjected to static loading undergo infinite displacements, even when the zero displacement boundary condition at infinity is enforced. However, the stress and strain fields are well behaved, and are of practical interest. This causes significant difficulty when analysis is attempted using displacement‐based numerical methods, such as the finite‐element method. To circumvent this difficulty problems of this nature are often changed subtly before analysis to limit the displacements to finite values. Such a process is unsatisfactory, as it distorts the solution in some way, and may lead to a stiffness matrix that is nearly singular. In this paper, the semi‐analytical scaled boundary finite‐element method is extended to permit the analysis of such problems without requiring any modification of the problem itself. This is possible because the governing differential equations are solved analytically in the radial direction. The displacement solutions so obtained include an infinite component, but relative motion between any two points in the unbounded domain can be computed accurately. No small arbitrary constants are introduced, no arbitrary truncation of the domain is performed, and no ill‐conditioned matrices are inverted. Copyright © 2002 John Wiley & Sons, Ltd.