Scaled Boundary Finite Element Research Articles

Adaptive scaled boundary finite element method for two/three-dimensional structural topology optimization based on dynamic responses

This paper presents an efficient solution for designing the topology of structures that can withstand dynamic loads. The method, called image/stereolithography (STL)-based adaptive scaled boundary finite element (SBFE), is a novel approach to topology optimization (TO) that is particularly effective for designing two/three-dimensional structures. The SBFE-based TO algorithm adopts a bi-evolutionary structural optimization (BESO) approach, which utilizes image/STL-based procedures to automatically adapt meshes and minimize mechanical compliance under varying volume fractions in the time domain. A key advantage of this approach is that it uses images of TO responses, rather than specific error functions, to guide the automatic adaptive mesh schemes. The method also incorporates a quadtree/octree mesh refinement to address the problem of hanging nodes, ensuring fast mesh convergence during the optimization process. Furthermore, a high-order implicit time integration technique provides a close approximation of equivalent static loads, allowing for the accurate mapping of displacement fields under actual dynamic responses at each time step. The effectiveness of this method is demonstrated through a variety of numerical examples and benchmarks.

Automatic Image-Based SBFE-BESO Approach for Topology Structural Optimization

This paper presents automatic image-based topology optimization with the highest mechanical stiffness. A so-called bi-evolutionary structural optimization is encoded within the scaled boundary finite element (SBFE) framework, incorporating digital image-based quadtree (2D) and octree (3D) decomposing mesh constructions. The information on both geometry and material distribution is transparently transferred using the standard tessellation language (STL) format of digital images, which can be directly modelled for stress analyses. The polytope SBFE method allows for a computationally advantageous model construction with hanging nodes. A convolution filter scheme is applied for colour intensity adjustment between solid and void regions, leading to multi-levels in the quadtree/octree hierarchy. Combined with the convolution filter method, the matrix precomputation technique, within the polytope SBFE decomposition framework enables the automatic adaptive (digital image-based) model construction from the limited number of master cells with balanced quadtree/octree meshes. Hence, the highly efficient analysis-ready design scheme for 2D and 3D structures is carried out. The proposed approach significantly reduces the number of degrees of freedom required, giving thus the computationally efficient efforts that are seen to enhance the high-resolution optimal topology of the practical-scale structures.

Application of scaled boundary finite element method in failure probability estimation of square nanographene sheet

In recent years, so much attention has been drawn to the field of nano computational material science. Due to the extensive applications of nanographene sheets in various industries such as aerospace industry, fracture analysis of square nanographene sheets is of crucial importance. Stochastic analysis is widely applied in fracture mechanics because it helps the results to be more realistic. In the present work, probabilistic fracture analysis of nanographene sheets is done using scaled boundary finite element (SBFEM) that is a novel semi-analytical method. In order to apply the stochastic method, we need to select certain statistical parameters, the ones considered here include crack length and failure stress. The results using the stochastic response method are compared with those of the Monte Carlo, which shows good agreement. Average, variance, third and fourth moments of stress and the fracture probability are calculated for all the bonds in two states of crack angle 0°, 90°. The results indicate that the maximum crack-tip fracture probability for central crack θ = 90° is 95%, while for θ = 0° equals 58%. For central crack θ = 90°, the failure probability in crack tip bonds, bonds adjacent to the crack edge and other bonds equal 95%, 88% and 65%, respectively. For nanographene sheet having the central crack θ = 90°, probabilistic density function is Gaussian, which reaches the highest point at the displacement being 0.990 pm. The third moment of the stress turned out to be zero, which shows that the input uncertain variables are all normal. The results obtained from computing the failure probability of all the bonds indicates that the highest failure probability is in the tip and edges of the crack. The stochastic fracture sensitivity versus crack length is calculated which shows that the maximum sensitivity in central crack happens when θ = 90° and crack length ranges between 0.3 and 0.4.

A coupled SBFETI-BDEs method for solving 2D static and dynamic contact problem with friction

In this study, we propose a coupled approach called the SBFETI-BDEs method for solving the 2D elastic frictional contact problem. The contact system is decomposed into multiple subdomains, similar to the finite-element tearing and interconnecting (FETI) algorithm. Each subdomain can be regarded as a scaled boundary finite element (SBFE), where only the boundaries are discretized. To avoid computing the global contact flexibility matrix using the force method, we consider the generalized inverse of the stiffness matrix for each subdomain as a subcontact-flexibility matrix. We uniformly formulate the contact equations, displacement continuity equations, and the equilibrium equations of force for the floating subdomains, as a B-differentiable equations (BDEs) system, which we solve using the B-differentiable damped Newton-Raphson method that possesses a global convergence property. The proposed method offers increased computational efficiency without compromising accuracy when compared to the original BDEs algorithm without domain decomposition. We present the dynamic scheme of the SBFETI-BDEs method within the framework of the HHT time-integration scheme, where the equilibrium equations of force for the floating subdomains disappear due to their nonsingular equivalent stiffness matrices. Numerical examples are provided to demonstrate the convergence, accuracy, efficiency, and capability in solving engineering problems of the proposed method.

Image based probabilistic slope stability analysis of soil layer interface fluctuations with Brownian bridges

Accurate interpretation of stratigraphic profiles, the phreatic surface and the spatial variability of geomaterials are essential to produce representative behaviour of geomechanical systems through numerical simulations. When considering slope stability, variations in soil layer boundaries and the phreatic surface may result in misleading metrics such as the Factor of Safety (FoS) and the Probability of Failure (PoF). This paper presents an image-based technique for generating continuous stratigraphic profiles, including random fluctuations based on Brownian motion. Brownian bridges are used to produce random walks between known points on both soil layer boundaries and the phreatic level that overlap with the slope profile image. Quadtree decomposition is used to discretise the stratigraphy and the phreatic level through an automatic process using generated digital images for mesh generation while also integrating material properties. The Scaled Boundary Finite Element (SBFEM) is used to analyse the slope stability problem. Images based on random walks along an unknown stratigraphic material boundary and the phreatic surface are randomly generated and have many random possibilities, which are used to undertake probabilistic analysis to obtain PoF. This process is complex when using numerical methods, such as the Finite Element Method, as it requires mesh generation from different slope profiles with alternating material interfaces at each probabilistic instance. When the Scaled Boundary Finite Element Method is applied, probabilistic numerical analyses can be fully automated for randomly generated material interfaces. The feasibility of the proposed method is illustrated through several cases of a slope with multiple material layers, in addition to a slope incorporating a Brownian bridge phreatic surface formulation.

A multiscale scaled boundary finite element method solving steady-state heat conduction problem with heterogeneous materials

A new numerical model is proposed by bridging small and large scales via the numerical base functions constructed by Scaled Boundary Finite Element Method (SBFEM) for the multiscale heterogeneous analysis of steady-state heat conduction problems. Utilizing the advantages of polygonal Scaled Boundary Finite Element (SBFE) and image-based quadtree gridding, the construction of base functions can be conveniently and efficiently conducted particularly for the heterogeneous media with complex geometries or singularity of heat flux at the small-scale, and the solution accuracy at the large-scale can be improved by increasing nodes of coarse element only without increasing new nodes inside. Consequently, the SBFEM based solutions at the large-scale can be solved with a reduced order and sufficient accuracy, and the solutions at the small-scale can be achieved using SBFEM based base functions. Numerical examples are provided and the effectiveness of the proposed approach is stressed at both the large and small scales.

A scaled boundary finite element partitioning based reduced order algorithm for the elastic analysis of cyclically symmetric structures

AbstractAn efficient reduced order algorithm is proposed for the elastic SBFE analysis for 2‐D cyclic symmetric structures with or without a common node. The general stiffness matrices of scaled boundary finite element (SBFE) is proved to be block‐circulant, and can be constructed via the basic region, instead of the whole computing domain. Thus, the expense on eigenvalue analysis required in generating stiffness matrix can be significantly reduced, and the solution scale can be reduced by partitioning the system equation into a series of small independent subproblems. Furthermore, the presented algorithm is combined with the Woodbury formula to reduce the computational cost on the analysis of incomplete cyclically symmetric structures, the original system equation is transformed into the equations with block‐circulant coefficient matrices, which is efficiently solved by partitioning algorithm. Four numerical examples are provided to demonstrate the effectiveness and advantages of the proposed approach.

Shock vibration and damage responses of nuclear island subjected to an aircraft crash

Although more attention has been given towards the analysis of large commercial aircraft impacts since the September 11 attacks, little work has been done on the vibration response of internal structures and equipment of nuclear power plants (NPPs) during an impact. With a focus on new NPPs, this paper develops a remeshing algorithm for contact surfaces, which utilizes the advantages of scaled boundary finite element (SBFE) polyhedral elements to fully realize the potential of Octree and structured grid discretization methods. The vibration response of nuclear island internal structures is examined. The influence of the interaction between the foundation and the structure on the overall dynamic response is analyzed. Meanwhile, different ranges of foundation size are investigated for balance between computational efficiency and accuracy. The vibration propagation mechanism is found by studying the influence of different artificial boundary types and damping conditions of foundation. The results show that the interaction has a significant influence on the vibration response, and that a foundation size of 1B (height of the steel containment vessel) on all sides provides the optimal balance between computational and accuracy requirements. Furthermore, the damping ratio of the foundation plays a key role in vibration propagation. Finally, the combined bonding method is highly flexible, and allows cross-scale integration of different mesh sizes.

A Scaled Boundary Finite-Element Method with B-Differentiable Equations for 3D Frictional Contact Problems

Contact problems are among the most difficult issues in mathematics and are of crucial practical importance in engineering applications. This paper presents a scaled boundary finite-element method with B-differentiable equations for 3D frictional contact problems with small deformation in elastostatics. Only the boundaries of the contact system are discretized into surface elements by the scaled boundary finite-element method. The dimension of the contact system is reduced by one. The frictional contact conditions are formulated as B-differentiable equations. The B-differentiable Newton method is used to solve the governing equation of 3D frictional contact problems. The convergence of the B-differentiable Newton method is proven by the theory of mathematical programming. The two-block contact problem and the multiblock contact problem verify the effectiveness of the proposed method for 3D frictional contact problems. The arch-dam transverse joint contact problem shows that the proposed method can solve practical engineering problems. Numerical examples show that the proposed method is a feasible and effective solution for frictional contact problems.

Revised SBFEM on arbitrary polygons and faceted polyhedrons with the second order completeness by elimination of bubble functions

By the theoretical analysis on the scaled boundary finite element (SBFEM), some appropriate 2D or 3D bubble functions are necessary in order that the elements can possess the second order completeness. However, the extra 2D bubble function is difficult to be adopted as facet shape function for 3D polyhedral element. In this paper, a revised-scaled-boundary element (denoted by 2DRSBd2) only corresponding to the boundary nodal displacements is constructed by elimination of the 2D bubble function derived from the constant right-hand term in 2D Poisson equation. This polygonal element 2DRSBd2 possesses the second order completeness and the third order convergence in L2-norm. Then we construct two polyhedral elements 3DSB-2DRSBd2 and 3DRSB-2DRSBd2 based on the polygonal element 2DRSBd2. The difference between the two 3D elements is: the 3DSB-2DRSBd2 element is constructed with three 3D bubble functions derived from the constant body loads in three dimensional elastostatic problem, while the 3D revised-scaled-boundary element 3DRSB-2DRSBd2 is constructed by elimination of those 3D bubble functions. Both elements 3DSB-2DRSBd2 and 3DRSB-2DRSBd2 possess the second order completeness and the third order convergence in L2-norm. Numerical examples verify that they have good accuracy and insensitive to mesh distortion.

SBFE Virtual Particle Boundary: A New Non-Reflecting Boundary in SPH

Smoothed Particle Hydrodynamics (SPH) has outstanding advantages in dealing with nonlinear problems. However, it is difficult to find an efficient and accurate non-reflecting boundary for SPH. In this paper, the scaled boundary finite element (SBFE) virtual particle boundary is proposed to model the non-reflecting characteristics of the boundary in SPH. It is implemented by 2–4 layers of SBFE virtual particles whose pressure and velocity are calculated by the Lagrange interpolation from the nearby SBFE nodes. The SBFE virtual particle boundary can effectively and accurately simulate the transmission process of pressure waves on the boundary, and eliminate the influence of the reflecting waves on pressure and velocity fields.

Plate-Bending Analysis by NURBS-Based Scaled Boundary Finite-Element Method

AbstractThis paper extends the idea of using nonuniform rational B-splines (NURBS) based shape functions in the scaled boundary finite-element method (SBFEM) to plate bending formulations, which in...

SBFE analysis of surface loaded elastic layered media with influence of surface/interface energy

A numerical technique, based upon a scaled boundary finite element (SBFE) method, is proposed for the analysis of a two-dimensional, infinite, elastic, layered medium under arbitrary surface loads and surface energy effects. The governing equations of the bulk layers and the material surface/interfaces were formulated from linear elasticity and Gurtin-Murdoch surface elasticity theories, respectively. The SBFE formulation was established in a general context to handle both homogenous and functionally graded bulk materials. Enforcing the continuity of the displacements and tractions along the material interfaces together with the surface/interface equations resulted in a system of non-homogeneous, linear, ordinary differential equations (ODEs) governing the unknown nodal functions of the layer medium. A general solution of the final system of ODEs was constructed via a standard procedure and then used to form a system of linear algebraic equations governing the nodal boundary data. To facilitate the treatment of surface loads over a finite region, the subdomain technique was applied to obtain the governing equations for the whole layered medium. A selected set of results is reported to demonstrate the accuracy, convergence, and performance of the proposed technique.

An accurate frequency-domain model of water interaction with cylinders of arbitrary shape during earthquakes

An accurate frequency domain model is proposed to analyze the seismic response of uniform vertical cylinders with arbitrary cross section surrounded by water. According to the boundary conditions and using the variables separation method, the vertical modes of the hydrodynamic pressure are firstly obtained. Secondly, the three-dimensional wave equation can be simplified to a two-dimensional Helmholtz equation. Introducing the scaled boundary coordinate, a scaled boundary finite element (SBFE) equation which is a linear non-homogeneous second-order ordinary equation is derived by weighted residual method. The dynamic-stiffness matrix equation for the problem is furtherly derived. The continued fraction is acted as the solution of the dynamic-stiffness matrix for cylinder dynamic interaction of cylinder with infinite water. The coefficient matrices of the continued fraction are derived recursively from the SBFE equation of dynamic-stiffness. The accuracy of the present method is verified by comparing the hydrodynamic force on circular, elliptical and rectangle cylinders with the analytical or numerical solutions. Finally, the proposed model is used to analyze the natural frequency and seismic response of cylinders.

Numerical prediction of effective properties for heterogeneous viscoelastic materials via a temporally recursive adaptive quadtree SBFEM

A new numerical method is presented for evaluating the effective properties of heterogeneous viscoelastic materials, and an efficient and high-fidelity Direct Numerical Simulation (DNS) is addressed by integrating the advantages of the quadtree Scaled Boundary Finite Element Method (SBFEM) and a temporally recursive adaptive algorithm. The quadtree technique that can be directly implemented with input images is utilized for the mesh generation of the Representative Volume Element (RVE) with complex constitutes, and the inconvenience caused by ‘hanging nodes’ and the interface conformity requirement in the mesh generation can be removed using a polygonal Scaled Boundary Finite Element (SBFE). The computing accuracy in the time domain can be stably achieved via a recursive adaptive process when the step sizes vary. By virtue of the proposed DNS, an inverse analysis based procedure is designed to estimate the effective viscoelastic properties, and two indicators are proposed to evaluate the effectiveness of estimation. Numerical verification is provided to demonstrate the effectiveness of the proposed approach, and the impacts of the distributions and shapes of inclusions, RVE sizes, and volume fractions are taken into account.

The static analysis of arch dam-foundation system by using scaled boundary finite element method

The scaled boundary finite element (SBFEM) as a semi-analytical numerical method has been used in many fields. SBFEM has the following advantages: reducing the spatial dimension by one, rigorously modelling the unbounded domain and involving no fundamental solution. The computational model based on SBFEM will be conducive to the reliability and efficiency of safety evaluation of arch dam. However, owing to the requirement of scaling inherently, the SBFEM cannot be immediately applied in the analysis of arch dam which belongs to irregular shell structure. This limitation can be overcome by employing the sub-structure method to subdivide the arch dam into several sub-structures which satisfy the scaling requirement. An example of static analysis of the arch dam-foundation system is presented to verify the effectiveness of the SBFEM sub-structure method. The result is helpful to establish the calculating model of arch dam-reservoir-foundation system based on scaled boundary finite element method.

Elasto-plastic fine-scale damage failure analysis of metro structures based on coupled SBFEM-FEM

This paper presents a new high-performance, fine-scale damage analysis method using three techniques. Firstly, efficient and detailed mesh discretization is performed via octree, which can easily accomplish cross scale and thus significantly alleviate the burden of fine modelling using octant cubes. Secondly, the introduction of coupled scaled boundary finite element (SBFEs) into FEM can offer more generality, flexibility and conveniences. Thirdly, the parameter matrices (e.g. stiffness, mass, damping) of geometrically similar elements generated by octree are proportionable and can be pre-computed. Therefore, by leveraging the similarity among elements for elasto-plastic problems will significantly improve computational efficiency. In order to demonstrate the effectiveness of the new approach, a fine-scale damage evolution of DaiKai subway was comprehensively modelled using different mesh sizes. Plastic damage and generalized plasticity models for soil and interfaces were used. Results from the case study demonstrated that octree possesses the advantages of high robustness, ability for automation and capacity to work with fewer elements during mesh generation. The coupled SBFEM-FEM and the use of element similarity markedly improved the computation cost (50% computation time was reduced). The presented method bypasses the need for undesired compromise between simulation accuracy and solver efficiency.

Wing Crack Propagation Simulation Using Scaled Boundary Finite Element Method

Considering the existence of side-face loads, a new scaled boundary finite element (SBFEM) shape function is used to study the problem of wing crack propagation. In the solution of crack contact, the contact interface constraint is introduced by Lagrange multiplier method. The polygon SBFEM remeshing technology is used to simulate the crack propagation problem with single and multiple wing crack. Compared with related experimental and numerical simulation results, the correctness and effectiveness of the new SBFEM shape function in solving the wing crack propagation are verified. It is found that under pressure, the wing crack propagates along the direction of the maximum principal compressive stress during initial expansion, and then the crack opens and gradually expands toward the pressure. Wing cracks only have contact phenomena on the initial crack surface during crack propagation. The contact force gradually decreases from the centre to the sides of the initial crack. The friction coefficient has a great influence on the crack propagation direction.

Global concurrent cross-scale nonlinear analysis approach of complex CFRD systems considering dynamic impervious panel-rockfill material-foundation interactions

Evaluating the seismic security of enormous structures is extremely vital, and this concern has continually motivated the trends of refined numerical simulations for years. However, the ability to perform a concurrent global refinement analysis of large-scale projects with complex spatial geometries and spanning huge scales has remained a formidable challenge in settings such as high concrete-faced rockfill dams. In this paper, a global concurrent cross-scale nonlinear analysis approach (GCCNA) benefitting from an efficient hybrid octree-based discretization technique is presented. Significantly, a polygon interface is constructed to automatically connect the cross-scale element and solve the interactions between the concrete-faced rockfill and foundation. A viscoelasticity polygonal artificial boundary element is subsequently developed to render the influence of radiation damping on an infinite foundation so that the travelling wave effect on the dynamic response and stabilization can be captured. A high-efficiency and economically time-consuming solution strategy is adopted, wherein the scaled boundary finite element is introduced to manage the minority polyhedrons in the generated octree model, and the numerous hexahedrons are assigned to the isoparametric element. The features of rapid discretization, high flexibility, extraordinary grid reconstruction and coupling with the conventional finite element are contained perfectly, which are demonstrated via the comprehensive elasto-plastic dynamic simulation of an extremely complicated practical constructed highest rockfill dam. The proposed approach has attractive potential and practicability for the efficient refinement analysis of complicated enormous engineering structures and can be readily extended to subterranean structures, nuclear plants, and architectural and aviation structures.

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.