Scaled Boundary Method Research Articles

A scaled boundary shell formulation in isogeometric analysis for static and dynamic analysis

AbstractIn modern applications of computer‐aided design (CAD) for the analysis of shell structures, isogeometric analysis is a powerful tool that integrates both design and analysis. An exact geometry description and a straightforward computation without loss of information are advantageous, especially for shell structures such as roofs, satellite hulls, or car bodies. In addition, the scaled boundary method provides a scale separation with a semi‐analytical solution procedure to consider a three‐dimensional linear elastic constitutive law. The presented approach deals with a scaled boundary solid shell formulation in the framework of isogeometric analysis. The formulation utilizes a normal scaling strategy which scales the shell along its normal vector at each point on the discretized bottom surface. This is fundamentally different from the well‐known radial scaling strategy, where each point on the problem domain is obtained from a fixed scaling center. This results in a separation of the analysis into an in‐plane direction and a scaling (normal) direction. By introducing the scaling, a scaled boundary differential equation is derived that is dependent on the scaling parameter only. Choosing a proper set of conditions, the differential equation can be solved by a Padé expansion. While the in‐plane direction is solved in a weak sense, the thickness direction along the normal vector can be solved analytically resulting in a semi‐analytical procedure. The isogeometric description of the CAD structure inherently yields the exact normal vector, its derivatives and a higher order continuity throughout the structure. Herein, the focus is on the solution technique in the thickness direction and the challenges are addressed. The power of the formulation is outlined in several numerical examples of static and dynamic analysis and a comparison to shell formulations in literature is provided.

A modified scaled boundary method to analyze structural elements

A modified scaled boundary method to analyze structural elements

A modified semi-weight function method for SIF calculation of arbitrary inclined interface crack with both single complex-root and double real-root singularities

An arbitrary inclined crack terminated at the interface usually has singularity of single complex-root or double real-root at crack tip. Currently, most methods of the stress intensity factor (SIF) calculation (e.g., Mellin transformation method, enrichment strategy method) only consider the single complex-root singularity, with difficulty in solving auxiliary functions. Although the scaled boundary method takes the double real-root singularity into account, it needs to set special finite elements at crack tip. In this study, based on the sub-matrix method and the reciprocal theorem of work, a modified semi-weight function method is proposed to calculate SIF of the arbitrary interface crack with consideration of both single complex-root and double real-root singularity, and also analyze the effect of bimaterial parameters (Gi), crack inclination angle (w) and relative length (a/L) on the SIFs. This modified method has the advantages of uniform auxiliary function (independent of crack size and external load) and simple finite element calculation (without special singularity element). Moreover, its validity is verified by good agreement of our calculation results with the available literature results and finite element results. The research results can provide a theoretical basis for anti-crack design of multi-layer caverns of the compressed air energy storage (CAES).

Shape sensing of plate structures through coupling inverse finite element method and scaled boundary element analysis

This study proposes an enhanced inverse finite element method (iFEM) to reconstruct the structural displacements based on strain data collected from single surface (top/bottom) of the structure. The method eliminates the drawback of existing iFEM formulations for which the strain sensors must be symmetrically installed on both top and bottom surfaces of the structure with respect to its mid-plane. Therefore, the present approach increases the practical application of shape sensing with a lower number of sensors. To this end, the least-squares variational principle of iFEM is established using scaled boundary method, which represents the displacement field of structure by scaling the boundary in the radial direction with a scaling factor. Moreover, a novel five-node inverse plate element is developed to discretize the geometry and approximate the kinematic variables of the present approach. This element increases the order of interpolation function and thus reduces the number of strain sensors used in shape-sensing process. Overall, the accuracy and applicability of the present iFEM is numerically and experimentally validated by performing shape sensing analyses of various plate structures.

A NURBS enhanced polygonal element formulation based on the scaled boundary method for the analysis of solids in boundary representation

AbstractThe contribution is concerned with a numerical element formulation for the nonlinear analysis of solid in boundary representation. Its results in a polygonal element with an arbitrary number of sides. Straight edges are possible as well as curved edges, which are described by e.g. Non‐Uniform Rational B‐Splines (NURBS). The element formulation is based on the so‐called scaled boundary finite element method (SBFEM). The basic idea is to scale the domain's boundary with respect to a scaling center in order to describe the interior domain. In contrast to SBFEM, the proposed method makes use of a numerical aproximation for the displacement response in scaling direction. This enables the analysis of geometrically and physically nonlinear problems in solid mechanics. The interpolation at the boundary in circumferential direction is independent of the interpolation in scaling direction. So, different basis functions can be used for each direction, e.g. NURBS basis functions in circumferential and Lagrange basis functions in radial direction. The advantage of the presented element formulation is the flexibility in mesh generation. The avoidance of so‐called hanging node problems as arbitrary element geometries are possible. For example, using Quadtree algorithms, a fast and reliable mesh generation can be achieved. Furthermore, in connection with trimming algorithms, the element formulation allows a precise representation of the geometry even with coarse meshes. Some benchmark tests are presented to evaluate the accuracy of the proposed numerical method against analytical solutions.

Element-Free Galerkin Scaled Boundary Method Based on Moving Kriging Interpolation for Steady Heat Conduction Analysis with Temperatures on Side-Faces

Element-Free Galerkin Scaled Boundary Method Based on Moving Kriging Interpolation for Steady Heat Conduction Analysis with Temperatures on Side-Faces

A polyhedral element formulation based on the scaled boundary method for the analysis of nonlinear problems in solid mechanics

AbstractIn this contribution a numerical element formulation for the nonlinear analysis of solids is proposed. The element is defined by a polyhedron with an arbitrary number of polygonal faces. It is based on the scaling concept, which is adopted from the so‐called scaled boundary finite element method (SBFEM). The SBFEM is a semi‐analytical formulation to analyze problems in linear elasticity. Within this method the fundamental concept is to scale the domain's boundary with respect to a scaling center in order to describe the interior domain. The present element formulation employs the scaling concept twice and in contrast to SBFEM, uses a numerical approximation for the displacement response in both scaling directions. This makes it suitable for the analysis of geometrically and physically nonlinear problems. The advantages of the proposed element formulation is its high flexibility in mesh generation. It avoids the hanging node problems as arbitrary element geometries are possible. For example, using Octree algorithms, a fast and reliable mesh generation can be achieved. In case of curved boundaries the element formulation allows a precise representation of the geometry even with coarse meshes. Some numerical examples are presented to evaluate the accuracy of the proposed numerical method against analytical solutions, and a comparison to standard element formulations is given as well.

Element-free Galerkin scaled boundary method based on moving Kriging interpolation for steady heat conduction analysis

This paper develops an element-free Galerkin scaled boundary method (EFG-SBM) for solving steady heat conduction problems, in which the circumferential direction is constructed by the moving Kriging (MK) interpolation based on the EFG approach. Because the MK interpolation satisfies the Kronecker delta property, it is more convenient in enforcing the essential boundary conditions than the traditional MLS-based EFG-SBM. As a newly boundary-type meshless method, EFG-SBM possesses advantages of EFG and scaled boundary finite element method (SBFEM). This method inherits the semi-analytical property of SBFEM by introducing the normalized radial coordinate system, in which the governing differential equations are weakened in the circumferential direction and solved analytically in the radial direction. Unlike the traditional SBFEM, the preprocessing and postprocessing processes of EFG-SBM are simplified and the more accuracy can be obtained because of higher continuity of the MK shape functions. Computation of EFG-SBM can be reduced since only the boundary needs to be discretized compared with the EFG approach. This proposed method is verified via four heat conduction examples including problems with thermal crack considering the prescribed heat flux and temperature on the side-face and unbounded domain. The numerical solutions show that EFG-SBM has higher accuracy and better convergence than the traditional SBFEM. An accurate smooth heat flux can be obtained directly without necessity of using the recovery procedure.

An interpolating element-free Galerkin scaled boundary method applied to structural dynamic analysis

• An efficient semi-analytical method, namely the IEFG-SBM is developed for structural dynamic analysis. • The proposed method unifies the advantages of the SBM and the IEFG method. • The IIMLS method is superior to the MLS approximation because its shape function has the delta function property . • The IIMLS method is also superior to the IMLS method because the weight function involved is nonsingular. • The improved continued fraction solution can represent the inertial effect at high frequencies with no internal mesh. An efficient semi-analytical method, namely the interpolating element-free Galerkin scaled boundary method (IEFG-SBM) is developed for structural dynamic analysis in this paper, which is based on boundary scattered nodes with no need of element connectivity. Since the shape functions of the improved interpolating moving least-squares (IIMLS) method satisfy the delta function property, the essential boundary conditions, as a result, can be enforced accurately without any additional efforts. Based on the improved continued fraction, the dynamic properties of a bounded domain are expressed by the high order static stiffness and mass matrices. This continued fraction solution is computationally more robust and the transient response can be obtained directly in the time domain using standard procedures in structural dynamics. It is testified from the computational results that the present method for structural dynamic analysis is quite easy to be implemented with high accuracy.

Scaled boundary point interpolation method for seismic soil-tunnel interaction analysis

The scaled boundary method (SBM) is an effective numerical approach for analyzing elasto-statics of bounded and unbounded media. To enhance abilities of the scaled boundary approach, this method can be coupled with mesh free technology. In this paper a point interpolation based SBM is proposed to analyze seismic soil-tunnel interaction problems. In the proposed approach, boundary of the domain is modelled with the scaled boundary point interpolation method while the interior domain is modelled by the conventional finite element method (FEM). This is the first time that a mesh-free scaled boundary method is used to analyze seismic problems. The presented method has some advantages over previously presented mesh-free SBMs. In the scaled boundary point interpolation method, the shape functions have the Kronecker delta function property and do not require radial basis functions to discretize the boundary of 2D problems. A shaking table test is designed and used to verify the proposed method. It is shown that the proposed numerical approach leads to results that are in a good agreement with those of the designed shaking table tests.

Numerical investigation of circle defining curve for two-dimensional problem with general boundaries using the scaled boundary finite element method

The scaled boundary finite element method (SBFEM) is applied to the static analysis of two dimensional elasticity problem, boundary value problems domain with the domain completely described by a circular defining curve. The scaled boundary finite element equations is formulated within a general framework integrating the influence of the distributed body force, general boundary conditions, and bounded and unbounded domain. This paper investigates the possibility of using exact geometry to form the exact description of the circular defining curve and the standard finite element shape function to approximate the defining curve. Three linear elasticity problems are presented to verify the proposed method with the analytical solution. Numerical examples show the accuracy and efficiency of the proposed method, and the performance is found to be better than using standard linear element for the approximation defining curve on the scaled boundary method.

Solving viscoelastic problems with cyclic symmetry via a temporally adaptive EFG-SB partitioning algorithm

Based upon a combination of a temporally-piecewise adaptive algorithm with an Element-Free Galerkin Scaled Boundary Method (EFG-SBM), a partitioning algorithm is presented for the two-dimensional viscoelastic analysis of cyclically symmetric structures. By expanding variables at a discretized time interval, the variations of variables can be described more precisely, and a space–time domain-coupled problem can be converted into a series of recurrent boundary value problems which are solved by an EFG-SBM-based partitioning algorithm via an adaptive computing process. Numerical examples are given to verify the proposed algorithm in terms of computing accuracy and efficiency.

Coupled interpolating element-free Galerkin scaled boundary method and finite element method for crack problems

The interpolating element-free Galerkin scaled boundary method (IEFG-SBM) is a semi-analytical method which only requires to discretize the boundary by the interpolating element-free Galerkin (EFG) method without fundamental solution. This method is ideally suited to solve problems containing infinite domain and singular physical fields. This study develops a novel method that couples the IEFG-SBM and the finite element method (FEM) for crack analysis in order to take the full advantages of both IEFG-SBM and FEM. The IEFG-SBM is adopted to model the domain close to the crack tip and the FEM is adopted in the rest of the domain. The corresponding displacement interpolation models are employed for each sub-domain respectively. Through continuity conditions on the interface between IEFG-SBM sub-domain and FEM sub-domain, the coupled formula of the proposed method can be easily derived. The proposed method is simple, effective, and easy to be programmed. Finally, two numerical examples are presented to demonstrate the validity of the proposed method.

Solution of steady-state thermoelastic problems using a scaled boundary representation based on nonuniform rational B-splines

ABSTRACTThis work explores the application of isogeometric scaled boundary method in the two-dimensional thermoelastic problems of irregular geometry. The proposed method inherits the advantages of both isogeometric analysis and scaled boundary finite element method and overcomes their respective disadvantages. In the proposed approach, the boundaries of the problem domain are discretized with nonuniform rational B-splines (NURBS) basis functions, while the temperature distributions inside the domain are represented by a sequence of power functions in terms of radial coordinate within the framework of scaled boundary finite element method. The resulting solution of the stress in radial direction can be computed analytically for the temperature changes. The construction of tensor product structure is circumvented for the two-dimensional problems as only the boundary information of the problem domain is required. Hence, the flexibility to represent the complex geometry can be significantly improved in the proposed method. Numerical examples are presented to validate the performance of the proposed method where it is seen that superior accuracy, efficiency, and convergence behavior can be achieved over the conventional scaled boundary finite element method.

A combination of isogeometric technique and scaled boundary method for the solution of the steady-state heat transfer problems in arbitrary plane domain with Robin boundary

The scaled boundary finite element method (SBFEM) has attracted considerable attention in recent years as a novel semi-analytical computational approach since only the boundary is discretized using finite element approach and the spatial dimension is reduced by one in this method. By introducing the radial and circumferential scaled boundary coordinates in this method, the reduction of spatial dimension is accomplished by using the conventional Lagrange functions to weaken the governing equations in the circumferential direction, while to work analytically in the radial direction. The NURBS-based isogeometric analysis (IGA) has remarkable advantages due to its integration of CAD/CAE, geometrical exact discretization for free-form shapes and numerical accuracy. Thus, an isogeometric analysis based on the framework of scaled boundary method (IGA-SBM) is proposed, which combines the concepts of IGA and SBFEM by employing NURBS to represent the unknown field variables in the circumferential direction. In this work, the IGA-SBM is extended to the solutions of the steady-state heat transfer problems in arbitrary plane domain enclosed by the complex boundary geometry. Four numerical examples are presented to illustrate that the excellent accuracy, computational efficiency and convergence performance of IGA-SBM. The numerical studies show that the heat transfer problems with complicated configuration can be more effectively handled by considering the combination of IGA and SBFEM.

An interpolating element-free Galerkin scaled boundary method for the elasticity problem

As a newly-developed semi-analytical method, the scaled boundary finite element method is very powerful to deal with singular and unbounded problems. By combining the element-free Galerkin (EFG) method with the scaled boundary method in the frame of improved interpolating moving least-squares (IIMLS) method, an interpolating element-free Galerkin scaled boundary method (IEFG-SBM) is firstly proposed to solve elasticity problems in this paper. In the IEFG-SBM, the solution in the radial direction is obtained analytically and only nodes are required to discretize the boundaries of the computational domain. In addition, higher accuracy and faster convergence are obtained due to the higher continuity of the shape functions in the circumferential direction. The IEFG-SBM does not need the fundamental solution and thus no singular integrations are involved. The shape function of the IIMLS method satisfies the Kronecker delta function property and thus the essential boundary conditions can be imposed directly as in the traditional finite element method. In comparison with the interpolating moving least-squares (IMLS) method proposed by Lancaster and Salkauskas, the key advantage of the IIMLS method is that it does not require singular weight function and thus any weight function used in the MLS approximation can also be applied in the IIMLS method. In addition, there are less unknown coefficients in the IIMLS method than in the conventional moving least-squares (MLS) approximation. Thus fewer nodes are required in the local influence domain and a higher computational accuracy can be reached in the IIMLS-based meshless method. At last, several numerical examples are presented to verify the effectiveness and accuracy of the developed method for the elasticity problem.

Steady heat conduction analyses using an interpolating element-free Galerkin scaled boundary method

Through use of the improved interpolating moving least-squares (IIMLS) shape functions in the circumferential direction of the scaled boundary method based on the Galerkin approach, an interpolating element-free Galerkin scaled boundary method (IEFG–SBM) is developed in this paper for analyzing steady heat conduction problems, which weakens the governing differential equations in the circumferential direction and seeks analytical solutions in the radial direction. The IIMLS method exhibits some advantages over the moving least-squares approximation and the interpolating moving least-squares method because its shape functions possess the delta function property and the involved weight function is nonsingular. In the IEFG–SBM, only a nodal data structure on the boundary is required and the primary unknown quantities are real solutions of nodal variables. Higher accuracy and faster convergence are obtained due to the increased smoothness and continuity of shape functions. Based on the IEFG–SBM, the steady heat conduction problems with thermal singularities and unbounded domains can be ideally modeled. Some numerical examples are presented to validate the availability and accuracy of the present method for steady heat conduction analysis.

A combination of EFG-SBM and a temporally-piecewise adaptive algorithm to solve viscoelastic problems

This paper combines Element-Free Galerkin Scaled Boundary Method (EFG-SBM) with a temporally-piecewise adaptive algorithm to solve viscoelastic problems. By expanding variables at a discretized time interval, the variations of variables can be described more precisely, and a space-time domain coupled problem can be converted into a series of recurrent boundary value problems which are solved by EFG-SBM via an adaptive computing process. Numerical tests including creep and relaxation are given to verify the proposed algorithm.

An EFG-SBM based partitioning algorithm for two-dimensional elastic analysis of cyclically symmetrical structures

Purpose – The purpose of this paper is to exploit the cyclic symmetry to reduce the computational expense of scaled boundary method (SBM) which may cumber its further application. Design/methodology/approach – A partitioning EFG-SB (Element-free Galerkin-SB) algorithm is proposed for the two-dimensional elastic analysis of cyclically symmetric structures. Findings – By utilizing the cyclic symmetry and partitioning algorithm, the whole computational cost can be significantly reduced. Three numerical examples are given to illustrate the advantages of the proposed algorithm. Originality/value – It is proved that the matrices of eigenvalue and system equations of EFG-SBM for cyclically symmetric structures are block-circulant so long as a kind of symmetry-adapted reference coordinate system is adopted. No matter whether displacement constraints are cyclically symmetric or not, the partition is available for the eigenvalue equations. Therefore the major computational cost can be saved via the proposed partitioning algorithm. This paper provides an efficient algorithm for the two-dimensional elastic analysis of cyclically symmetric structures using EFG-SBM. A higher computing efficiency can be expected since the proposed partitioning algorithm facilitates parallel processing.

An EFG-SBM based partitioning algorithm for heat transfer analysis of cyclically symmetrical structures

An Element-free Galerkin scaled boundary method (EFG-SBM) based partitioning algorithm is proposed for the 2D steady-state heat transfer analysis of cyclically symmetrical structures. The eigenvalue matrices of EFG-SBM for the rotationally periodic system are proved to be block-circulant under a symmetry-adapted reference coordinate system. Furthermore, both the eigenvalue and system equations can be partitioned into a number of smaller independent problems which are solved by a partitioning algorithm. Numerical examples are given to verify the proposed algorithm in term of computing accuracy and efficiency.