This study adapts the flexible characteristic of meshfree method in analyzing three-dimensional (3D) complex geometry structures, which are the interlocking concrete blocks of step seawall. The elastostatic behavior of the block is analysed by solving the Galerkin weak form formulation over local support domain. The 3D moving least square (MLS) approximation is applied to build the interpolation functions of unknowns. The pre-defined number of nodes in an integration domain ranging from 10 to 60 nodes is also investigated for their effect on the studied results. The accuracy and efficiency of the studied method on 3D elastostatic responses are validated through the comparison with the solutions of standard finite element method (FEM) using linear shape functions on tetrahedral elements and the well-known commercial software, ANSYS. The results show that elastostatic responses of studied concrete block obtained by meshfree method converge faster and are more accurate than those of standard FEM. The studied meshfree method is effective in the analysis of static responses of complex geometry structures. The amount of discretised nodes within the integration domain used in building MLS shape functions should be in the range from 30 to 60 nodes and should not be less than 20 nodes.

Meshfree methods, that have been developed recent years, are believed to show more advantages than finite element method in dealing with problems of fluid flows, large deformations, crack growth along the complex path, fragment-impact, complex shape structures and so on [

Meshfree methods have been implemented in various fields and problems of engineering, in which problems of liquid and/or interaction between solid and liquid are solved by using SPH method. Ren et al. [

EFG is one of the most known and robust meshfree methods. It is a development of Galerkin method applied on partial differential equations. EFG analyses the weak form formulations of displacement field similarly to standard FEM. Nevertheless, a number of discretised nodes within structure domain are used for building approximation functions instead of FEM elements. Compared to traditional FEM, EFG method has many advantages. Firstly, the problem of meshing structure can be eliminated. Secondly, EFGM performance does not depend on nodal distribution. Additionally, EFGM does not require any special treatment for structure with incompressible material [

The process of forming shape functions is the main feature indicating the discrepancy between traditional finite element methods and meshless methods. Many meshfree shape functions have been successfully developed, which include moving least squares approximation, point interpolation, partition of unity (PU) and HP-clouds approximation [

The objective of this study is to present the application of EFGM using moving least square approximation in analyzing elastostatic responses of three-dimensional structure with complex geometry. Effect of spherical support domains with pre-defined number of nodes, which are used for building MLS approximation, on convergence and accuracy of results is also investigated. The number of nodes in each domain ranges from 10 to 60. Each step increases by 10 nodes. The efficiency of the EFG method is verified by comparing the elastostatic responses of a complex concrete block with the results determined by standard FEM and commercial software ANSYS. Whereas, standard FEM uses linear tetrahedral elements and ANSYS uses 10-node tetrahedral elements. All the coding in this study including EFGM and standard FEM are performed using MATLAB program.

The EFG method is a typical meshless method solving the Galerkin weak form by only using a set of distributed points. The shape functions of EFG method are formulated relying on MLS approximation over a local support domain, which contents a group of nodes. The EFGM is one of the most extensively applied meshless methods in many boundary value problems of solid mechanics. There are three components used to construct the moving least square approximations, which include a weight function; a basis, which is linear polynomial in this study; and coefficients depending on position. The weight function does not vanish only in support domain of current integration node. Connectivity of discretized nodes in meshfree structure is defined by overlapping of the nodal support domains. In general, the support domains can be any geometric shape and size. However, in this study, the spherical domains are taken into considering for its numerical simplicity.

Even though EFG is a meshless method, a background mesh is needed for solving the integration of Galerkin weak form. The MLS constructing process in 3D is described in this section.

Let us consider a spherical domain _{i} corresponding to

The moving least square approximation ^{h}(

where

In this study, 3D linear basic function is used as:

in which moment matrix

with _{i} is the weight function at node

and

Substituting

where

In this work, a 4th order spline weight function is used as follows:

where _{i} = |_{i} – _{i} and _{i} is the radius of the support of point _{i} for weight function.

The meshfree method is used to analyse three-dimensional solid structure using displacement based formulation. The codes are compiled in MATLAB program based on following steps:

Step 1

The domain of problem is discretised into arbitrarily distributing nodes:

Numbering discretised nodes, creating background cells for integration, which are tetrahedral cells in this study.

Assigning the physical properties of the analysed structure. The concrete material has Young’s modulus

Step 2

The boundary value problem is presented through the differential equation with desired boundary conditions. For the fixed boundary, the displacements of corresponding DOFs of boundary are assigned as zeroes.

Step 3

Assembling global stiffness matrix constructed from weak form governing differential equation includes:

Generating the Gauss points based on the background tetrahedral cells.

Defining support domain for each Gauss point (looping over). A number of nodes in each support domain are pre-defined ranging from 10 to 60 nodes. In each step an increase of 10 nodes is applied for investigating the effect of support domain in the studied problems.

Within the support domain, the derivatives of shape functions can be determined as in following formulations:

The shapes functions of

where

and

The partial derivatives of the shape functions are calculated by re-writing

where

Analysing the partial derivatives of

with subscript, “,

Hence,

After obtaining the partial derivatives of shape functions, the stiffness matrix is assembled similarly to standard finite element method.

The assembling process of surface load matrix is similar to stiffness matrix, but the shape functions are used instead of their partial derivatives.

Step 4

The above discrete nodal equations of stiffness and force matrices are assembled into global matrices.

Step 5

The results of nodal displacements can be obtained by solving the standard discrete equations

Step 6

The nodal stresses are determined by performing post-processing of the displacements obtained in Step 5.

The studied structure is a 3D concrete block of an interlocking step seawall (

The displacement and stress responses are estimated by using EFG method in MATLAB program. The efficiency of EFG method is evaluated by making a comparison between the obtained solutions and those of standard FEM and ANSYS software.

The results shown in

The following analysis uses 40-node domain for investigating elastostatic responses along edge 1–2 of the concrete block obtained by EFGM compared with FEM. The convergence studies of displacement and stress responses are performed and shown in

The distribution of normal stress

The EFG meshfree method using three-dimensional moving least square approximation was successfully applied in analyzing three-dimensional complex geometry structures, namely interlocking concrete blocks for step seawall structure. The elastostatic responses of the block were obtained relying on Galerkin weak form formulation over local support domains, which are defined by a collection of discretised nodes in a spherical region. The nodal amount ranging from 10 to 60 was also investigated because of its crucial role in MLS interpolation. The accuracy and efficiency of the studied method were validated by making comparison of the solutions analysed by FEM using linear shape functions on tetrahedral elements and well known commercial software, ANSYS. Some conclusions are pointed out as follows:

The elastostatic responses of the studied concrete block obtained by EFG method converge faster and are more accurate than those of standard FEM.

The number of nodes in support domain should be greater than 20 and should be in the range from 30 to 60.

The singular moment matrix may occur if the nodal number in support domain is less than 20.

Because of just using a number of discretised nodes, the application of EFG method in analysing 3D complex structure is simpler compared to regular FEM.

The authors acknowledge the help of ours colleague.