An overview on the development of hybrid fundamental solution based finite element method (HFS-FEM) and its application in engineering problems is presented in this paper. The framework and formulations of HFS-FEM for potential problem, plane elasticity, three-dimensional elasticity, thermoelasticity, anisotropic elasticity, and plane piezoelectricity are presented. In this method, two independent assumed fields (intraelement filed and auxiliary frame field) are employed. The formulations for all cases are derived from the modified variational functionals and the fundamental solutions to a given problem. Generation of elemental stiffness equations from the modified variational principle is also described. Typical numerical examples are given to demonstrate the validity and performance of the HFS-FEM. Finally, a brief summary of the approach is provided and future trends in this field are identified.
A novel hybrid finite element formulation, called the hybrid fundamental solution based FEM (HFS-FEM), was recently developed based on the framework of hybrid Trefftz finite element method (HT-FEM) and the idea of the method of fundamental solution (MFS) [
The HFS-FEM mentioned above inherits all the advantages of HT-FEM over the traditional FEM and the boundary element method (BEM), namely, domain decomposition and boundary integral expressions, while avoiding the major weaknesses of BEM [
This paper is organized as follows: in Section
The Laplace equation of a well-posed potential problem (e.g., heat conduction) in a general plane domain
For convenience, (
In this section, the procedure for developing a hybrid finite element model with fundamental solution as interior trial function is described based on the boundary value problem defined by (
Similar to the method of fundamental solution (MFS) in removing singularities of fundamental solution, for a particular element
Intraelement field and frame field of a HFS-FEM element for 2D potential problems.
4-node 2D element
8-node 2D element
In implementation, the number of source points is taken to be the same as the number of element nodes, which is free of spurious energy modes and can keep the stiffness equations in full rank, as indicated in [
The corresponding outward normal derivative of
In order to enforce the conformity on the field variable
Typical quadratic interpolation for frame field.
For the boundary value problem defined in (
Illustration of continuity between two adjacent elements “
A modified variational functional is developed as follows:
To show that the stationary condition of the functional (
As for the continuous requirement between two adjacent elements “
If the following expression:
For the proof of the theorem on the existence of extremum, we may complete it by the so-called “second variational approach” [
With the intraelement field and frame field independently defined in a particular element (see Figure
To enforce interelement continuity on the common element boundary, the unknown vector
Generally, it is difficult to obtain the analytical expression of the integral in (
For the
The calculation of vector
Considering the physical definition of the fundamental solution, it is necessary to recover the missing rigid-body motion modes from the above results. Following the method presented in [
In linear elastic theory, the strain displacement relations can be used and equilibrium equations refer to the undeformed geometry [
The constitutive equations for the linear elasticity are given in matrix form as
For elasticity problem, two different assumed fields are employed as in potential problems: intraelement and frame field [
With the assumption of intraelement field in (
The unknown
As in Section
By applying the Gaussian theorem, (
As in Section
For the same reason stated in Section
In this section, the HFS-FEM approach is extended to three-dimensional (3D) elastic problem with/without body force. The detailed 3D formulations of HFS-FEM are firstly derived for elastic problems by ignoring body forces, and then a procedure based on the method of particular solution and radial basis function approximation are introduced to deal with the body force [
Let (
Geometrical definitions and boundary conditions for a general 3D solid.
The constitutive equations for linear elasticity and the kinematical relation are given as
For a well-posed boundary value problem, the boundary conditions are prescribed as follows:
The inhomogeneous term
From the above equations it can be seen that once the particular solution
For body force
To solve this equation, the displacement is expressed in terms of the Galerkin-Papkovich vectors
After obtaining the particular solution in Sections
The intraelement displacement fields are approximated in terms of a linear combination of fundamental solutions of the problem as
Intraelement field and frame field of a hexahedron HFS-FEM element for 3D elastic problem (the source points and centroid of 20-node element are omitted in the figure for clarity and clear view, which is similar to that of the 8-node element).
According to (
To link the unknown
Typical linear interpolation for the frame fields of 3D brick elements.
In the absence of body forces, the hybrid functional
Substituting (
Considering a surface of the 3D hexahedron element, as shown in Figure
For the
As in Section
Thermoelasticity problems arise in many practical designs such as steam and gas turbines, jet engines, rocket motors, and nuclear reactors. Thermal stress induced in these structures is one of the major concerns in product design and analysis. The general thermoelasticity is governed by two time-dependent coupled differential equations: the heat conduction equation and the Navier equation with thermal body force [
Consider an isotropic material in a finite domain
For a well-posed boundary value problem, the following boundary conditions, either displacement or traction boundary condition, should be prescribed as
For the governing equation (
RBF is to be used to approximate the body force
The body force
To solve (
To solve (
Considering the temperature gradient plays the role of body force, we can approximate
Once we have obtained the particular solutions of (
In materials science, composite laminates are usually assemblies of layers of fibrous composite materials which can be joined together to provide required engineering properties, such as specified in-plane stiffness, bending stiffness, strength, and coefficient of thermal expansion [
In the literature, there are two main approaches dealing with generalized two-dimensional anisotropic elastic problems. One is Lekhnitskii formalism [
In the Cartesian coordinate system (
For the generalized two-dimensional deformation of anisotropic elasticity
To find the fundamental solution needed in our analysis, we have to first derive the Green’s function of the problem: an infinite homogeneous anisotropic elastic medium loaded by a concentrated point force (or line force for two-dimensional problems)
Therefore, fundamental solutions of the problem can be expressed as
A typical composite laminate consists of individual layers, which are usually made of unidirectional plies with the same or regularly alternating orientation. A layer is generally referred to the global coordinate frame
For the two coordinate systems mentioned in Figure
Schematic of the relationship between global coordinate system (
The intraelement displacement fields for a particular element
The corresponding stress fields can be expressed as
With the assumption of two distinct intraelement field and frame field for elements, we can establish the modified variational principle based on (
Using Gaussian theorem and equilibrium equations, all domain integrals in (
Piezoelectric materials have the property of converting electrical energy into mechanical energy and vice versa. This reciprocity in the energy conversion makes them very attractive for using in electromechanical devices, such as sensors, actuators, transducers, and frequency generators. To enhance understanding of the electromechanical coupling mechanism in piezoelectric materials and to explore their potential applications in practical engineering, numerous investigations, either analytically or numerically, have been reported over the past decades [
For a linear piezoelectric material in absence of body forces and electric charge density, the differential governing equations in the Cartesian coordinate system
For the transversely isotropic material, if
For the piezoelectric problems, HFS-FEM is based on assuming two distinct displacement and electric potential (DEP) fields: intraelement DEP field
The intraelement DEP field
Making use of (
From (
For the two-dimensional piezoelectric problem under consideration, the frame field is assumed as
Based on the assumption of two distinct DEP fields, the Euler equations of the proposed variational functional should also satisfy the following interelement continuity requirements in addition to (
Since the stationary conditions of the traditional potential or complementary variational functional cannot satisfy the interelement continuity condition required in the proposed HFS-FEM, new modified variational functional should be developed. In the absence of the body forces and electric charge density, the hybrid functional
It can be proved that the stationary conditions of the above functional (
The element stiffness equation can be generated by setting
The order of magnitudes of the material constants and the corresponding field variables in piezoelectricity have a wide spectrum as large as 1019 in SI unit. This will lead to ill-conditioned matrix of the system [
Reference values for material constants and field variables in piezoelectricity derived from basic reference variables:
Displacement |
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Stress |
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Electric induction |
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Compliance |
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Impermeability |
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Electric field |
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Piezoelectric strain constant |
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Several numerical examples are presented in this section to illustrate the application of the HFS-FEM and to demonstrate its effectiveness and accuracy. Unless otherwise indicated, mesh convergence tests were conducted for the reference solutions obtained from ABAQUS in the following examples.
An isotropic cubic block, with dimension 10 × 10 × 10 and subject to a uniform tension as shown in Figure
Cubic block under uniform tension and body force: geometry, boundary condition, and loading.
Cubic block with body force under uniform distributed load: convergent study of (a) displacement
Cubic block under uniform tension and body force: (a) mesh 1 (4 × 4 × 4 elements), (b) mesh 2 (6 × 6 × 6 elements), and (c) mesh 3 (10 × 10 × 10 elements).
Contour plots of (a) displacement
In this example, a long circular cylinder with axisymmetric temperature change in domain is considered. Both inside and outside surfaces of the cylinder are assumed to be free from traction. The temperature
(a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurations of a quarter of circular cylinder (128 eight-node elements).
Figure
(a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius.
Figure
Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of the calculated results in postprocessing).
As shown in Figure
(a) Schematic of the 3D cube under arbitrary temperature and body force, (b) mesh used by HFS-FEM (125 20-node brick elements), and (c) mesh for ABAQUS (8000 C3D20R elements).
Figure
(a) Displacement
A finite composite plate containing an elliptical hole (Figure
(a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension, and its mesh configurations for (b) HFS-FEM, 1515 quadratic elements, and (c) ABAQUS, 9498 quadratic elements.
Figure
Variation of hoop stresses along the rim of the elliptical hole for different fiber orientation
Contour plots of stress components around the elliptic hole in the composite plate: (a)
Figure
Variation of SCF with the lamina angle
In this example, a multi-inclusion problem is investigated to show the capability of the HFS-FEM to deal with both isotropic and anisotropic materials in a unified way. As shown in Figure
(a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions.
In general, the Stroh formalism is suitable for the anisotropic material with distinct material eigenvalues, and it fails for the degenerated materials like isotropic material with repeated eigenvalues
Table
Comparison of displacement and stress at points A and B.
Items | Points | HFS-FEM (272 elements) | ABAQUS (272 elements) | ABAQUS (30471 elements ) |
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Disp. |
A | 0.04322 | 0.04318 | 0.04335 |
B | 0.03719 | 0.03721 | 0.03744 | |
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C | 0.03062 | 0.03076 | 0.03091 | |
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Stress |
A | 10.0446 | 9.9219 | 9.9992 |
B | 9.8585 | 9.8304 | 9.9976 | |
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C | 20.6453 | 13.7302 | 23.6625 |
The variation of displacement component (a)
Consider an infinite piezoelectric plane with a circular hole as shown in Figure
Properties of the material PZT-4 used for the model.
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(a) An infinite piezoelectric plate with a circular hole subjected to remote stress. (b) Mesh used by HFS-FEM. (c) Mesh used by ABAQUS.
Figure
Distribution of (a) hoop stress
Variation of (a) normalized stress
It can be seen from Figure
In this paper, we have reviewed the HFS-FEM and its application in engineering applications. The HFS-FEM is a promising numerical method for solving complex engineering problems. The main advantages of this method include integration along the element boundaries only, easily adopting arbitrary polygonal or even curve-sided elements and symmetric and sparse stiffness matrix, and avoiding the singularity integral problem as encountered in BEM. Moreover, as in HT-FEM, this method offers the attractive possibility to develop accurate crack singular, corner, or perforated elements, simply by using appropriate special fundamental solutions as the trial functions of the intraelement displacements. It is noted that the HFS-FEM has attracted more attention of researchers in computational mechanics in the past few years, and good progress has been made in the field of potential problems, plane elasticity, piezoelectric problems, and so on. However, there are still many possible extensions and areas in need of further development in the future: to develop various special-purpose elements to effectively handle singularities attributable to local geometrical or load effects (holes, cracks, inclusions, interface, corner, and load singularities), with the special-purpose functions warranting that excellent results are obtained at minimal computational cost and without local mesh refinement, to extend the HFS-FEM to elastodynamics, fluid flow, thin and thick plate bending, and fracture mechanics, to develop efficient schemes for complex engineering structures and improve the related general purpose computer codes with good preprocessing and postprocessing capabilities, to extend this method to the case of multifield problems such as thermoelastic-piezoelectric materials and thermomagnetic-electric-mechanical materials, and to develop multiscale framework across from continuum to micro- and nanoscales for modeling heterogeneous materials.
The authors declare that there is no conflict of interests regarding the publication of this paper.