Application of the Double Approximation Method for Constructing Stiffness Matrices of Volumetric Finite Elements

Introduction. In finite element modeling of the stress-strain state of massive bodies, volumetric finite elements (FE) in the form of parallelepipeds (hexahedra), prisms and tetrahedra are used, the construction of stiffness matrices of which is usually performed using isoparametric technology [1-5]. At the same time, it is known that multilinear isoparametric FE, when using a single-layer scheme, do not satisfactorily model bending deformations even with a significant thickening of the mesh [6][7]. The core of this problem is the effect of “locking” the element due to the so-called deformation of the “false shift” [8][9]. To “improve” isoparametric FE, an apparatus of incompatible elements created through introducing additional out-of-node degrees of freedom, or auxiliary approximating polynomials, is used [8]. At the same time, the most effective way to solve the problem of “jamming” of the FE is the use of the moment scheme of the finite element method, whose theoretical foundations were developed by A.S. Sakharov [7]. Subsequently, this approach was called the double approximation method (DAM) [6]. Conceptually, the DAM is based on a separate representation of the distribution functions of displacements and deformations inside the element. The objective of this study is to construct on the basis of MDA and test new volumetric multilinear FE that allow simulating the behavior of various structures under different types of external actions.

Materials and Methods. Let us consider a family of volumetric FE consisting of eight-node and six-node elements in global Cartesian axes , = 1, 2, 3 (Fig. 1). Geometry and displacements of the FE are presented in the following form:

;

,

where

, — nodal coordinates and displacements;

— “shape functions” representing the product of one-dimensional Lagrange linear polynomials;

, , — local, in general, nonorthogonal coordinates of FE;

— number of element nodes.

For the basic eight-node element (Fig. 1 a) = 8, “shape functions” are defined by formula:

, (1)

Here, — coordinates of nodes in local axes. We set values in matrix form:

.

The relationship between the covariant components of the strain tensor in the local basis and displacements in the global axes has the form [7]:

where

;

.

The relationship between vector of deformations and vector of nodal displacements is represented in matrix form:

,

where block matrix:

;

submatrix:

,

.

Expressions for the vector columns of the considered FE have the following form [10]: an eight-node element (Fig. 1 a)

(2)

Six-node element (Fig. 1 b)

. (3)

The notation is introduced here:

,

;

.

;

;

.

Expressions for “shape functions of a six-node FE obtained on the basis of the polynomial (1) using the degeneration principle have the following form:

;

;

;

;

;

.

Formulas (2) and (3) were the basis for constructing stiffness matrices of the FE under study. The corresponding software was developed on the basis of the Microsoft Visual Studio computing platform and the Intel® Parallel Studio XE compiler with the built-in Intel® Visual Fortran Composer XE text editor. The processes of storing and processing the global stiffness matrix were implemented in terms of sparse matrices [11]. To visualize the results of calculations, the descriptor graphics of the MATLAB computer system was used.

Research Results. The investigation on the accuracy and convergence of the developed finite element algorithm was carried out on test examples with an analytical solution. The test examples show numerical solutions obtained using the developed elements and the SOLID185 element of the ANSYS Mechanical software package similar in dimension [5][11]. Below are examples selected so that they contain a combination of bending deformations and rigid displacements of FE.

The convergence results are presented in Table1.

Table 1 shows the value of deflection f in the numerator, and relative error in the denominator.

The visualization of displacements , obtained on the basis of DAM and ANSYS on a 2×2×64 grid is shown in Figures 3 and 4. Note that the field of vertical displacements obtained using ANSYS does not reflect the zones with extreme value = –0.00314 m shown in Figure 3 by the arrow.

In this example, the importance of taking into account rigid displacements is particularly clearly seen.

,

where

= 0.00126;

— cylindrical stiffness;

In this example, ¼ part of the plate was considered taking into account the conditions of symmetry. The convergence results in the form of graphs for single-layer and double-layer models are shown in Figures 5 and 6. Here and further, layers mean the breakdown of the plate into FE by thickness. In these figures, values 1, 2, 3, 4 of parameter 1, 2, 3, 4 correspond to the grids: 4×4, 8×8, 16×16, 32×32. The graphs below show the results of the solution obtained using ANSYS (line 1) and DAM (line 2). The horizontal line indicated by the number 3 corresponds to the exact solution.

It follows from Figure 5 that with a single-layer breakdown scheme, the relative error values on a 32×32 grid are: SOLID185 — 16%; DAM - 10.5%. When using a two-layer scheme (Fig. 6) on a 32×32 grid, we have: SOLID185 — 36%; DAM — 2.8%.

FE patch testing was performed for the 16×16×2 breakdown scheme with grid distortion (Fig. 7). The patch test results in the form of deflection distribution patterns are shown in Figures 8 and 9.

As can be seen from the Figures, the distortion of the grid, when using SOLID185, causes a more noticeable asymmetry of field , than when using DAM. At the same time, the value of the maximum deflection for FE DAM = 0,001263 m coincides with the exact solution.

Example 3. A round plate, rigidly pinched along the contour and loaded with a uniformly distributed load. Radius and thickness of the plate: R = 1 m; h = 0.01 m. The mechanical constants are similar to the data in Example 2.

The exact value of the deflection in the center of the plate is determined by formula [12]:

.

The convergence results in the form of graphs for sampling the ¼ part of the plate according to version 1 with single-layer and two-layer breakdown schemes are presented in Figures 11, 12, and Figures 13, 14, respectively

In these figures, values 1, 2, 3, 4 of parameter S 1, 2, 3, 4 correspond to the grids: 4×4, 8×8, 16×16, 32×32. As in Example 1, line 1 corresponds to a SOLID185-based solution, and line 2 corresponds to DAM. The horizontal line indicated by the number 3 corresponds to the exact solution.

It follows from the above graphs that an element constructed according to the moment scheme on a 32×32×2 grid of version 2 has a relative error of 4 %.

Visualization patterns of the field of distribution of vertical displacements for DAM and SOLID185 with a radial layout of the ¼ plate (32×32×2 grid) are shown in Figures 15 and 16, respectively.

From the above figures, it can be seen that despite the qualitative coincidence of patterns , relative errors for the maximum deflection are: DAM — 3.8%; SOLID185 — 71%. Such a significant error, when using SOLID185, is because the developers applied “shape functions” similar to the functions used for the eight-node element to approximate the geometry and displacements of the six-node element, i.e., without the “degeneration” principle [7].

Discussion and Conclusion. Stiffness matrices of volumetric multilinear finite elements constructed on the basis of the double approximation method make it possible to simulate the stress-strain state of building structures of arbitrary geometry under various types of external actions. The fundamental difference between the proposed concept and the previously known finite element technologies is that the displacements, in this case, are set in global coordinates, and the components of the strain tensor are determined in local, in general, nonorthogonal axes.

The test examples show that the volumetric finite elements, constructed by the double approximation method, have stable convergence and compete successfully with an element of a similar type SOLID185 of the ANSYS Mechanical computing complex.

The developed mathematical software can be introduced into domestic import-substituting software complexes implementing the finite element method in the form of the displacement method.