INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 33, 1997-2014 (1992)

A THREE-DIMENSIONAL BEM SOLUTION FOR PLASTICITY USING REGRESSION INTERPOLATION

WITHIN THE PLASTIC FIELD

ANIL GUPTA

Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, M A , U.S.A.

H U G O E. DELGADO

Wyman Gordon Company, North Grafon, M A , U.S.A.

JOHN M. SULLIVAN, JR.*

Mechanical Engineering Department, Worcester Polytechnic Institute, Worcesler, MA, U.S.A.

SUMMARY

This paper presents an improved solution of three-dimensional plasticity problems using the boundary element method (BEM). The BEM formulation for plasticity requires volume as well as boundary discretiz- ations. An initial stress formulation is used to satisfy the material non-linearity. Conventionally, the plastic field in the volume element (or cell) is interpolated based on the value of plastic stress at the nodes of the cell. In this paper, the distribution of the plastic field in the cell is based on a number of points interior to the cell. The plastic field is described using regression interpolation polynomials through these interior points. The constitutive relation is satisfied at each interior point. The number of points can be varied in each cell, thus allowing for adaptive volume cells. The plastic stresses are computed at the interior points only, therefore, the need for surface stress computation (which uses numerical derivatives at the surface) is completely eliminated. Three-dimensional applications are used to compare the present regression interpolation procedure with the conventional method for elasto-plasticity problems. In all variations of the applications studied regression interpolation based on interior points provided superior results to those determined via the conventional nodal interpolation method.

INTRODUCTION

BEM has reached a state of relative perfection for linear analysis, but non-linear response simulations are still being investigated with varying degrees of success by many researchers. The earliest plasticity formulation with BEM was presented by Swedlow and Cruse,' wherein the initial plastic strain field was incorporated in the governing integral equation through a volume integral. Ricardella' implemented the algorithm for two-dimensional elasto-plasticity using constant domain cells for the plastic strain field. To compute stresses, Ricardella evaluated the displacement integral analytically before differentiating it for stresses. Mendel~on,~ Mukherjee4 and others implemented the plasticity formulation and computed interior stresses by directly differentiating the integral equation for displacement. The strongly singular nature of the volume

* Correspondence Coordinator

0029-598 1/92/13 1997-18$09.00 0 1992 by John Wiley & Sons, Ltd.

Received 22 August 1990 Revised 25 February 1991

1998 A. GUPTA, H. E. DELGADO AND J. M. SULLIVAN, JR

integral involving the plastic field was not addressed in these solutions. However, Bui5 differenti- ated the integral equation for displacement to compute interior stresses by taking the convective differentiation (Leibnitz formula) which incorporated the singular nature of the plastic field.

The works of Banerjee and co-workers,6- s3 Mukherjee and co-workers,14- s 6 Brebbia et al.,17 and others have demonstrated the potential benefits of using BEM for elasto-plasticity, but it still has not achieved the level of effectiveness sufficient to be considered a serious competitor to a fully developed FEM technique. Albeit, many of these researchers have pointed out positive attributes of the BEM method.

For non-conservative force fields like plasticity, the BEM formulation retains the surface and volume integrals. The need for volume discretization to solve plasticity problems appears to give BEh4 an appearance of the finite element method (FEM), but it differs from FEM in one important aspect. The degrees of freedom in FEM are those associated with the nodes of the volume elements. whereas the degrees of freedom in BEM are limited to the nodes of surface elements only. In BEM, there is no required correlation between the surface and volume discretization of the analysis domain The volume cells (as opposed to finite elements in FEM) are needed only to compute the spread of the plastic zone as the solution evolves. This independency of the volume cells to the response prediction makes BEM very attractive for large deformation and highly non-linear analysis. The volume cells can distort severely (surface elements will not necessarily distort to a similar extent) as the response evolves without affecting the response predictability, thus avoiding the need for frequent re-meshing.

Traditionally, the plastic effect within a volume cell is interpolated based on plastic stresses at the cell nodes. An accurate plastic field requires small volume cells in areas of high plastic stress gradients. Also, for the cell nodes lying on the surface, the ‘integral equation’ can not be used to compute the stress field because the kernels are hypersingular on the surface. Therefore, an indirect scheme based on numerical differentiation of the displacement field at the surface element connected to the cell node must be used to compute the surface stresses. The accuracy of the computed stresses 1s a function of the connected element sizes and the relative accuracy of the field variables (displacement and traction) at the nodes of the elements.

Numerous situations exist where the surface discretization does not conform to the volume mesh. A surface may require refinement to simulate large surface stress gradients compared to mild distortions within the interior. This situation requires a finer surface mesh (leading to more collocation points) to simulate the incremental response and results in non-conforming surface and volume meshes. Alternatively, the meshing strategy of the surface may be triangular and independent of a volume brick deployment. The net result is the creation of surface nodes which are unattached to the volume discretization. For such nodes the accuracy of the singular volume integral may be compromised because a large solid angle is subtended at the opposite faces of the volume cell. It is shown in this paper that the error with the conventional procedure from such unattached nodes is cumulative and results in an erroneous collapse load.

In the present formulation the interpolation points are moved inside the volume cell. The number of such sampling points can be varied in each cell to account for the variation of the plastic field or the size of the cell. The interpolation through these points is based on a multi- dimensional polynomial regression. The interior version of the boundary integral equation can be used to compute stresses at the interior points as the kernels are tractable inside the domain even in the presence of a plastic field. For simplicity, in this paper, the integral equation is not used to compute interior stresses but instead the stresses are computed from differentiation of the interpolated displacement field (as in FEM). The present formulation allows for larger cells (by varying the number of interior points) and it avoids the accumulation of error due to non- conforming surface and volume meshes.

3-D BEM SOLUTION FOR PLASTICITY 1999

In the present discussion the plastic stress field is interpolated, based on plastic stresses at 3 x 3 x 3 Gauss point locations (27 stress points), with SO independent regression coefficients (a complete second order least-squared polynomial regression). Since the stresses are always computed inside the cell, the problem of non-unique material properties for the nodes lying at the interface of two zones is eliminated. The conventional procedure requires some kind of averaging procedure for this situation.

SMALL STRAIN ELASTO-PLASTIC BEM FORMULATION

The following steps describe, in brief, a boundary element formulation for solving an elasto- plasticity problem. Please refer to References 12 and 17 for more details.

The required governing equations for BEM elasto-plasticity are:

1. Equilibrium equation: The basic governing equation for elastic or elasto-plastic analysis is

where 6, is the stress rate (or increment) w.r.t. pseudo time and xj is the j : ’ component of the current co-ordinates.

2. Strain rate decomposition: The total strain rate is assumed to be divisible into elastic and plastic strain rates, i.e.

dij = d; + 25 = symmetric part of deformation gradient (2) ++-) 1 ddj dd j 2 ax, a x i

where dij, i; and 2; are the total strain rate, elastic strain rate and plastic strain rate, respectively. The displacement rate is tii and xi is the current co-ordinate.

3. Hooke’s stress-strain law for elastic strain rate: The elastic strain rate is related to the elastic stress rate as

6.. LJ = EJ + 2p95 (34 where 6,, is the Kronecker delta and and ,LL = E/2(1 + 1,); E is the Young’s modulus and v is the Poisson’s ratio.

and p are the Lame’s constants given as 1, = 2pv/(l - 2v)

Substituting (2) in (3a) we get

6, = 6 i j A ( d k k - i [ k ) + 2p(dij - i;) (3b) 4. Prandtl-Reuss stress-strain law for plastic strain rate: The plastic strain rate is related to the

instantaneous stress deviation as

i?. [ J = dps.. 1.l (4) where sij is stress deviation and d p is a positive scalar factor of proportionality (eliminated through the yield condition) and in general depends on time t (strain hardening).

5. Weak statement of equilibrium: The weighted integral of the equilibrium equation over the domain is forced to be zero, i.e.

2000 A. GUPTA, H. E. DELGADO AND J. M. SULLIVAN, JR.

where wij is the weighting function and is taken to be the Green’s function for the equilibrium differential operator. Integrating (5) by parts twice we get the direct boundary element formula- tion for an elastic system in the presence of an initial stress/strain field,

[Gij(k t)ii(x) - Fij(x, t)Gi(x)l dS(x)

where t i i , ii and 6; are the displacement rate, traction rate and initial stress rates, respectively. Also Gij, Fij and Bikj are the fundamental displacement, fundamental traction and fundamental strain, respectively, at location ‘x’ due to a unit load placed at location t.

Equation (6) provides a relation that can be used to compute the displacement rates at any interior point due to known and unknown values of boundary traction rates, displacement rates and the prescribed distribution of initial stress rates over the volume. Since the unknown initial stress rate components cannot be established a priori they are calculated using an iterative algorithm.

Equation (6) can be changed to a singular formulation by taking the load point t onto the boundary point to, i.e.

Edij - Cij(tO)I~i(to) =Z CGij(x, t~ ) i i (x ) - Fij(x9 to)&(x)I dS

where C i j ( t o ) is the discontinuity or jump term arising from the treatment of the improper integral involving Fij.

The solution strategy for an elastic-plastic response can be described briefly in the following five steps.

1 . 2.

3. 4. 5.

Compute the elastic response of the body under applied incremental loads. Satisfy the constitutive relation of the material and compute the fictitious stress/strain field to balance the incremental loads. Compute the elastic response due to the fictitious field. Repeat 1-3 until the fictitious field is negligible. Update the material state at each cell node and repeat the above process for all incremental loads. The converged solution is the sum of all the responses.

BOUNDARY INTEGRATION

The boundary (surface) integrals are discretized using 6-noded isoparametric continuous triangu- lar elements and/or 8-noded continuous isoparametric quadrilateral elements. The integration strategy used is a function of the nature of the kernel (integrand).

(a) Non-singular-Integration is performed based on a few preselected sets of Gauss points. The choice of a particular set depends on the relative proximity of the collocation node and the element being integrated. This procedure avoids the element subdivision technology1* at the expense of a slight inaccuracy bfl realizes substantial savings in computer time.”

3-D BEM SOLUTION FOR PLASTICITY 2001

(b) Weakly singular (removable singularity)-Singularity is of order l/r. It is removed by changing the variable of integration to a polar form.” Again, the Gauss points are preselected to save computing time.

(c) Strongly singular-These integrals exist in the sense of the Cauchy principal value and are computed indirectly by forcing the integral equation to satisfy the rigid body modes.

VOLUME INTEGRATION

The volume integrals are discretized using 20-noded isoparametric continuous volume cells (bricks). Again, the integration methodology is a function of the nature of the kernel.

(a) Non-singular-The volume cell is subdivided into sub-cells depending upon the relative proximity of the collocatiqn node and the element being integrated. Sub-division depends on the order of available Gauss point quadrature and the specified error bound.’** 2o Collocation nodes lying very close to the volume cell result in excessive sub-division (quasi-singular) to satisfy a uniform error bound. Such a condition is more likely to occur for a collocating surface node unattached to the volume discretization.

(b) Weakly singular-These kernels have a singularity of order l/r2 which can be removed by using spherical co-ordinates or by mapping the pyramid shaped sub-cells into a cuboid such that the Jacobian of transformation is proportional to r2 . In this work the latter approach is used to remove the singularity.

The brick cell (integration domain) is subdivided into pyramids with the vertex at the collocation point and the face of the cell as the base. Each pyramid is mapped into a unit cube. The transformation Jacobian for the mapping is of order r2 . The integration accuracy for a given number of Gauss points depends on the solid angle subtended at the vertex of each pyramid. Collocation nodes unattached to the volume mesh subtend a large solid angle at the vertex. Increasing the order of Gauss points is not a solution, which we demonstrate in this work. One possible solution is to subdivide the pyramid into sub-pyramids. But, this will lead to a complex computer code as well as excessive computer time.

INTERIOR STRESS COMPUTATION

To compute interior stresses at the preselected interior locations in the presence of a plastic field the following two procedures are used:

1. Integral equation-The interior version of the governing boundary integral equation (equation (6 ) ) is used to compute stresses at an interior location. The strongly singular integrals are computed indirectly using the response due to a constant plastic field. In the regression interpolation the strongly singular kernels are those associated with the constant term of the regression equation, whereas in the conventional formulation the diagonal coefficients corres- ponding to the internal cell nodes are singular in nature.

This integral equation is used to compute interior stresses in the conventional formulation (presented for comparison) but it is not used in the present proposed modification, for simplicity. (The integral equation formulation would increase the accuracy of the computed stresses. However, for the tests demonstrated herein this enhancement was not necessary.) The proposed modification makes use of the shape function interpolation procedure (explained below). References 21 and 22 describe in detail the numerical implementation of the integral equation procedure.

2002 A. GUPTA, H. E. DELGADO AND J. M. SULLIVAN. JR

2. Shape function interpolation-Strains are calculated exactly as in FEM, i.e. using the nodal displacements at the nodes of the volume cell and differentiating the interpolated displacement field,

where 5 is the natural co-ordinate with u and x interpolated as 20 20

In matrix form for i = 1 equation (8) can be written

For small rotation analysis, strain can be written as the symmetric part of the deformation tensor,

1 aui auj & . . = - -+-

I~ 2 0 a.xj axi

Elastic stresses are then obtained from Hooke's law (equation (3a)).

(9)

LOCATION O F INTERIOR POINTS

In the present study the interior stresses are computed at the 3 x 3 x 3 Gauss point locations. However, alternate interior locations could have been selected. In fact, the interior point resolution can be varied depending on cell size. In either case, the constitutive relation is satisfied at all interior locations. Barlow has shown that Gauss point locations provide excellent stress computation results.23

REGRESSION PROCEDURE FOR PLASTIC STRESS FIELD

A least-square regression is a statistical procedure that fits a curve through a data set such that the square of the deviation from the predicted and the actual values in the data set is minimized. In the present formulation, the plastic stress computations are carried out at the 3 x 3 x 3 Gauss point locations in the isoparametric cuboidal cell element. This plastic stress data set is used in a regression procedure to obtain a complete second order regression model. For each stress component (six total) in a particular cell, the matrix formulation of regression procedure can be written asz4

Xb = Y

Premultiplying both sides by X', we get

X'Xb = XTY

3-D BEM SOLUTION FOR PLASTICITY

where

2003

and ::: 0::

Y =

Premultiplying both sides by (XTX)-' yields

b = [(X'X)-'X"] Y

b = [K]Y

The matrix K remains constant for a preselected number of interior points. Since I( is independent of cell, stress component and iteration, it can be computed once and saved for further use.

A particular stress Component in isoparametric co-ordinates in terms of regression coefficients is given as

011 == hi + b 2 1 + b3q + b,y + b 5 i 2 + b6y2 + b7y2 + bsjq + 6911) + bloYY (12) In this expression, gl1 is a particular component of stress at any interior point in the intrinsic cell. The b's in equation (12) are the regression coefficients based on the same component of stress at 27 interior points. Thus, for each cell six different sets of coefficients corresponding to six stress components exist.

NUMERICAL RESULTS

The boundary integral equation (6) with regression interpolation for the plastic stress field is solved to obtain the elasto-plastic response. The usual procedure of assembling the discretized integral equation for known and unknown boundary quantities is performed. The plastic kernel ' B is computed once and saved for subsequent use in each iteration. An associated flow rule with von Mises yield criteria and isotropic hardening rule is used for plasticity theory. A modified Newton-Raphson method solves the non-linear response. An acceleration parameter based on the history of plastic stresses in the previous increment is applied for the current increment.

Example A

Thick pressurized cylinder. A 15" sector of the cylinder is considered for analysis (Figure 1). The ratio of outer to inner radius is two. The sector is analysed under plane strain boundary

2004 A. GUPTA, H. E. DELGADO AND J. M. SULLIVAN, JR

Figure I . Mesh A of a thick pressurized cylinder

Figure 2. Mesh B of a thick pressurized cylinder with refined boundary mesh

conditions. Appropriate boundary conditions are applied to simulate the symmetry condition. The pressure at the inner radius is increased in small uniform steps (such that the dynamic effects need not be considered) till collapse. Small displacement theory is assumed to be valid throughout the analysis. In reality the displacements are considerable. All comparisons are matched to results obtained from a highly refined FEM model using the small displacement assumption. The FEM solution is obtained from a commercial code (ABAQUSZ5) using twenty 4-noded plane strain elements with reduced integration.

The constitutive model was an elastic-perfectly plastic system. The material data used were: modulus of elasticity, E = 2600 psi; Poisson’s ratio, v = 0.3; yield stress, cry = 400 psi; and plastic strain hardening parameter, H’ = 00 psi.

Case study A . I . T W 0 variations and the basic BEM model (Figure 1, Mesh A) of the cylinder are analysed to study the effect of mismatch (incompatibility) of boundary mesh and the volume mesh. The variations are shown in Figures 2 and 3 and are labelled as Mesh B and Mesh C. The

3-D BEM SOLUTION FOR PLASTICITY 2005

0 0 Nodes of Volume Cell

Nodes unattached to Volume Cell

1 Refined Boundary Mesh

Mesh C

Figure 3. Mesh C of a thick pressurized cylinder with refined boundary mesh

Table I. Mesh configurations for thick pressurized cylinder model

Boundary elements Volume elements _______ Mesh

type Number Type Number Type

A 22 Quadrilateral 5 Brick B 42 Quadrilateral 5 Brick c 128 Triangle 5 Brick

difference between the three meshes is in the number and type of boundary elements (Table I). The volume elements (five 20-noded brick elements) are identical in all the three meshes.

BEM analyses with the conventional approach on the three meshes show substantial differ- ences. Only Mesh A yielded a correct collapse response. Mesh B and Mesh C show a steadily rising pressure displacement response without collapse. The BEM analysis with the regression approach obtained the same collapse pressure for all meshes.

In both approaches, volume integration is carried out using 4 x 4 x 4 Gauss quadrature in each sub-cell with the boundary node as the apex for the singular integration.

The erroneous results in the conventional approach are a function of the inaccurate singular volume integration from the extra boundary nodes (nodes which lie on the face of a volume cell). The inaccurate integration results in erroneous responses at the nodes which propagate to the next iteration. Although increasing the Gauss quadrature can reduce the error somewhat, the computational requirements are considerable. Gauss quadrature of 13 x 13 x 13 in a sub-cell is 40 times the computational effort of a 4 x 4 ~ 4 analysis. Alternatively, the subdivision of the pyramid sub-cell retains the CPU efficiency, albeit with a more complex code.

The conventional approach can be modified (called modified conventional in this work) by computing stresses at the nodes (on the surface) of the volume cell from the displacement field, interpolated based solely on the nodes of the volume cell. This requires an additional boundary connectivity array to be used for the boundary stress recovery at the volume cells. The numerical

2006 A. GLJPTA, 11. E. DELGADO AND J. M. SULLIVAN, JR

error from singular volume integration is still present in the system but its propagation is constrained. This modified BEM strategy is a compromise between the additional accuracy due to more degrees of freedom provided from the extra nodes and the effect of error (singular volume integration) at the extra nodes on the volume cell nodes.

Table I1 compares the singular integration for the outermost cell. Figure 4, with 4 x 4 x 4 Gauss quadrature against that of 13 x 13 x 13 Gauss quadrature. It is apparent that the volume cell

Table 11. Singular volume integration for various node locations (Figure 4) on the outermost cell

Singular integration x 10-1

Gauss quadrature/sub-cell

Nodeno. 4 x 4 ~ 4 1 3 x 1 3 ~ 1 3 %Deviation

N1 N2 N3 N4 N5 N6 N7 N8

* N9 *NlO *N11

3.8 1 5.68 3.83 6.76 3.83 5.68 3.8 1 6.65 9.20

10.08 9.3 1

3.8 1 5.67 3.83 6.85 3.83 5.67 3.81 6.70 9.83

1041 9.74

0.00 0.18 0.00 1.31 000 0.18 0.00 0.75 6.41 3.17 4.4 1

*Surface nodes unattached to the volume mesh

0 0 Nodes of Volume Cell

Nodes unattached to Volume Cell

Figure 4. An outermost 20-noded volume cell of a 15 degree sector of a pressurized cylinder with three unattached surface nodes on the face ABCD

3-D BEM SOLUTION FOR PLASTICITY 2007

nc.des can be integrated relatively accurately with 4 x 4 x 4 quadrature, but extra nodes need a higher quadrature.

Table TI1 lists the predicted collapse pressure as a function of mesh configuration and BEM formulation. Fagalre 5 shows the displacement at Node 1 of the pressurized cylinder for Mesh C with thre: : ~ ~ ~ r o ~ c ~ ~ e s The displacement from BEM is compared against a FEM solution.

able !All. Predicted collapse pressure for thick pressurized cj linder \%ith various approaches of BEM

coilqxe pessure (psi)

Gauss quadfatiire, sub-cell for ioluisle ictsp ation

- - -_-

-~ HEM Mesh --- --

approach type 4 X 4 X 4 8 x 8 ~ 8

A 310 310 Conventional B - 310

C - -

Modified conventional c 321 320

A 365 315 Regression B 315 315

c 315 320

"--'Collapse pressure not obtained; FEM collapse prcssure = 318 psi

4vv

360

320

2x0

240 0.20 0.30 0.40 0.50

DISPLACEMENT @ NODE 1 ( in. )

Figure 5 . Displacement at Node 1 for a pressurized cylinder with various BEM approaches

2008 A. GUPTA, H E. DELGADO AND J. M. SULLIVAN, JR.

Case study A.2. The conventional BEM provides acceptable results for a topological conform- ing discretiLation. Displacement results for two representative nodes of conforming mesh A are shown in Figure 6. Although the conventional results agree closely with the highly refined FTM solution, they do not match the performance of our regression BEM model, Figure 6. Similarly, the regression BEM model provided somewhat superior results for traction than the csnven- tional BEM model, Figure 7. A refined BEM model with 10 volume cells served as our basis for traction comparison to avoid extrapolation of FEM results for stresses at the Gauss points to the node.

0.1 0.2 0.3 0.4 0.5

DISPLACEMENT @ NODE 1 & NODE 21 ( in. )

Figure 6. Displacement at Nodes 1 and 21 for a thick pressurized cylinder

230

210

1 90

170

I50 240 260 280 300 320

INERNAL PRESSURE (psi )

Figure 7. Traction at Node 1 for a thick pressurized cylinder

3-D BEM SOLUTION FOR PLASTICITY 2009

Example B

Thin perforated aluminium strip. A quarter symmetry model of a thin perforated aluminium strip was discretized for the analysis, Figure 8. There is a total of 17 volume cells (20-noded bricks). A uniaxial tension of 140 psi is applied slowly in 30 equal increments (after the first yield). Material characteristics used were: elastic modulus, E = 70000 psi; Poisson’s ratio, v = 0.2; yield

Figure 8. Discretized quarter symmetry model of a thin perforated aluminium strip in tension with 17 volume cells

Figure 9. Fine FEM mesh of a thin perforated aluminium strip

2010 A GIJPTA, H E UELGADO AND J M SIJLLIVAW, JR

stress, CT,, = 243 psi; plastic strain hardening modulus, H' = 2240 psi. The results obtained from BEM with the two approaches are compared against the results from FEM (using ABAQUS with 20-noded brick elements and small strain theory), Figure 9.

Case study B.1. A BEM nlodel (Figurc 10) with triangular elements on part of the surface (enclosed volume undergoing plastic deformation) and quadrilateral elements on the remaining sections is used to study the effect of extra nodes on displacement. A representative example of this effect is shown in Figure 11 for a symmetry node (node 62 of Figurc 14) The BEM with regression approach agrees very closely with FEM results. However, the coriveniional BEM failed to converge with 4 x 4 x 4 Gauss quadrature. Increasing the order of Gauss quadrature to

0 0 Nodes of Vollune Celt

Nodes unattached to Volume Cell

A Typical Boundary Element

Figure 10. Discretized quarter symmetry model of a thin perforated aluminium strip in tension with 17 volume cells and refined boundary mesh

1 40

Figure 11.

100 0.3 0.5 0.7 0.9 1.1 1.3

R I S F L A ~ ~ M ~ ~ @ NODE 62 ( x 0.1 in. )

Comparison of displacement at Node 62 for the refined mesh of perforated alumimum strip

3-L) BEM SOL UTION FOR PLASTli'I TY 201 1

13 x 13 x 13 does show an improvement but its performance is markedly below that of the regression BEM.

C'ase mmdy B.2. The numerical accuracy of both BEM approaches is compared on a conform- ~nig mesh. Figure 8 Since this mesh does not contain extra nodes, the variation in the results is s r k 4 y due to dlflerent approaches used to compute the plastic field. Figures 12 and 13 show displacement profiles for nodes I , 7, 62 and 70. Although the four profiles adhere closely to the F E M results, it IS the regression BEM approach that clearly tracks the FEM results with fidelity.

I40

130

1 20

110 0.2 0.4 0.6 0.8

D I S P L A C E ~ E ~ ~ @ NODE 1 & NODE 7 ( x 0.1 in. )

T m r ? 12 Displacement in Y-direction at Noder 1 and 7 for a perforated aluminium strip with 17 cells

N62 - NODE 62

N l 0 - NODE I 0

0.4 0.6 0.8 1.0 1.2 DISPLACEMENT @ NODE 62 & NODE 70 ( x 0.1 in. )

Figure 1.3. Displacement in X-direction at Nodes 62 and 70 for a perforated aluminium strip with 17 cells

2012 A. GUPTA, H. E. DELGADO AND J. M. SULLIVAN, JR.

Figure 14. Discretized quarter symmetry model of a thin perforated aluminium strip in tension with 14 volume cells

140

130

120

110 0.3 0.5 0.7 0.9 1.1

DISPLACEMENT @ NODE 62 ( x 0.1 in. )

Figure 15. Displacement in X-direction at Node 62 for a perforated aluminium strip with 14 cells

140

130

120

110 0.4 0.6 0.8 1.0 1.2

DISP~ACEMENT @ NODE 90 ( x 0.1 in. )

Figure 16. Displacement in X-direction at Node 70 for a perforated aluminium strip with 14 cells

3-D BEM SOLUTION FOR PLASTICITY 2013

Case study B.3. The effectiveness of regression interpolation for large cell sizes was tested with the same quarter symmetry model but with fewer cells (14 cells), Figure 14. Again there were no extra nodes, thus the only difference in the result is due to the different interpolaton of plastic field. Displacement results for representative nodes are shown in Figures 15 and 16. The enhanced accuracy of the regression approach over the conventional approach of BEM is evident.

CONCLUSION

This paper presented an improved approach for the analysis of three-dimensional plasticity problems with BEM using regression interpolation of the plastic field. The regression fit through :he interior points is a straightforward procedure. The computationally efficient K matrix is generated only once. The procedure does not involve the computation of surface stresses. The number of interior stress points can be varied to account for the size of the cell or the variation of the plastic field. Regression interpolation is flexible in the choice of the number of coefficients. The number of coefficients can be increased to a higher order, second order is used in this paper, for more accurate plastic stress field interpolation (the effect of using higher order regression interpolation is not presented in this work).

The propagation of erroneous results from the conventional approach, due to inaccurate volume integration at surface nodes unattached to the volume mesh, is constrained through the use of a modified conventional and/or regression approach.

Additionally, it is expected that the use of an interior version of the integral equation to compute stresses in the regression based interpolation scheme will increase the accuracy of response further, especially in the regions of high stress gradients.

REFERENCES

1. J . L. Swedlow and T. A. Cruse, ‘Formulation of boundary integral equation for three-dimensional elasto-plastic flow’,

2. P. Ricardella, ‘An implementation of the boundary integral technique for planar problems of elasticity and elasto-

3. A. Mendelson, ‘Boundary integral methods in elasticity and plasticity’, N A S A TND-7418, 1973. 4. S. Mukherjee, ‘Corrected boundary integral equation in planar thermoelastoplasticity’. Int. J . Solids Struct., 13,

5. H. D. Bui, ‘Some remarks about the formulation of three-dimensional thermoelastoplastic problems by integral equations’, Int . J . SoIids Struct., 14, 935-939 (1978).

6. P. K. Banerjee, D. N. Cathie and T. C. Davies, ‘Two and three-dimensional problems of elasto-plasticity’, in De~;elopments in Boundary Element Methods, Vol. I , Applied Science Publishers, London, 1979.

7. P. K. Banerjee and D. N. Cathie, ‘A direct formulation and numerical implementation of the boundary element method for two-dimensional problems of elasto-plasticity’, Int. J . Mech. Sci., 22, 233-245 (1980).

8. D. N. Cathie and P, K. Banerjee, ’Boundary element methods for plasticity, creep including a viscoplastic approach’, Res. Mechanica, 4, 3-22 (1982).

9. P. K. Banerjee and T. G. Davies, ‘Advanced implementation of the boundary element methods for three-dimensional problems of elasto-plasticity and viscopiasticily’, in Development in Boundary Element Methods, Vol. 3, in P. K. Banerjee and S. Mukherjee (eds.), Applied Science Publishers, London, 1984.

10. P. K. Banerjee and S. T. Raveendra, ‘Advanced boundary element analysis of two- and three-dimensional problems of elasto-plasticity’, Inr. j . numer. metkods eny., 23, 985-1002 (1986).

11. D. P. Henry, Jr. and P. K. Banerjee, ‘A new BEM formulation for two- and three-dimensional elastoplasticity using particular integrals’. Int. j . numer. methods eng., 26, 2079-2096 (1988).

12. P. K. Banerjee and R. Butterfieid, BoundarJ; Element Methods in Engineering Science, McGraw-Hill, London, 1981. 13. P, K. Banerjee, R. B. Wilson and N. Miller, ‘Development of a large BEM system for three-dimensional inelastic

analysis’, in Advanced Topics in Boundary Element Analysis, T. A. Cruse el al. (eds.), AMD-Vol. 72, ASME, New York, pp. 1-20.

f n t . J . Solids Struct., 7 , 1673-1683 (1971).

plasticity’, Ph.D. Thesis, Cdrnegie-Melton University, Pittsburgh, , PA, USA., 1973.

331-335 (19771.

14. S. Mukherjee, Boundary Element Methods in Creep and Fracture, Applied Science Publishers, London, 1982. 15. V. Sarihan and S. Mukherjee, ‘Axisymme~ric viscoplastic deformation by the boundary element method‘, f n t . J . Solids

Struct., 18, 1113-1128 11982).

2.[. I4 A. 6;‘IiP’TAi H. R. DELGADO AND J . M . SIJLLIVAN, JR

16. Vl. I.lor,!aria and S. Mukherjee. ‘Improved boundary integral equation method for time dependent inelastic deliirmation in metals’, Int. j . mmer. rnetfzods my., 15, 97-111 (1980).

i y . C. A. Brebbia, .I. C. F. Telles and L. C. Wrobel, Boundary EIement Techniques, Springer-Verlag. Berlin, 1984. 18 J. c‘. Latchat and J, 0. Watson, ‘Effective numerical treatment of boundary integral equation: a formulation for

three-dimensional elasto-statics’, Int. .j. numer. methods eng., 10, 991-1005 (1976). :<I. f H. Kane, A. Gupta and S. Saigal, ‘Reusable intrinsic sample point (RISP) algorithm for the elficient numerical

integration of three dimensional curved boundary elements’, Inf. j . numer. methods. eng., 28, 1661 -1676 (1989). 20. G. G. Mustoe. ‘Advanced integration schemes over boundary elements and volume cells for two and three-

dimensional noniinear analgris’. in P. K. Ranerjee and S. Mukherjee (eds.), Detielopments in Boundii~,y Element Mrtlzods: Vol. 3. Applied Science Publishers, London, 1984.

21. Wyman-Gordw report, a Research Project Progress Summary, Aug. 10, 1988, Attn. Hugo. E. Delgado, Wyman-

Brebbia and S . A. Orsmg (eds.), The Boundary Element Method Applied to Inelastic ProbkFns, nrth G:a!ion, MA-01336, 1988.

Ineci-ing, Sp:.inger-:Tcr!ag, Berlin, 1983. 23. 3 . Barlow. ‘0p:irnal stress 1ucation:i in Finite Elemcnt Models’, Int. j . numer. metftods eng., 10, 243-251 (1976).

crman and M. H. Kutner, Appiicd Linear Sfatistical Models, 2nd edn, R. D. Irwin, Homewood,

25. H. $3. Hibbitt, B. K . KarlssoTI and E. P. Sorensen, ABAQUS-User’s .ManuaE, Version 4.6, HKS, Inc., Providence, Illinois. 1985.

Rhode Island. 1987.