Yunhua Luo
Concrete Division, Institute of Concrete Structures and Building Materials University of Karlsruhe, 76128 Karlsruhe, Germany
e–mail: gd92@rz.uni-karlsruhe.de
ABSTRACT
Summary A gradient-based adaptive procedure is proposed in this paper. The relative error in strain energy from two adjacent adaptation stages is used as a stop-criterion. The refinement criterion is based on the gradient of strain energy density. The strategy was implemented in the Element-Free Galerkin method.
Numerical investigation was conducted to study the performance of the proposed procedure.
General idea In numerical methods such as finite element methods, meshless methods and so on, mesh refinement based on an adaptivity mechanism is becoming a standard procedure, in order to achieve a prescribed accuracy with a minimal number of nodes or to capture a local structural behavior. Most conventional adaptive methods [1, 2] were developed aiming at the finite element methods and it is for this reason, some of them have limitations or are inconvenient in application to meshless methods.
In this paper, a gradient-based adaptive procedure is suggested and implemented in the Element-Free Galerkin method. The two main ingredients in an adaptive procedure are: when to stop and how to refine. For a given structure under a set of given loads with a given load path, at an equi-librium point the total strain energy stored in the structure is definite. As a basic requirement on an adaptive procedure, approximate strain energy must converge to its real value. Therefore the relative variation in strain energy can be used as a stop-criterion. The other ingredient is how to refine meshes. The aim of refinement is to give a better approximation to unkown fields such as displacements, strains, stresses and so on, and thus give a better approximation to strain energy.
Based on this reasoning, the gradient of strain energy density can be used as a guide to refine meshes.
Stop-criterion The principle of minimum potential energy is used here as a starting point.
The total potential energy in a load-bearing structure can be written as
= s
+ e
= 1
2 Z
T
"d Z
p
T
ud Z
S q
T
uds (1)
whereand"are stress and strain vectors.uis the displacement vector.pandqare respectively body force and surface traction.is the domain occupied by the structure andSthe surface where traction is exerted.sand e represent strain energy and external potential energy. For a given structure under a set of given loads with a given load path, at an equilibrium point the total strain energy stored in the structure is definite, although it is unknown in most cases, except an analytical solution to the problem is available.
In pursuing a better approximate solution, an adaptive procedure produces a series of approxi-mate strain energy,~si (i=1, 2,), corresponding to a series of meshesMi (i=1;2;). As a basic requirement on an adaptive procedure, the above series~s
i
This leads to the following equivalent relation
j
Equation (2) provides a criterion to stop an adaptation loop, which is similar to a conventional global error estimator defined by the energy-norm of errors from strains or stresses [2], except that the more exact solution is from a subsequent (better) mesh rather than from the same mesh by a recovery or a projection procedure.
Refinement-criterion The general idea for mesh-refienment in a gradient-based method can be briefly stated as: A larger gradient needs a richer mesh andvice versa. A coarse mesh may not be able to produce accurate solutions to displacements or stresses, but might give ade-quate information about how these fields vary in the problem domain. As regard to variation from which field should be considered, there are quite several choices, displacement, strain or stress and so on. But strain energy density is a reasonable choice for the following reasons: strain energy density is scalar and independent of coordinate system; information about other fields such as dis-placement, strain, stress, load distribution and so on are comprehensively embodied in the strain energy density.
To measure the the richness of mesh with respect to the variation of strain energy density, mesh in-tensityrd is introduced and defined as the ratio of the variation (gradient) of strain energy density to mesh density. whereGSED is the gradient of strain energy density.DM is mesh density defined as the number of nodes in a unit area or volume. The maximum and minimum ofrdare, respectively, denoted as
R
maxandRmin. The criterion for mesh-refinement is set up as
R
Numerical results The above adaptive procedure was implemented in the Element-Free Galerkin method [3, 4] and is obviously applicable to finite element methods in principle. Numerical inves-tigation was conducted to study the performance of the proposed procedure. Results from one example, among many others, are presented here. The cantilever beam shown in Fig. 1 was simu-lated as a plane strain problem. The beam has a lengthL=10and a width2b=2. The material of the beam is elastic, with Young’s modulusE =1000and Poisson’s ratio =0:3. The beam is constrained at its left end and loaded at right end by a distributed shear force with a resultantQ.
y
Figure 1: A cantilever beam simulated as a plane stress model
0 500 1000 1500 2000 2500 3000 3500 4000 4500
0.425
Transverse displacement at point (A)
Adaptive refinement Uniform refinement analytical solution
0 500 1000 1500 2000 2500 3000 3500 4000 4500
0.435
Figure 2: Convergence in displacement at pointAand in total strain energy
To avoid singularity distributed loads are also applied at left end. The convergences in transverse displacement at pointA, cf Fig. 1, and in strain energy are shown in Fig. 2, and compared with analytical solutions. Fig. 3 displays mesh configurations from different stages.
Conclusions A gradient-based adaptive procedure is proposed in this paper. The relative er-ror in total strain energy from two adjacent adaptation stages is used as a stop-criterion. Mesh refinement is guided by the gradient of strain energy density. The procedure was implemented in the Element-Free Galerkin method. Numerical investigations show that the approximate strain energy,~s
i
, steadily converges to its ’real’ value,s, with the increase of node number, or more exactly with the decrease of maximal mesh intensity; Approximate fields such as displacements and stresses converge to their corresponding ’real’ fields with the decrease of error in total strain energy. The gradient of strain energy density is very effective as a guide for mesh refinement; For problems free of singularity the accuracy of approximate fields can always be improved by reduc-ing the maximal mesh intensity; For problems with sreduc-ingularity, convergence in strain energy can
27 nodes
50 nodes
122 nodes
247 nodes
Figure 3: Node configurations from different adaptation stages be guaranteed by suitably controlling the minimal nodal distance.
REFERENCES
[1] I. Babuska and W. C. Rheinboldt. Error estimates for adaptive finite element computations.
SIAM J. Numer. Anal. 1978; 15:736-754.
[2] O. C. Zienkiewicz and J. Z. Zhu. A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Meth. Engng. 1987; 24:337-357.
[3] Y. H. Luo and U. H¨aussler-Combe. An adaptivity procedure based on gradient of energy den-sity and its application in meshless methods. International Workshop on Meshfree Methods for PDEs, Bonn, Sept. 11-14, 2001.
[4] Y. H. Luo and U. H¨aussler-Combe. A gradient-based adaptation procedure and its implemen-tation in Element-Free Galerkin method. Manuscript submitted to Int. J. Numer. Meth. Engng., 2001.