Comput. Mech. (2006) 38: 533–550 DOI 10.1007/s00466-005-0003-z ORIGINAL PAPER G. R. Liu · K. Y. Dai · X. Han · Y. Li A mesh-free minimum length method for 2-D problems Received: 7 March 2005 / Accepted: 18 August 2005 / Published online: 19 October 2005 © Springer-Verlag 2005 Abstract A mesh-free minimum length method (MLM) has been proposed for 2-D solids and heat conduction problems. In this method, both polynomials as well as modified radial basis functions (RBFs) are used to construct shape func- tions for arbitrarily distributed nodes based on minimum length procedure, which possess Kronecker delta property. The shape functions are then used to formulate a mesh-free method based on weak-form formulation. Both Gauss inte- gration (GI) and stabilized nodal integration (NI) are em- ployed to numerically evaluate Galerkin weak form. The numerical examples show that the MLM achieves better accu- racy than the 4-node finite elements especially for problems with steep gradients. The method is easy to implement and works well for irregularly distributed nodes. Some numerical implementation issues for MLM are also discussed in detail. Keywords Mesh-free method · Meshless method · Minimum length method · Radial basis function (RBF) · Interpolation function 1 Introduction Mesh-free methods have achieved remarkable progress in re- cent years to avoid the problems related to the creation and application of predefined meshes in the traditional numerical methods, such as the finite element method (FEM), the fi- nite difference method (FDM). In general mesh-free methods developed so far can be categorized into three main groups. The first group includes mesh-free methods based on strong- form equations, in which discretization is formed directly G. R. Liu (B )· K. Y. Dai · Y. Li Center for ACES, Department of Mechanical Engineering, National University of Singapore, Singapore 119260, Singapore E-mail: mpeliugr@nus.edu.sg Tel.: +65-6874 6481 Fax: +65-6779 1459 X. Han College of Mechanical and Automobile Engineering, Hunan University, Changsha 410082, P. R. China from governing differential equations, such as the smooth particle hydrodynamics (SPH) method [6, 13], the general fi- nite difference method [9] and other mesh-free collocation methods. The second group covers the mesh-free methods based on weak-form formulation, such as element-free Galer- kin (EFG) method [2], reproducing kernel particle meth- ods (RKPM) [15], meshless local Petrov-Galerkin (MLPG) method [1], radial point interpolation method (RPIM) [10, 17] and other boundary-type mesh-free methods. The third category involves mesh-free methods combining both weak- and strong-form formulations, such as the newly developed mesh-free weak-strong form method [11, 14]. One attractive feature of strong-form methods is the high computational efficiency as no numerical integration is re- quired. However, straightforward imposition of Neumann boundary condition often leads to unstable results. By com- parison, mesh-free weak-form methods can usually achieve higher accuracy than strong-form methods especially in deal- ing with problems in solids and structural mechanics. In addi- tion, they can treat Neumann boundary condition more eas- ily and the results are more stable. Hence weak-form method is considered in this work. One of the most important is- sues in mesh-free methods is the construction of mesh-free shape functions. The developed shape functions should have some primary merits, such as desirable accuracy, very easy to implement, valid and stable for arbitrary nodal distribution. There are two widely used methods: moving least-squares (MLS) method and radial point interpolation method (RPIM). One successful application of MLS is its integration in the so-called EFG method [2]. EFG is a viable method which has very good accuracy and convergence rate and a high res- olution of localized gradient can be achieved. As the MLS uses excessive nodes which leads to shape function lacking in delta function property. Hence its approximation function does not pass through the data points, which consequently complicate the imposition of essential boundary conditions. RPIM based on weak form was then proposed to overcome this difficulty (see, e.g., [17]). This method uses radial basis functions (RBFs) augmented with polynomials to interpolate the data points exactly and the generated shape functions pos-