SUMMARYThe Element-Free Galerkin (EFG) method allows one to use a nodal data structure (usually with an underlying cell structure) within the domain of a body of arbitrary shape. The usual EFG combines Moving Least-Squares (MLS) interpolants with a variational principle (weak form) and has been used to solve two-dimensional (2-D) boundary value problems in mechanics such as in potential theory, elasticity and fracture. This paper proposes a combination of MLS interpolants with Boundary Integral Equations (BIE) in order to retain both the meshless attribute of the former and the dimensionality advantage of the latter! This new method, called the Boundary Node Method (BNM), only requires a nodal data structure on the bounding surface of a body whose dimension is one less than that of the domain itself. An underlying cell structure is again used for numerical integration. In principle, the BNM, for 3-D problems, should be extremely powerful since one would only need to put nodes (points) on the surface of a solid model for an object.Numerical results are presented in this paper for the solution of Laplace's equation in 2-D. Dirichlet, Neumann and mixed problems have been solved, some on bodies with piecewise straight and others with curved boundaries. Results from these numerical examples are extremely encouraging.
Accurate imposition of essential boundary conditions in the Element Free Galerkin (EFG) method often presents difficulties because the Moving Least Squares (MLS) interpolants, used in this method, lack the delta function property of the usual finite element or boundary element method shape functions. A simple and logical strategy, for alleviating the above problem, is proposed in this paper. A discrete norm is typically minimized in the EFG method in order to obtain certain variable coefficients. The strategy proposed in this work involves a new definition of this discrete norm. This new strategy works very well in all the numerical examples, for 2-D potential problems, that are presented here. In addition to the discussion of boundary conditions, some recommendations are also made in this paper regarding strategies for refinements in order to improve the accuracy of numerical solutions from the EFG method.
The Element‐Free Galerkin (EFG) method allows one to use a nodal data structure (usually with an underlying cell structure) within the domain of a body of arbitrary shape. The usual EFG combines Moving Least‐Squares (MLS) interpolants with a variational principle (weak form) and has been used to solve two‐dimensional (2‐D) boundary value problems in mechanics such as in potential theory, elasticity and fracture. This paper proposes a combination of MLS interpolants with Boundary Integral Equations (BIE) in order to retain both the meshless attribute of the former and the dimensionality advantage of the latter! This new method, called the Boundary Node Method (BNM), only requires a nodal data structure on the bounding surface of a body whose dimension is one less than that of the domain itself. An underlying cell structure is again used for numerical integration. In principle, the BNM, for 3‐D problems, should be extremely powerful since one would only need to put nodes (points) on the surface of a solid model for an object. Numerical results are presented in this paper for the solution of Laplace's equation in 2‐D. Dirichlet, Neumann and mixed problems have been solved, some on bodies with piecewise straight and others with curved boundaries. Results from these numerical examples are extremely encouraging. © 1997 by John Wiley & Sons, Ltd.
Thermal stresses were studied in freezing of biomaterials containing significant amounts of water. An apparent specific heat formulation of the energy equation and a viscoelastic model for the mechanics problem were used to analyze the transient axi-symmetric freezing of a long cylinder. Viscoelastic properties were measured in an Instron machine. Results show that, before phase change occurs at any location, both radial and circumferential stresses are tensile and keep increasing until phase change begins. The maximum principal tensile stress during phase change increases with a decrease in boundary temperature (faster cooling). This is consistent with experimentally observed fractures at a lower boundary temperature. Large volumetric expansion during water to ice transformation was shown to be the primary contributor to large stress development. For very rapid freezing, relaxation may not be significant, and an elastic model may be sufficient.
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