Solution of solid mechanics using equilibrated basis functions in three-dimensional space Daniyal Afifi d.afifi@cv.iut.ac.ir Date of submission: September 19, 2020 Degree: M.Sc. Language: Farsi Department of Civil Engineering Isfahan University of Technology, Isfahan, Iran Supervisor: Assis. Prof. Nima Noormohammadi (noormohammadi@cc.iut.ac.ir) Supervisor: Prof. Bijan Boroomand (boromand@cc.iut.ac.ir) In this paper, solution of 3-D elasticity and harmonic problems is considered by using Equilibrated Basic functions (EqBFs) in homogenous and heterogenous media. The methods using basis functions, including EqBFs, treat satisfaction of the Partial Differential Equation (PDE) and the boundary conditions in separate steps, thus reducing the overall solution effort. Besides the mesh generation procedure is omitted by only considering a boundary point set. A weighted residual approach in weak form has been used for satisfaction of the homogeneous PDE independent from imposition of the boundary conditions. By assuming a fictitious cubic domain that surrounds the main domain of the problem, all related parameters may be decomposed into a combination of 1-D components, which breaks the main 3-D integrals into a combination of 1-D pre-evaluated normalized ones stored as library values, resulting in a drastic increase in the speed of calculations and omission of the usual numerical integration progress visible in most of the numerical techniques. In spite of high convergence rate and excellent accuracy for simple benchmark examples, spectral methods including EqBFs cannot be effectively adapted to large scale problems due to emergence of ill-conditioning in the resulted matrix equations, therefore a Meshless Local Equilibrated Basis Function (MLEqBFs) method is developed as well. The method considers some nodes for definition of the Degrees of Freedom (DOFs) as displacement components throughout the problem domain. Each node corresponds to a local sub-domain called cloud, which includes some other nodes than the main central one, resulting in the overlap of the clouds. The mentioned overlap between adjacent clouds spreads the effective continuity of both the solution function as well as its derivatives in the form of stress components all over the solution domain, which is an advantage with respect to the commonly used formulations like the well-known finite element method. Boundary conditions are also applied over a set of boundary points completely independent of the domain nodes, granting the method the ability of application for arbitrarily shaped domains without the drawback of irregularity in the domain node grid. From this point of view, the proposed method may be counted for the so called fixed grid techniques. The presented examples also prove the capabilities of the method. Keywords: 3-D Elasticity and Harmonic, Equilibrated basis functions, Meshless method, Weighted residual approach.