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This chapter develops a set of two-step identification methods for social interactions models with unknown networks, and discusses how the proposed methods are connected to the identification methods for models with known networks. The first step uses linear regression to identify the reduced forms. The second step decomposes the reduced forms to identify the primitive parameters. The proposed methods use panel data to identify networks. Two cases are considered: the sample exogenous vectors span Rn (long panels), and the sample exogenous vectors span a proper subspace of Rⁿ (short panels). For the short panel case, in order to solve the sample covariance matrices’ non-invertibility problem, this chapter proposes to represent the sample vectors with respect to a basis of a lower-dimensional space so that we have fewer regression coefficients in the first step. This allows us to identify some reduced form submatrices, which provide equations for identifying the primitive parameters.

Numerical results generated by a highly efficient finite‐difference method (originated by Keller for aerodynamical flows at the California Institute of Technology in 1970), and a robust double shooting Runge‐Kutta‐Merson scheme are presented for the boundary layer equations representing the convection flow of a viscous incompressible fluid past a hot vertical flat plate embedded in a non‐Darcy porous medium. Viscous dissipation due to mechanical work is included in the temperature field equation. The computations for both solution techniques are compared at the leading edge (ξ = 0.0) and found to be in excellent agreement. The effects of the viscous heating parameter (Ec), thermal conductivity ratio (λ) and a Darcy porous parameter (Re/GrDa) on the fluid velocities, temperatures, local shear stress and wall heat transfer rate are discussed with applications to geothermal and industrial flows.

This article describes a method for mesh generation, suitable for applications of the finite‐element method, which proceeds fully automatically from a geometric model of the object provided by a CAD‐system. It first generates a coarse mesh which is then adapted to fit the finite‐element problem. A resulting system of equations can be solved by a Gaussian‐type matrix method with as few computations as are necessary for a well‐banded matrix, but without the need for node or element numbering.

Derives a first order perturbation algorithm for the computation of mean values and (co‐) variances of the transient temperature field in conduction heated materials with random field parameters. Considers both linear as well as non‐linear heat conduction problems. The algorithm is advantageous in terms of computer time compared to the Monte Carlo method. The computer time can further be reduced by appropriate transformation of the random vectors resulting from the discretization of the random fields.

Computational algorithms for finite element dynamic analysis of large‐scale structural problems that exploit both concurrent and parallel features of multiple instruction multiple data streams computers are presented. A new computer program architecture is used in which large finite element domains are automatically divided into subdomains. The number of subdomains generated is equal to the number of available processors. The spatial solution is obtained using a basis of orthogonal vectors. The temporal solution is computed exactly. Discussion is focused on the concurrent generation of global Ritz vectors. Examples run on a hypercube multiprocessor confirm the potential of the proposed scheme.

The Semiloof shell element stiffness and mass matrices are analysed. Various integration rules for the stiffness matrix are used, and the influence of these rules on the existence of mechanisms and on the element spectra is studied. Some methods for lumping the mass matrix are attempted with special reference to a method imposing a given behaviour of the spectra of eigenvalues.

This paper sets out to present developments of a numerical model of squeeze casting process.

The entire process is modelled using the finite element method. The mould filling, associated thermal and thermomechanical equations are discretized using the Galerkin method. The front in the filling analysis is followed using volume of fluid method and the advection equation is discretized using the Taylor Galerkin method. The coupling between mould filling and the thermal problem is achieved by solving the thermal equation explicitly at the end of each time step of the Navier Stokes and advection equations, which allows one to consider the actual position of the front of the filling material. The thermomechanical problem is defined as elasto‐visco‐plastic described in a Lagrangian frame and is solved in the staggered mode. A parallel version of the thermomechanical program is presented. A microstructural solidification model is applied.

During mould filling a quasi‐static Arbitrary Lagrangian Eulerian (ALE) is applied and the resulting temperatures distribution is used as the initial condition for the cooling phase. During mould filling the applied pressure can be used as a control for steering the distribution of the solidified fractions.

The presented model can be used in engineering practice. The industrial examples are shown.

The quasi‐static ALE approach was found to be applicable to model the industrial SQC processes. It was found that the staggered scheme of the solution of the thermomechanical problem could parallelize using a multifrontal parallel solver.

This paper aims to propose a novel direct method for indefinite algebraic linear systems. It is well adapted for sparse linear systems, such as those of two-dimensional (2-D) finite elements problems, especially for coupled systems.

The proposed method is developed on an example of an indefinite symmetric matrix. The algorithm of the method is given next, and a comparison between the numbers of operations required by the method and the Cholesky method is also given. Finally, an application on a magnetostatic problem for classical methods (Gauss and Cholesky) shows the relative efficiency of the proposed method.

The proposed method can be used advantageously for 2-D finite elements in stepping methods without using a block decomposition of matrices.

This method is advantageous for direct linear solving for 2-D problems, but it is not recommended at this time for three-dimensional problems.

The proposed method is the first direct solver for algebraic linear systems proposed since more than a half century. It is not limited for symmetric positive systems such as many of direct and iterative methods.

The purpose of this study is to investigate the post-buckling analysis of functionally graded columns by using three analytical, approximate and numerical methods. A pre-defined function as an initial assumption for the post-buckling path is introduced to solve the differential equation. The finite difference method is used to approximate the lateral deflection of the column based on the differential equation. Moreover, the finite element method is used to derive the tangent stiffness matrix of the column.

The non-linear buckling analysis of functionally graded materials is carried out by using three analytical, finite difference and finite element methods. The elastic deformation and Euler-Bernoulli beam theory are considered to establish the constitutive and kinematics relations, respectively. The governing differential equation of the post-buckling problem is derived through the energy method and the calculus variation.

An incremental iterative solution and the perturbation of the displacement vector at the critical buckling point are performed to determine the post-buckling path. The convergence of the finite element results and the effects of geometric and material characteristics on the post-buckling path are investigated.

The key point of the research is to compare three methods and to detect error sources by considering the derivation process of relations. This comparison shows that a non-incremental solution in the analytical and finite difference methods and an initial assumption in the analytical method lead to an error in results. However, the post-buckling path in the finite element method is traced by the updated tangent stiffness matrix in each load step without any initial limitation.