In this article one of the greatest generalized methods for establishing the equilibrium equations of a rigid body and the set of rigid bodies is proposed. It is related to six equations of moments of force about six the edges of a reference tetragon. It is possible to obtain different alternatives by substituting the force moment-equation for the force project-equation. Four different forms of equilibrium are established. It is important writing equilibrium equations of bodies possible to apply the special software as Mathcad, Maple.

*Keywords:* equilibrium of a rigid body, equilibrium of mechanical system, matrix method, frame tetragon, generalized tetragon frame of axis.

Ali Kaveh, Kambiz Koohestani. Formation of graph models for regular finite element meshes. (Received in the final form September 21, 2009) CAMES 2009 (16)

Graph theory has many applications in structural mechanics and there are also numerous topological transformations which make the related problems simpler. The skeleton graph and natural associate graph of finite element models are among such transformations. These transformations can efficiently be used for nodal and element ordering of regular finite element models. Natural associate graph and its mesh basis play a key role in optimal finite element analysis by combinatorial force method. In this paper, an efficient method is presented for generation of skeleton graph, natural associate graph as well as their mesh bases for finite elements models, using graph and digraph products.

*Keywords:* finite elements, graph products, digraph products, associate graph, mesh bases.

S.Y. Fialko. A block sparse shared-memory multifrontal finite element solver for problems of structural mechanics. (Received in the final form July 17, 2009) CAMES 2009 (16)

The presented method is used in finite-element analysis software developed for multicore and multiprocessor shared-memory computers, or it can be used on single-processor personal computers under the operating systems Windows 2000, Windows XP, or Windows Vista, widely popular in small or mediumsized design offices. The method has the following peculiar features: it works with any ordering; it uses an object-oriented approach on which a dynamic, highly memory-efficient algorithm is based; it performs a block factoring in the frontal matrix that entails a high-performance arithmetic on each processor and ensures a good scalability in shared-memory systems. Many years of experience with this solver in the SCAD software system have shown the method's high efficiency and reliability with various large-scale problems of structural mechanics (hundreds of thousands to millions of equations).

*Keywords:* finite element method, large-scale problems, multifrontal method, sparse matrices, ordering, multithreading.

Ashok Dargar, Ali Hasan, Rasheed Ahmed Khan. A method of identification of kinematic chains and distinct mechanisms. (Received in the final form November 19, 2009) CAMES 2009 (16)

A new method is proposed to identify the distinct mechanisms derived from a given kinematic chain in this paper. The kinematic chains and their derived mechanisms are presented in the form of a flow matrix. Two structural invariants, sum of the absolute values of the characteristic polynomial coefficients (SCPC) and maximum absolute value of the characteristic polynomial coefficient (MCPC) are determined using the software MATLAB. These invariants are used as the composite identification number of a kinematic chain and mechanisms and clearly identify the distinct mechanisms derived from the family of 1-F, 8-links and 10-links KC as well as 2-F, 9-links simple joined KC. This study will help the designer to select the best possible mechanism to perform the specified task at the conceptual stage of design. The proposed method does not require any test for isomorphism separately. Some examples are provided to demonstrate the effectiveness of this method.

*Keywords:* kinematic chain (KC), distinct mechanism (DM), flow matrix, SCPC, MCPC.

Mark Azzam1, Jan-Christoph Haag, Peter Jeschke. Application concept of artificial neural networks for turbomachinery design. (Received in the final form August 19, 2009) CAMES 2009 (16)

This paper presents the results of an extensive investigation evaluating and improving the development of artificial neural network (ANN) models for turbomachinery design purposes. A set of 1100 differing axial compressor geometries based on 5 single-stage compressor rigs was prepared. Computations with the mean line analysis tool **AXIAL™** took place to determine the according compressor maps defined by 15 operating points each. The challenge of ANN model development in terms of dimensionality reduction (feature selection), data normalization, defining the networks necessary plasticity, and network training is discussed using the example of three different models. As a result, the first model is able to predict the total pressure loss of the rotor blade row with a mean magnitude of the relative error (MMRE) of 3.6%. The second model predicts the total pressure ratio with an average accuracy of 0.8%. The third and last model was trained to predict basic geometrical parameters by presenting the load level and the performance data as an input. The achieved MMRE varied between 2.4% and 5.6% in respect of the particular output variable. The results show that ANNs are applicable to develop efficient models for turbomachinery design and analysis purposes, respectively.