In finite element analysis, treating boundary condition having multi freedom constraints is needed to produce modified system of equation based on master stiffness equation considering multi freedom constraint. Generally the operation of imposing multi freedom constraints can be developed using Master-Slave Elimination, Penalty Augmentation or Lagrange Multiplier Adjunction methods. The master-slave method is useful only for simple cases but exhibits serious shortcomings for treating arbitrary constraints. The Penalty Augmentation method and Lagrange Multiplier Adjunction are better in many applications. But there are not free of disadvantages. The penalty method has difficulty of choice of weight values that balance solution accuracy with the violation of constraint conditions. The multiplier method is sensitive to the degree of linear independence of the constraints, and the bordered stiffness is singular in the case of the dependent constrains. To reduce the disadvantages of these methods, this paper presents a new method for treating multi freedom constraint. Proposed method is similar to penalty function method and it is called “modified penalty function method”. The concept of the modified penalty function method is based on the constructing equivalent solver system of equations from traditional modified system using regulatory parameters. The basic of method for selecting the regulatory parameters is reducing the violation of modified system. For solving equivalent system the authors proposed an iterative algorithm for seeking the solution. The calculation programs are established based on proposed algorithm. The calculation results using the proposed method matched with the calculation results using other methods.
This paper is concerned with the treatment of nonlinear multi-freedom and multi-point boundary condition in finite element analysis of frame system. The treatment of boundary constraints is required to produce modified system of equation based on master stiffness equations considering nonlinear multi freedom constraints. The nonlinear constraints considerably increases the difficulty in constructing and solving the modified system of equations. Generally, the operation of imposing multi-freedom constraints can be developed using master-slave elimination, penalty augmentation or Lagrange multiplier adjunction methods. The master-slave method is useful only for simple cases but exhibits serious shortcomings for treating arbitrary constraints. The penalty method has difficulty in selecting appropriate weight values that balance solution accuracy with the violation of constraint conditions. In present work the Lagrange multiplier adjunction methods is employed and endowed with possibility of substitution and works particularly well for nonlinear constraints. The incremental-iterative algorithm based on Crisfield arc-length method is proposed to solve the nonlinear modified system of equation. Based on the presented algorithm, the paper proposed calculation procedure and established programs for determining internal forces and displacements of frames having nonlinear multi-freedom constrains condition. The numerical test examples are presented to investigate load-displacement and load-internal relationship of system having nonlinear multi freedom constraints. The calculation results show the efficiency and convergence of proposed algorithm.
This paper focuses on the treatment of nonlinear multi freedom constraints using an augmented Lagrangian method in finite element analysis of frames. The process of imposing boundary constraints is developed by changing the assembly stiffness equation to produce a modified system of equation considering nonlinear multi freedom constraints. For imposing the nonlinear constraints two better methods are the penalty augmentation method and Lagrange multiplier adjunction method. But there are not free of disadvantages. Using penalty method has a disadvantage in the choice appropriate weight values that balance solution accuracy with the violation of constraint conditions. Using the Lagrange multiplier adjunction method requires additional unknowns, and more complicated storage allocation procedures. This research proposes the connection between these methods using the augmented Lagrangian method for imposing the nonlinear multi freedom constraints in finite element analysis of frame. Based on the Newton Raphson method the incremental-iterative algorithm for solving the nonlinear balanced equations is established.
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