Derivation of Tangent matrix for nonlinear transient problems ?

Yousef Bahrambeigi

Hello Ravi Varma

You can use this Refrences:

https://math.stackexchange.com/questions/2528174/jacobian-for-non-linear-transformations

https://www.lehigh.edu/~wes1/apci/24feb00-s.pdf

https://www.mm.bme.hu/~gyebro/files/ans_help_v182/ans_thry/thy_tool10.html

Article Jacobian matrix: a bridge between linear and nonlinear polyn...

https://faculty.etsu.edu/knisleyj/multicalc/Chap3/Chap3-5/part1.htm

Qamar Ul Islam

Ravi Varma Deriving the tangent matrix for nonlinear transient problems in the context of finite element analysis (FEA) involves linearizing the governing equations and can be quite complex. Below are the general steps involved in the derivation:

Nonlinear Transient Problem Formulation: Start with the governing equations that describe the transient behavior of your system. These equations often involve time derivatives, material properties dependent on temperature (as in your laser sintering case), and possibly other nonlinear terms.

Time Discretization: Discretize the time domain into discrete time steps (e.g., using implicit or explicit time integration schemes). This leads to a time-stepping algorithm.

Linearization: At each time step, linearize the nonlinear terms by Taylor series expansion. The key is to linearize with respect to the unknowns (e.g., temperature field, displacement field) at the current time step.

Assemble Tangent Matrix: The linearized equations can be assembled into a system of linear equations. The tangent matrix (also called the Jacobian matrix) represents the coefficients of these linear equations. It relates the increments in the unknowns to the increments in the loads.

Solve Linearized Equations: Solve the linearized system of equations, typically using a solver like a direct solver or an iterative solver. This provides the increments in the unknowns.

Update Solution: Update the solution at the current time step using the increments obtained from the linearized equations.

Repeat: Repeat steps 3-6 for each time step until the simulation reaches the desired final time.

Unfortunately, providing a complete derivation and test cases within this format is challenging due to the complexity of nonlinear transient FEA. However, I can suggest some resources where you can find detailed derivations and examples:

Textbooks: Look for textbooks on finite element analysis and nonlinear finite element methods. Books by authors like O.C. Zienkiewicz and J.N. Reddy are good starting points.

Research Papers: Explore academic papers and research articles in the field of computational mechanics. Papers related to thermal analysis, materials with temperature-dependent properties, and nonlinear FEA will be relevant.

Commercial FEA Software Manuals: Manuals and documentation provided by commercial FEA software (e.g., Abaqus, ANSYS) often include detailed explanations of the theory and implementation.

Online Courses: Consider enrolling in online courses or MOOCs related to finite element analysis. These courses often cover the derivation of tangent matrices and provide practical examples.

Remember that implementing and verifying such complex simulations can be challenging, and it's important to validate your results against analytical solutions or experimental data when available.

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