## Time-dependent dynamic problem with nonlinear stiffness using ode45

I am solving a time-dependent finite element problem with nonlinearities in the stiffness matrix which takes the following form:

where the dot symbolizes derivative with respect to time, U is the displacement vector, M is the mass matrix, C is the damping matrix, K(U) is the stiffness matrix which depends on U, and F is an external force. I would like to use ode45 to solve the dynamic problem. The general flow for solving this problem would be the following:

Calculate the stiffness matrix K using the latest displacement U (Initial condition if it is the first time step).

Formulate the state-space equations using K matrix, M matrix, C matrix, and F vector.

Integrate the state-space equations with ode45 to caclulate the new displacement U.

Recalculate the K matrix with the new displacement U. Check if the residual error meets the preselected tolerance.

If the error is met, update velocities and accelerations, and go to the next time step. If the error is not met, then go back to step 1 using this latest K matrix.

I am familiar with solving this problem without the nonlinearity in K. However, I am unsure how to update the K matrix at every time step within the ode45 framework.

Thank you for the guidance!I am solving a time-dependent finite element problem with nonlinearities in the stiffness matrix which takes the following form:

where the dot symbolizes derivative with respect to time, U is the displacement vector, M is the mass matrix, C is the damping matrix, K(U) is the stiffness matrix which depends on U, and F is an external force. I would like to use ode45 to solve the dynamic problem. The general flow for solving this problem would be the following:

Calculate the stiffness matrix K using the latest displacement U (Initial condition if it is the first time step).

Formulate the state-space equations using K matrix, M matrix, C matrix, and F vector.

Integrate the state-space equations with ode45 to caclulate the new displacement U.

Recalculate the K matrix with the new displacement U. Check if the residual error meets the preselected tolerance.

If the error is met, update velocities and accelerations, and go to the next time step. If the error is not met, then go back to step 1 using this latest K matrix.

I am familiar with solving this problem without the nonlinearity in K. However, I am unsure how to update the K matrix at every time step within the ode45 framework.

Thank you for the guidance! I am solving a time-dependent finite element problem with nonlinearities in the stiffness matrix which takes the following form:

where the dot symbolizes derivative with respect to time, U is the displacement vector, M is the mass matrix, C is the damping matrix, K(U) is the stiffness matrix which depends on U, and F is an external force. I would like to use ode45 to solve the dynamic problem. The general flow for solving this problem would be the following:

Calculate the stiffness matrix K using the latest displacement U (Initial condition if it is the first time step).

Formulate the state-space equations using K matrix, M matrix, C matrix, and F vector.

Integrate the state-space equations with ode45 to caclulate the new displacement U.

Recalculate the K matrix with the new displacement U. Check if the residual error meets the preselected tolerance.

If the error is met, update velocities and accelerations, and go to the next time step. If the error is not met, then go back to step 1 using this latest K matrix.

I am familiar with solving this problem without the nonlinearity in K. However, I am unsure how to update the K matrix at every time step within the ode45 framework.

Thank you for the guidance! nonlinear-dynamics, ode45 MATLAB Answers — New Questions