user: I am trying to implement a POD galerkin reduced order model for finite element analysis. Can you explain in detail the method? starting with the definition of the scalar product with the mass matrix, then describing the POD and finally explaining how to project the linear and non linear operators in the reduced basis.

assistant: The Proper Orthogonal Decomposition (POD) Galerkin Reduced Order Model (ROM) is a technique used to reduce the computational complexity of Finite Element Analysis (FEA). It involves the use of a low-dimensional subspace of the original FEA space to approximate the solution of the full problem. The method can be summarized in three steps: (1) construction of the reduced basis using POD, (2) projection of the linear operators into the reduced basis, and (3) projection of the nonlinear operators into the reduced basis.

Scalar Product with the Mass Matrix:
The POD-Galerkin ROM method starts with defining a scalar product with the mass matrix of the system. The mass matrix is a positive definite matrix that relates the velocity and acceleration of the system. The scalar product is defined as:

⟨u,v⟩_M = ∫ Ω u v dΩ

where Ω is the domain of the system, u and v are functions defined on Ω, and M is the mass matrix. This scalar product defines the norm of the system and the inner product used in the POD method.

Proper Orthogonal Decomposition (POD):
The POD is a mathematical technique used to extract the dominant features of a set of data. In the context of FEA, POD is used to extract the dominant modes of the system from a set of snapshots of the solution. These snapshots are obtained by solving the FEA problem for different input parameters or initial conditions. The POD method then extracts the dominant modes of the system by performing a Singular Value Decomposition (SVD) on the snapshot matrix. The SVD of the snapshot matrix X is given by:

X = UΣV^T

where U and V are orthogonal matrices, and Σ is a diagonal matrix containing the singular values of X. The dominant modes of the system are contained in the first few columns of U and V, corresponding to the largest singular values in Σ.

Projection of Linear Operators:
After the reduced basis is constructed using the POD method, the linear operators of the system can be projected into the reduced basis. The linear operators are the stiffness matrix K and the damping matrix C, which are defined by:

Ku = f
Cv = -Ku_dot

where u and v are the displacement and velocity vectors of the system, f is the force vector, and u_dot is the time derivative of u. To project the linear operators into the reduced basis, we define the reduced stiffness and damping matrices, K_r and C_r, as:

K_r = Φ^T K Φ
C_r = Φ^T C Φ

where Φ is the matrix containing the reduced basis vectors, and Φ^T is its transpose. The reduced system can then be written as:

K_r u_r = f_r
C_r v_r = -K_r u_r_dot

where u_r and v_r are the reduced displacement and velocity vectors, and f_r is the reduced force vector.

Projection of Nonlinear Operators:
The projection of the nonlinear operators into the reduced basis is more complicated than the projection of the linear operators. The nonlinear operators are the result of the nonlinear terms in the governing equations, and they cannot be projected directly into the reduced basis. Instead, the nonlinear terms are approximated using a Galerkin projection, which involves projecting the nonlinear terms onto the reduced basis. The Galerkin projection of the nonlinear term is given by:

N(u) ≈ Φ^T N(Φu)

where N(u) is the nonlinear term of the governing equation, Φ is the matrix containing the reduced basis vectors, and Φu is the approximation of the full solution u in the reduced basis. The projected nonlinear term is then added to the reduced linear system, resulting in the reduced nonlinear system:

K_r u_r = f_r + Φ^T N(Φu_r)
C_r v_r = -K_r u_r_dot

where N(Φu_r) is the Galerkin projection of the nonlinear term onto the reduced basis.

In summary, the POD-Galerkin ROM method involves the construction of a reduced basis using the POD method, followed by the projection of the linear and nonlinear operators into the reduced basis. The reduced system can then be solved using standard numerical techniques. The method provides a significant reduction in computational cost compared to the full FEA problem, while maintaining a high degree of accuracy.