Reduced Order Modeling in Finite Element Analysis

In my previous blog article (All models are wrong ??), we discussed how all models have their own set of assumptions and drawbacks that make them deviate from the physical reality. In this article, I wanted to briefly discuss the topic of reduced-order modeling in an effort to find a middle ground in modeling approaches (ranging from detailed physics-based models to system-level modeling where each component is treated as a black box defined by a set of empirical relations).

One of the major challenges in mechanics is to capture the flexibility/compliance of the system. Though we have closed-form equations for simple structures such as beams, cylinders, etc based on principles of solid mechanics. These are not convenient to use while dealing with real-life systems due to their complex geometric arrangement, and the potential for contacts to exsist. The majority of these modeling challenges were addressed with the advent of the Finite Element Modeling (FEM) technique in 1977. Discretizing any given component into finite elements and assembling their stiffness provides the independence to analyze any complex geometry. That being said, one always faces a tradeoff between accuracy and analysis time in FEM. i.e., to achieve better simulation accuracy, a fine mesh is required (more number of nodes), which results in higher computational times. This article makes an effort to discuss different strategies out there which one could employ to achieve accurate results without compromising on computational efficiency.

Firstly let's start our discussion by considering a static analysis. Static/Guyan condensation is the most commonly used procedure for reducing the size of a FE model by retaining only the important nodes of interest from the full-blown FE model. This is done by relating the retained DoFs (X_r) to the full DoF's (X_f) as described in Eqn.1.

(1)

To obtain the transformation matrix (T) let's consider the standard equation of statics, partitioned according to retained (X_r) and deleted (X_f) DoFs,

(2)

Where F_r and F_d denote the external forces applied along the retained (r) and deleted (d) DoFs, respectively. The deleted DoF's (Phi_d) can be expressed in terms of the retained DoF's (Phi_r), using the second relation in Eqn.2 (K_dr*Phi_r + K_dd*Phi_d = F_d), and the fact that external forces on the deleted nodes must be zero (F_d = 0),

(3)

Equating the strain energy (0.5*(Phi_f^T)*K_f*Phi_f), of the full FE model to that of the reduced model (0.5*(Phi_r^T)*K_r*Phi_r), expressions for the reduced stiffness (K_r) matrix are obtained as shown in Eqn.4,

(4)

This condensation technique, being derived based on the static equilibrium conditions, is highly accurate, and its accuracy is independent of the number of retained nodes when used for static analysis (i.e. no inertial terms). One could use the same transformation matrix (T) to condense the mss matrix (M_r) as shown in Eqn. 5 and perform a dynamic analysis as well. But one should be aware of the accuracy of this technique as this procedure would omit several critical mode shapes of the system. The accuracy of this procedure is limited based on the number of retained nodes and their positions. Higher number of retained nodes and their choice such that they take up a significant amount of inertia and applied load compared to the deleted nodes improves the technique's accuracy.

(5)

To improve upon the drawbacks of Guyan reduction, one could make use of dynamic condensation techniques, which not only takes into account the structure's static response but also its modal response to capture its flexibility. This is achieved by expressing the total system deformations as a function of retained nodes and modal wights (q) as shown in Eqn. 6.

(6)

Here the deleted DoFs are estimated as, Phi_d = -(K_dd^-1)*K_dr*Phi_r + Phi_mode*q, where the first term captures static response at deleted DoFs for enforced displacement Phi_r at the retained nodes, while the second term is the modal influence on the response of the deleted DoFs (where, Phi_mode is the modal vector matrix and q is an array containing the modal weights). The number of modes captured determines the level of accuracy of the condensation process.

There are various versions of the dynamic condensation based on the boundary conditions applied for determining the mode shapes of the structure. The main intention of this article was to introduce the concept of reduced-order modeling pertaining to both static and dynamic analysis.