Sensitivity Calculation

The sensitivity of an output y (u) to an input u is defined as the change in y due to a unit change in u.

Around an operating point

(y_{0}, u_{0})

one can write the Taylor’s series as:(1)

y = y_{0} + (\frac{\partial y}{\partial u}) (u - u_{0}) + O (b) = y_{0} + (\frac{\partial y}{\partial u}) Δ u + O (b)

The quantity $(\frac{\partial y}{\partial u})$ is called the first order sensitivity of the quantity y to the input u.

Sensitivity analysis is the study of how the change or uncertainty in the output of a mathematical model or system (y) can be apportioned to different sources of change or uncertainty in its inputs (u).

For Design Sensitivity Analysis, the quantity (u) is the design b. Here we are asking the question: How does the response y change for a given change in the design b.

In a multibody simulation, the response $y$ is typically a function of the system states $x$ and $\dot{x}$ , and, perhaps, explicitly on the design b. The system states $x$ consist of (a) Displacements, (b) Velocities, (c) Lagrange Multipliers (or constraint reaction forces), (d) User defined differential equations, (e) User defined algebraic equations originating from Variables and LSE/GSE/TFSISO outputs, and, (f) Internally created intermediate states that simplify computation.

In mathematical terms:(2)

\begin{array}{l} y = y (x, \dot{x}, b) \\ ∴ \frac{Δ y}{Δ b} = (\frac{\partial y}{\partial x}) \frac{\partial x}{\partial b} + (\frac{\partial y}{\partial \dot{x}}) \frac{\partial \dot{x}}{\partial b} + \frac{\partial y}{\partial b} + higher order terms \end{array}

The equations of motion provide an implicit relationship between (x and b) and ( $\dot{x}$ and (x, b)). The quantities $\frac{\partial x}{\partial b}, \frac{\partial \dot{x}}{\partial b} and \frac{\partial y}{\partial b}$ need to be computed first. Once these are known, the design sensitivity, $\frac{Δ y}{Δ b}$ , can be computed.

The calculation of $\frac{Δ y}{Δ b}$ is called Design Sensitivity Analysis (DSA) in MotionSolve. This is a new analysis method in MotionSolve. It always accompanies a regular analysis, such as static analysis, quasi-static analysis, kinematic analysis or dynamic analysis.

The job of the regular analysis is to compute the states x, ẋ and the outputs y for a given design b.

Once these are known, the DSA analysis will compute the sensitivity, $\frac{Δ y}{Δ b}$ . When there are N_y responses and N_b design variables, $[\frac{Δ y}{Δ b}]$ is a matrix of dimension N_y x N_b.

There are three well-known methods for computing design sensitivity: Finite Differencing, Direct Differentiation, and Adjoint Approach.