Design variables whose values should be changed

From the viewpoint of supporting design, detecting inconsistent inequalities is not sufficient help to a designer. Especically in embodiment design, it is quite important for a designer to know what variable values are not appropriate. This section describes the method of finding out design variables whose values should be changed.

Suppose that $c(t_{1},\ldots,t_{r})=0$ is an equation showing inconsistency among inequalities. Assignment of a concrete value to a design variable is expressed as a univariate linear equation. That is, univariate linear equation ``x<sub>k</sub>=a'' means ``a is assigned to x_k''. Thus, by searching for univariate linear equations used for computing $c(t_{1},\ldots,t_{r})=0$, design variables whose values should be changed can be found out. If ``x<sub>k</sub>=a'' is used to obtain $c(t_{1},\ldots,t_{r})=0$, ``x_k=a'' has something to do with the inconsistency. Otherwise, ``x_k=a'' has nothing to do with it.

Let C be the whole constraint set including univariate linear equations. Obviously, $c(t_{1},\ldots,t_{r})=0$ belongs to the ideal defined by C. If $c(t_{1},\ldots,t_{r})=0$ does not belong to the ideal defined by $C\setminus\{x_{k}=a\}$, it is concluded that x_k=a is used to compute $c(t_{1},\ldots,t_{r})=0$. Otherwise, x_k=a is not necessary to obtain $c(t_{1},\ldots,t_{r})=0$. Therefore, design variables whose values should be changed are found out according to the following procedures.

Method of detecting design variables whose values should be changed

1.

Let C be the whole constraint set Eq. (2), and U be a set of univariate linear equations included in C. Suppose that $c(t_{1},\ldots,t_{r})=0$ is the equation which shows inconsistency among inequalities in Eq. (1). Let E be an empty set.

2.

  If U is empty, return E.

3.

Let x_k=a be the first element in U, and U be $U\setminus\{x_{k}=a\}$.

4.

The Gröbner base G of $C\setminus\{x_{k}=a\}$ is computed.

5.

Let $c^{\prime}$ be a normal form of $c(t_{1},\ldots,t_{r})$ by G. If $c^{\prime}$ is not 0, let E be $E\cup\{x_{k}=a\}$.

6.

Go back to Step 2.

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