Application of Generic Constraint Solving Techniques for Insightful Engineering Design

Hiroyuki Sawada

Digital Manufacturing Research Center (DMRC)National Institute of Advanced Industrial Science and Technology

(AIST)

25 October, 2004

Contents

1. Background of Research

2. Research Approach

3. New Constraint Solving Methods

4. Prototype System: DeCoSolver (Design Constraint Solver)

5. Design Example: Heat Pump System Design

6. Conclusion

Background of Research

Design Process: Process of decision making

Introducing many design parameters

Difficulties in gaining an insight into underlying relationships among design parameters including:

Critical design parameters for the performance

Trade-off between design requirements

Less optimal and/or inferior design solution

Aim of Research

Overcoming the above difficulties by applying generic constraint solving techniques based on Groebner basis (GB) and Quantifier Elimination (QE)

Research Approach

(1) Formalisation of design as a process of defining constraints and solving a design problem using constraints

(2) Development of new constraint solving methods based on symbolic algebra overcoming conventional difficulties in:

Analysing incomplete design solutions

Detecting underlying conflicts between constraints

Establishing explicit relationships between design parameters

Advantages of this Approach

(1) Generic Constraint Based Approach

Design support system is generic enough to deal with multidisciplinary design problems involving mechanics, electrics, thermodynamics, hydrodynamics, etc.

(2) Rigorous Constraint Solving Methods

All the results are guaranteed to be correct mathematically.

Robotic System

Thermal System

New Constraint Solving Methods

Necessary information for design decision making(1) Possible numerical values for design parameters [Thornton et al. 96](2) Optimized numerical solutions [Thompson 99](3) Conflicts in a design solution [Oh et al. 96](4) Fundamental relationships among design parameters [Hoover et al. 94]

New constraint solving methods providing the above information

f1(x1, ..., xn) = 0, …, fp(x1, ..., xn) = 0, g1(x1, ..., xn) 0, …, gq(x1, ..., xn) 0,h1(x1, ..., xn) 0, …, hr(x1, ..., xn) 0.

Preprocess: Inequalities equations by introducing slack variables

f1(x1, ..., xn) = 0, …, fp(x1, ..., xn) = 0, g1(x1, ..., xn) s1= 1, …, gq(x1, ..., xn) sq= 1,

h1(x1, ..., xn) = t1, …, hr(x1, ..., xn) = tr,t1 0, …, tr 0.

Let A be the region represented by the converted formulae.

(1) Possible numerical values for design parameters

Objectives1. to make clear whether there exists a design solution2. to compute numerical solutions when solutions do exist

Target function u(x, s, t)1. u(x, s, t) is continuous in (x, s, t)-space. 2. u(x, s, t) has the minimum value in (x, s, t)-space. 3. If at least one parameter of (x, s, t) becomes positive or negative infinite, u(x, s, t) becomes positive infinite.

A is not empty. u(x, s, t) has the minimum value in A.A is empty. u(x, s, t) does not have the minimum value in A.

Possible numerical values can be obtained by computing the minimum value of u(x, s, t) in A.

(2) Optimized numerical solutions (Minimization of the given objective function u(x, s, t))

Assumption: u(x, s, t) is a polynomial function.

Suppose u(x, s, t) = P(x, s, t)/Q(x, s, t). Minimizing u(x, s, t) is equivalent to minimizing v under the condition P(x, s, t) = Q(x, s, t)v.

u(x, s, t) is continuous and differential. The minimum point can be obtained by computing its extreme points. Lagrange Multiplier method is employed to compute the extreme points algebraically.

Objectivesto identify a set of inequalities that cannot be satisfied simultaneously

Finding out inequalities that cannot be satisfied simultaneously

(a) Computing c1(ti1, ..., tim

) = ... = ck(ti1, ..., tim

) = 0

(b) {c1(ti1, ..., tim

) = ... = ck(ti1, ..., tim

) = 0, ti1 0, ..., tim

0 } has

no solution.

　　 ti1 0, ..., tim

0 cannot be satisfied simultaneously.

　　 hi1 (x1, ..., xn) 0, …, him (x1, ..., xn) 0 cannot be satisfied

　　　 simultaneously.

(c) Checking all the possible combinations {ti1, ..., tim

}.

(3) Conflicts in a design solution

Equations (displayed as curves) ： Computing the Groebner basis

Inequalities (displayed as regions)(a) The Groebner basis is computed to obtain a set of equations consisting of xi, xj and slack variables t1, …, tr. Let Gp be the obtained equation set. (b) The partial solution space is represented by the following logical formula.

(t1, …, tr){Gp {t1 0, …, tr 0}}

　　 Quantifier Elimination can obtain a set of inequalities consisting of

xi and xj.

(4) Fundamental relationships among design parameters

Objectives1. Establishing explicit relationships among design parameters2. Showing such explicit relationships as a partial design solution space in the form of two-dimensional graph

Prototype System: DeCoSolver (Design Constraint Solver)

Constraint Editor Defining Constraints

Component Library Database of commonly used Components

Context-Tree Is-a Hierarchy of Design Alternatives

Product Explorer Results of Analysis

Solver Handler Interface to Constraint Solver

Low pressure saturated gas & liquid

High pressure saturated gas & liquid

Expansion Valve

(Isenthalpic Expansion)

Condenser

Compressor(Adiabatic

Compression)

Evaporator

Hot water supplyfor a bath

(45 C, 3.0 l/s)

Hot spring(30 C, 3.0 l/s)

Drain(5 C, 2.4 l/s)

Low pressure gas

High pressure gas

Drain water(20 C, 2.4 l/s)

Tc: Condensation Temp.Ac: Heat Transfer Area

Te: Evaporation Temp.Ae: Heat Transfer Area

Pd: Discharging Press.: Compression Ratio

Qr: Mass flow rate of Refrigerant

Design Example: Heat Pump System

Conventional difficulties

Loop structure of the heat pump system Non-linearity of thermodynamic properties

Complicatedly coupled design parameter relationships

Difficulty in gaining insights into underlying relationships among design parameters

Design by trial and errors without insights

Design procedure with DeCoSolver

(1) Constructing the product model

(2) Drawing graphs between design parameters to gain insights into underlying relationships

(3) Determining design parameter values based on the gained insights

Drag & Drop

Drawing lines

(1) Constructing the product model

Tc: Condensation Temp.

Tc: Condensation Temp.

Ae: Heat Transfer Area of Evaporator

Ac: Heat Transfer Area of Condenser

Qr: Mass Flow Rate of Refrigerant

Te: Evaporation Temp. Pd: Discharging Press. : Compression Ratio

(2) Drawing graphs between design parameters

Gained Insights(1) As Tc increases, Te and Pd also increase almost linearly. (2) Qr and are almost unchanged. (3) As Tc increases, Ac decreases non-linearly. (4) As Tc increases, Ae increases non-linearly.

Violated Constraint

Numerical Values of Clicked PointTe: Evaporation Temp.

A small heat transfer area leads to a small equipment. Total heat transfer area, Ac + Ae, should be minimised.

(3) Determining design parameter values based on the gained insights

Ac+Ae: Total Heat Transfer Area [m2]

Tc: Condensation Temperature [K]

Minimum Point Tc = 323 [K] As = 29.7626 [m2]

Other design parameter values will be determined.

Conclusion: Advantages of DeCoSolver

Generic and Rigorous Constraint Solving Methodsbased on Groebner basis and Quantifier Elimination

All the analysis results are guaranteed to be correct mathematically.

Computational mistakes due to numerical errors or computational convergence problems are completely excluded.

Incomplete design solutions in multidiscipline can be analysed.

Deep and accurate insights into a design problem as well as design solutions