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

f 1 (x 1,..., x n ) = 0, …, f p (x 1,..., x n ) = 0, g 1 (x 1,..., x n )  0, …, g q (x 1,..., x n )  0, h 1 (x 1,..., x n )  0, …, h r (x 1,..., x n )  0. Preprocess: Inequalities  equations by introducing slack variables f 1 (x 1,..., x n ) = 0, …, f p (x 1,..., x n ) = 0, g 1 (x 1,..., x n )  s 1 = 1, …, g q (x 1,..., x n )  s q = 1, h 1 (x 1,..., x n ) = t 1, …, h r (x 1,..., x n ) = t r, t 1  0, …, t r  0.  Let A be the region represented by the converted formulae.

(1) Possible numerical values for design parameters Objectives 1. to make clear whether there exists a design solution 2. 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.

Objectives to identify a set of inequalities that cannot be satisfied simultaneously Finding out inequalities that cannot be satisfied simultaneously (a) Computing c 1 (t i 1,..., t i m ) =... = c k (t i 1,..., t i m ) = 0 (b) {c 1 (t i 1,..., t i m ) =... = c k (t i 1,..., t i m ) = 0, t i 1  0,..., t i m  0 } has no solution.  t i 1  0,..., t i m  0 cannot be satisfied simultaneously.  h i 1 (x 1,..., x n )  0, …, h i m (x 1,..., x n )  0 cannot be satisfied simultaneously. (c) Checking all the possible combinations {t i 1,..., t i m }. (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 x i, x j and slack variables t 1, …, t r. Let G p be the obtained equation set. (b) The partial solution space is represented by the following logical formula.  (t 1, …, t r ){G p  {t 1  0, …, t r  0}} Quantifier Elimination can obtain a set of inequalities consisting of x i and x j. (4) Fundamental relationships among design parameters Objectives 1. Establishing explicit relationships among design parameters 2. 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 supply for 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. 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 Point Te: 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 [m 2 ] Tc: Condensation Temperature [K] Minimum Point Tc = 323 [K] As = [m 2 ]  Other design parameter values will be determined.

Conclusion: Advantages of DeCoSolver Generic and Rigorous Constraint Solving Methods based 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