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Preface 1

Nomenclature 4

1．Introduction 7

1.1 Research background 7

1.2 Review of research for thermoforming 8

1.3 Descriptions for contents 12

2．Simulation method for storing stage 15

2.1 Governing equation and method for analysis 15

2.1.1 Convective heat transfer 16

2.1.2 Conductive heat transfer 16

2.2 Time-dependent temperature distribution 18

3．Simulation and optimization methods for preheating process 20

3.1 Analysis model 23

3.2 Development of analysis code suitable for preheating process 24

3.2.1 Radiative heat transfer 24

3.2.2 Convective heat transfer 31

3.2.3 Conductive heat transfer 32

3.3 Steady optimization for heater power distribution 34

3.3.1 Basic theory for optimization 34

3.3.2 Steady heater power distribution 38

3.3.3 Effect of initial temperature of sheet 44

3.3.4 Definition of heating time and heater power 46

3.4 Unsteady optimization for variation of heater power 54

3.4.1 Effect of variation of heater power 54

3.4.2 Unsteady optimization using response surface method 55

3.4.3 Unsteady optimization using analytic method 70

3.4.4 Unsteady optimization using approximate analytic method 76

3.4.5 Unsteady optimization using finite difference method 98

4．Simulation method for cooling process 105

4.1 Prediction method of cooling time 106

4.2 Development of 1-d analysis code 107

4.2.1 2-d analysis 107

4.2.2 Conversion from 2-d to 1-d problem 109

4.2.3 1-d FDM analysis code 110

4.3 Specification of 1-d analysis conditions 112

5．Conclusions 115

Bibliography 118