Simulation of thermoelastic behaviour of spacecraft structures : fundamentals and recommendations

Foreword
Preface
Acknowledgements
Perspective
Contents
Acronyms and Abbreviations
	Abbreviations
	Symbols
	Greek Symbols
1 Thermoelastic Verification
	1.1 The Thermoelastic Problem
	1.2 Structure of This Book
2 Occurrence of Thermoelastic Phenomenon in Spacecraft
	2.1 Introduction
	2.2 Hubble Space Telescope
	2.3 Korean Observation Satellite
	2.4 Gaia
	2.5 Surface Water and Ocean Topography (SWOT)
	2.6 PLATO
3 Physics of Thermoelastics
	3.1 Introduction
	3.2 Coefficient of Thermal Expansion
	3.3 Young's or Elasticity Modulus
	3.4 Constitutive Laws of Linear Thermoelasticity
		3.4.1 General 3-D Constitutive Laws of Linear Thermoelasticity
		3.4.2 1-D Stress–Strain Relation
		3.4.3 Plane Stress State
		3.4.4 Plane Strain State
	3.5 Summary Governing Equilibrium and Constitutive Equations
		3.5.1 Equilibrium
		3.5.2 Strain–Displacement Relations
		3.5.3 Constitutive Law
4 Modelling for Thermoelastic
	4.1 Introduction
	4.2 What Is a Thermal Gradient?
	4.3 What to Model?
	4.4 Structural and Thermal Modelling for Thermoelastic: An Integrated Process
	4.5 Integrated Model Convergence Checks
	4.6 Modelling Features
		4.6.1 Features and How These Are Commonly Modelled
		4.6.2 Assessment of a Box on a Plate
		4.6.3 Simplifying Feature Modelling: Preserve the Physics
	4.7 Need for Automation of the Analysis Chain
	4.8 Summary and Recommendations
		4.8.1 Which Deformations Cause Degradation of Performance of Instruments?
		4.8.2 Which Mechanisms Can Make the Degradation of Performance of Instruments Happen?
		4.8.3 What Is Needed to Simulate the Thermoelastic Mechanisms?
		4.8.4 Mesh Resolution and Level of Detail
		4.8.5 Temperature Mapping
		4.8.6 Selection of Worst Cases
		4.8.7 Uncertainties
		4.8.8 Concluding Recommendations
5 Thermal Modelling for Thermoelastic Analysis
	5.1 Introduction
	5.2 Space Thermal Environment
		5.2.1 On Ground Phase
		5.2.2 Launch and Ascent Phase
		5.2.3 Orbital Phase
		5.2.4 Direct Solar Flux
		5.2.5 Planet Reflected Solar Flux (Albedo)
		5.2.6 Planet Flux, Infrared Radiation
		5.2.7 Internal Dissipation
	5.3 Heat Transfer Mechanisms
		5.3.1 Conduction
		5.3.2 Contact Conductance
		5.3.3 Convection
		5.3.4 Thermal Radiation Heat Transfer
	5.4 Spacecraft Thermal Modelling with the Lumped Parameter Method
		5.4.1 Thermal Network Modelling with the Lumped Parameter Method
		5.4.2 Thermal Node in a Thermal Lumped Parameter Model
		5.4.3 Geometric Mathematical Model
		5.4.4 Thermal Mathematical Model
	5.5 Thermal Transient Analysis
		5.5.1 Transient Phenomena in Space Thermal Analysis
		5.5.2 Solution Approach for Thermal Transient Problems
	5.6 Thermoelastic Analysis for Transient Problems
	5.7 Thermal Analysis for Thermoelastic Versus Thermal Control
		5.7.1 Objectives of Thermal Analysis for Thermal Control
		5.7.2 Objectives of Thermal Analysis for Thermoelastic
		5.7.3 Selection of Worst Case Temperature Fields
		5.7.4 Thermal Mesh Convergence for Thermoelastic
		5.7.5 Level of Detail in Models for Thermoelastic
		5.7.6 Thermal Analysis Uncertainties for Thermoelastic
		5.7.7 Concluding Thermal Analysis for Thermal Control Versus Thermoelastic
6 Structural Modelling for Thermoelastic Analysis
	6.1 Introduction
	6.2 The Finite Element Method for Thermoelastic Simulations
	6.3 Characteristics of Finite Elements for Thermoelastic Analysis
	6.4 Elastic Finite Elements
		6.4.1 0-D, Scalar Element
		6.4.2 1-D, Rod Element
		6.4.3 1-D, Bar and Beam Element
		6.4.4 2-D, Membrane Element
		6.4.5 2-D, Plate, Shell, Sandwich Element
		6.4.6 3-D, Volume (Solid) Element
	6.5 Constraint Equations and Rigid Elements
		6.5.1 Principle of Constraint Equations
		6.5.2 The Interpolation Element
		6.5.3 The Rigid Body Element
	6.6 Boundary Conditions
		6.6.1 Iso-static Supports
		6.6.2 Statically Indeterminate Supports
		6.6.3 Intertia Relief Method
	6.7 Refurbishing a Dynamic Finite Element Model for Thermoelastic
		6.7.1 Introduction
		6.7.2 Required Mesh Resolution for Dynamic and Thermoelastic Models
		6.7.3 Finite Element Models for High-Frequency Response Analysis
		6.7.4 Simulation of Joints
		6.7.5 Check on Adequacy of Rigid Body Elements for Thermoelastic
	6.8 Finite Element Model Health Checks Thermoelastic FE Models
		6.8.1 Introduction
		6.8.2 Strain Energy as Rigid Body
		6.8.3 Free Iso-thermal Expansion
7 Transfer of Thermal Analysis Results  to the Structural Model
	7.1 The Interface Problem
	7.2 Thermal Lumped Parameter Node Versus Finite Element Node
	7.3 Building Correspondence Between Models
	7.4 Temperature Mapping Methods
		7.4.1 Geometric Temperature Interpolation Method
		7.4.2 Centre-Point Prescribed Temperature Method
		7.4.3 Patch-Wise Temperature Application Method
		7.4.4 Prescribed Average Temperature Method
	7.5 Comparing Mapping Methods on a 1-D Problem
		7.5.1 One-Dimensional Model Description
		7.5.2 Temperature Mapping Results
		7.5.3 Thermoelastic Responses
		7.5.4 Conclusion of One-Dimensional Problem
	7.6 Benchmarking of Temperature Mapping Methods  on a Two-Dimensional Problem
		7.6.1 Geometry, Mesh and Boundary Conditions
		7.6.2 Temperature Field to Be Mapped
		7.6.3 Reference Temperature, Displacement and Stress
	7.7 Comparing Performances of Mapping Methods
		7.7.1 Performance Criteria for the Mapping Methods
		7.7.2 Qualitative Comparison of the Mapped Temperature Fields
		7.7.3 Average Temperature Comparison
		7.7.4 Displacement Comparison
		7.7.5 Stress Comparison
		7.7.6 Concluding the 2-D Benchmark Model
	7.8 Summary Temperature Mapping/Interpolation Methods
8 Prescribed Average Temperature Method
	8.1 Introduction
	8.2 Relating Thermal Nodes and FEM Nodes
	8.3 Creation of Consistent Values of A-Matrix Coefficients with a Finite Element Code
	8.4 Coupling TMM to the FE Model
	8.5 Evaluating PAT Method Results
	8.6 Mathematical Models Checks for PAT Method
		8.6.1 Introduction
		8.6.2 Conduction FE Model Health Check
		8.6.3 Checking A-Matrix Input to the PAT Method
	8.7 Effect of Incomplete Correspondence
9 Generation of Linear Conductors for Lumped Parameter Thermal Models
	9.1 Need for Automated Conductor Generation
	9.2 Calculation of a Single Linear Conductor with a Conduction FE Model
		9.2.1 Calculation of a Conductor Through Reduction  of the Conduction Matrix
		9.2.2 Conductor Calculation Through Steady-State Thermal Analysis
		9.2.3 Far Field Method for Generation of 1-D Linear Conductors
	9.3 PAT-Based Methods for Generating TMM Conductors
		9.3.1 Extracting Conductors from Lagrange Multipliers Λ
		9.3.2 Reduction of FE Model Conduction Matrix
		9.3.3 Consideration for the Use of the PAT-Based Conductors
10 Estimating Uncertainties  in the Thermoelastic Analysis  Process
	10.1 Uncertainties in the Thermoelastic Analysis Process
		10.1.1 Uncertainties from the Thermal Analysis
		10.1.2 Uncertainties from the Temperature Mapping Process
		10.1.3 Uncertainties from the Thermoelastic Structural Response Analysis
		10.1.4 Uncertainties from the Instrument Performance Impact Analysis
	10.2 Use of Factors of Safety for Covering the Uncertainties
	10.3 Uncertainty Assessment of Thermoelastic Analysis Using Probabilistic Analysis
	10.4 Monte Carlo Simulation Method
	10.5 Modified MCS, Latin Hypercube Sampling Method
	10.6 The Rosenblueth 2k+1 Point Estimates Probability Moment Method
	10.7 Sensitivity Analysis
Appendix A Detailed Description of ``Box on Plate'' Experiment
A.1  Background and Research Question
A.2  Description of Numerical Experiment
A.3  Model Description
A.4  Analysis Approach
A.5  Detailed Results and Evaluation
A.5.1  Structure of this Section
A.5.2  Global Observations
A.5.3  Understanding the Effect of Different Box Configurations
A.6  Simplified Modelling of an Electronic Box  on a Sandwich Plate: Preserve the Physics
A.7  Conclusions
Appendix B One-Dimensional (1-D) Conduction Finite Element
B.1  Introduction
B.2  General Heat Transfer Equations
B.2.1 General Finite Element Matrix Derivation
B.3  1-D Conduction Rod Finite Element
B.4  Assembly of System Matrices
Appendix C One-Dimensional (1-D) Thermoelastic Finite Element
C.1  Introduction
C.2  Equilibrium Equations for a One-Dimensional Rod
C.3  Shape Functions for One-Dimensional Element
C.4  Galerkin's Weighted Residual Method
C.5  Iso-parametric Formulation
C.6  Virtual Work
C.7  Virtual Work Applied to an Iso-parametric Linear Rod Element
C.8  Assembly of the System Equation of the Finite Element Model
Appendix D Theory of Introduction Multipoint Constraint Equations in Thermoelastic Problems
D.1  Introduction
D.2  Use of Lagrange Multipliers
D.3  Elimination of Dependent Degrees of Freedom
Appendix E Solutions
References
Index