Slide 1 of 18 Commonality Analysis of Families of Motivation CA Template Physical Models for use in Scientific Family of Material Models Computing Data Definition Goal Statement Assumptions Theoretical Model Variabilities Spencer Smith, Jacques Carette and John McCutchan Dependency Graphs Example Concluding Remarks Department of Computing and Software McMaster University First International Workshop on Software Engineering for Computational Science and Engineering
Outline Slide 2 of 18 Motivation Motivation 1 CA Template Family of Material Models Commonality Analysis Template 2 Data Definition Goal Statement Assumptions Theoretical Model A Family of Material Behaviour Models 3 Variabilities Dependency Graphs Terminology Definitions Example Concluding Goal Statement Remarks Assumptions Theoretical Model Variabilities Dependency Graphs Example
Motivation Slide 3 of 18 Motivation • Requirements documentation of physical models CA Template • Allows judgement of quality Family of • Improves communication Material Models Data Definition • Between domain experts Goal Statement Assumptions • Between domain experts and programmers Theoretical Model • Explicit assumptions Variabilities Dependency Graphs • Range of applicability Example • A family approach, potentially including a DSL to allow Concluding Remarks generation of specialized programs • Improves efficiency of product and process • Facilitates reuse of requirements and design, which improves reliability • Improves usability and learnability • Clarifies the state of the art
Background Slide 4 of 18 Motivation CA Template • Program family idea since the 1970s (Dijkstra, Parnas, Family of Weiss, Pohl, ...) - variabilities are often from a finite set Material Models Data Definition of simple options Goal Statement Assumptions • Families of algorithms and code generation in SC Theoretical Model Variabilities (Carette, ATLAS, Blitz++, ...) - not much emphasis on Dependency Graphs Example requirements Concluding Remarks • Previous work on requirements for SC • Template for a single physical model • Template for a family of multi-purpose tool • Need a requirements template for a family of physical models
Background Slide 5 of 18 Motivation CA Template • Program family idea since the 1970s (Dijkstra, Parnas, Family of Weiss, Pohl, ...) - variabilities are often from a finite set Material Models Data Definition of simple options Goal Statement Assumptions • Families of algorithms and code generation in SC Theoretical Model Variabilities (Carette, ATLAS, Blitz++, ...) - not much emphasis on Dependency Graphs Example requirements Concluding Remarks • Previous work on requirements for SC • Template for a single physical model • Template for a family of multi-purpose tool • Need a requirements template for a family of physical models
1 Reference Material: a) Table of Contents b) Table of Units c) Table of Symbols c) Abbreviations and Acronyms e) Types Slide 6 of 18 2 Introduction: a) Purpose of the Document b) Motivation Scope of the Family c) Organization of the CA Template Document Family of Material Models Data Definition 3 General System Description: a) Potential System Goal Statement Assumptions Contexts b) Potential User Characteristics c) Theoretical Model Potential System Constraints Variabilities Dependency Graphs Example 4 Commonalities: a) Background Overview b) Concluding Terminology Definition c) Goal Statements d) Remarks Assumptions e) Theoretical Models f) Derived Quantities 5 Variabilities 6 Dependence Graphs 7 Sample Family Members 8 References
1 Reference Material: a) Table of Contents b) Table of Units c) Table of Symbols c) Abbreviations and Acronyms e) Types Slide 7 of 18 2 Introduction: a) Purpose of the Document b) Motivation Scope of the Family c) Organization of the CA Template Document Family of Material Models Data Definition 3 General System Description: a) Potential System Goal Statement Assumptions Contexts b) Potential User Characteristics c) Theoretical Model Potential System Constraints Variabilities Dependency Graphs Example 4 Commonalities: a) Background Overview b) Concluding Terminology Definition c) Goal Statements d) Remarks Assumptions e) Theoretical Models f) Derived Quantities 5 Variabilities 6 Dependence Graphs 7 Sample Family Members 8 References
A Family of Material Models Slide 8 of 18 y Motivation CA Template F Family of Material Models x Data Definition H 0 H W Goal Statement W 0 Assumptions Theoretical Model L 0 ∆ L Variabilities z L Dependency Graphs Example Concluding σ σ Remarks λ 1 σ y 1 1 λ 2 σ y 2 2 3 λ 3 σ y 3 ǫ ǫ
Terminology Definitions Slide 9 of 18 Label: D YieldFunction Symbol: F = F ( σ , κ ) Motivation CA Template Type: ( tensor2DT × R ) → R Family of Related: D Stress, D HardeningParameter Material Models Data Definition Sources: ... Goal Statement Assumptions Descrip: The yield function defines a surface F = 0 Theoretical Model Variabilities in the six dimensional stress space ... Dependency Graphs Example Concluding Remarks ∂ Q ∂ σ F = 0 Q = 0
Goal Statement Label: G StressDetermination Slide 10 of 18 Descrip: Given the initial stress and the deforma- Motivation tion history of a material particle, deter- CA Template mine the stress within the material parti- Family of Material Models cle. Data Definition Goal Statement Refine: T ConstitEquation Assumptions Theoretical Model Variabilities y Dependency Graphs σ yy Example Concluding σ yx Remarks σ yz σ xy σ xx σ zy x σ xz σ zz σ zx z
Assumptions Slide 11 of 18 Label: A AdditivityPostulate Related: D StrainRate Motivation ǫ e + ˙ ǫ vp Equation: ǫ = ˙ ˙ CA Template Family of with the following types and units Material Models ǫ : tensor2DT (1/t) (1/s) ˙ Data Definition Goal Statement ǫ e : tensor2DT (1/t) (1/s) ˙ Assumptions Theoretical Model ǫ vp : tensor2DT (1/t) (1/s) ˙ Variabilities Dependency Graphs Descrip: The total strain rate ( ˙ ǫ ) is assumed to de- Example ǫ e ) and viscoplastic Concluding compose into elastic ( ˙ Remarks ǫ vp ) strain rates. ( ˙ Rationale This is a standard assumption for elasto- plastic and elastoviscoplastic materials. The appropriateness of this assumption is born out by the success of theories built upon it. Source: [6, page 339]; [7, page 181]
Theoretical Model Slide 12 of 18 Motivation Label: T ConstitEquation CA Template Related: A CauchyStress, A DeformationHistory, Family of A PerzynaConstit, A AdditivityPostulate, Material Models Data Definition A ElasticConstit, A DescriptionOfMotion, Goal Statement Assumptions V MaterialProperties Theoretical Model Variabilities Input: σ 0 : tensor2DT (StressU) (Pa) Dependency Graphs Example t begin : R (t) (s) Concluding t end : R (t) (s) Remarks ǫ ( t ) : { t : R | t begin ≤ t ≤ t end : t } → ˙ tensor2DT (1/t) (1/s) mat prop val : string → R E : R + (StressU) (Pa) ν : poissonT (dimensionless)
Theoretical Model Continued Slide 13 of 18 Label: T ConstitEquation Output: σ ( t ) : { t : R | t begin ≤ t ≤ t end : t } → Motivation tensor2DT such that CA Template Family of � ǫ − γ < φ ( F ( σ , κ )) > ∂ Q ( σ ) � Material Models σ = D ˙ ˙ Data Definition ∂ σ Goal Statement Assumptions Theoretical Model Variabilities and σ ( t begin ) = σ 0 , the components of σ Dependency Graphs Example have the units of StressU (Pa) Concluding Derive: The governing differential equation Remarks ǫ e is found by first solving for ˙ in A AdditivityPostulate and then ... Descrip: The theoretical model is only completely defined once the associated variabili- ties (V MaterialProperties) that define the material have been set. ... History: Created – June 14, 2007
Variabilities Slide 14 of 18 Motivation CA Template Family of Material Models • F = F ( σ , κ ) : R 6 × R → R Data Definition Goal Statement • Q = Q ( σ ) : R 6 → R Assumptions Theoretical Model • κ = κ ( ǫ vp ) : R 6 → R Variabilities Dependency Graphs Example • φ = φ ( F ) : R → R Concluding Remarks • γ : R • mat prop names : set of string
Dependency Graph Slide 15 of 18 G_Stress Determination Motivation T_Constit Equation CA Template Family of Material Models A_Cauchy A_Deformation A_Additivity A_Perzyna Data Definition A_ElasticConstit Stress History Postulate Constit Goal Statement Assumptions Theoretical Model Variabilities Dependency Graphs Example Concluding Remarks D_Plastic D_YieldFunction A_Isothermal A_Isotropic Potential D_Hardening Parameter D_Stress D_StrainRate A_NoDistrib A_Continuum A_Small A_Cartesian A_Description Moments Hypothesis DefGradients Coords OfMotion D_DescriptionOf Motion
Dependency Graph Between Commonalities and Variabilities Slide 16 of 18 Motivation CA Template V_Material Family of V_Description V_MatName Properties Material Models Data Definition Goal Statement Assumptions Theoretical Model Variabilities V_Plastic V_Hardening V_Fluidity V_YieldFunction V_Phi Dependency Graphs Potential Parameter Parameter Example Concluding Remarks V_StressState V_StrainState T_Constit Equation D_Stress D_StrainRate
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