Contents

• INTRODUCTION
  • Transient Analysis: First-Order Parabolic Systems
    • Review
    • Brief overview of common algorithms for transient analysis
  • Dynamic Analysis: Second-Order Hyperbolic Systems
    • Review
  • Towards Formally Standardized Theoretical Design Developments and Evolution of a Generalized -Family of Integral/Integration Operators for Transient Analysis
    • Theoretical basis: a theory of development/evolution leading to the theoretical design of transient/dynamic algorithms
  • Transient/Dynamic Field Problems: A Prelude to the Theoretical Design Developments, Evolution, Classification and Characterization of Time Discretized Operators
    • Theoretical developments: a standardized formal theory of design of computational algorithms, and development/evolution, classification and characterization
    • Classification of time discretized operators
• TRANSIENT ANALYSIS: FIRST-ORDER PARABOLIC SYSTEMS
  • Time Discretized Operators: Type 1 Classification
    • Introducing the notion of transformation to modal basis
    • Theoretical Details: The Generalized Explicit VIrtual-Pulse [VIP] Time-Integral Methodology for a Class of Nonlinear Transient Thermal Problems
    • Without introducing the notion of transformation to modal basis
  • Time Discretized Operators: Type 2 Classification
    • Case of : L-stable Nth-order Integration Operators [LNInO]: inverse operator approximation
    • Case of : Alternative and/or existing variants pertaining to Type 2 classification of time discretized operators
    • Numerical examples
  • Time Discretized Operators: Type 3 Classification
    • Formal development of time integration operators from time integral operators: development of Single-Step Integration Operators [SSInO]
    • Algorithms: transient analysis
    • Development of Multi-Step Integration Operators [MSInO]
    • Finite Element in Time Integration Operators [FETInO]
• DYNAMIC ANALYSIS: SECOND-ORDER HYPERBOLIC SYSTEMS
  • Time Discretized Operators: Type 1 Classification
    • Single field form: introducing the notion of transformation to modal basis
    • Implementation of the generalized VIP representation
    • The original VIrtual-Pulse (VIP) time integral methodology of computation structural dynamic problems
    • Single field form: without introducing transformation to modal basis
    • Two field form: without introducing transformation to modal basis
  • Time Discretized Operators: Type 2 Classification
    • Generic design and theoretical developments towards a new generation of a generalized family of time discretized operators for dynamic problems
    • Exact weighted time fields: two-field form
    • Case of and : approximations to the exact weighted time fields which still preserve the theoretical representation: Type 2 classification for dynamic systems
    • Implicit representations
    • Explicit representations
    • Numerical examples
    • Case of : Alternative and/or existing variants pertaining to Type 2 classification of time discretized operators
    • Implicit time discretized operators: second-order accurate representations
    • Hamiltonian equation: elasto-dynamics and multi-body-dynamics
    • Extensions of the implicit time discretized operator for nonlinear dynamic systems
    • Trapezoidal rule and mid-point rule for the two-field representation
    • Numerical examples: linear dynamic system
    • Explicit representations of time discretized operators: second-order accurate representations
    • On the design of an L-stable second-order accurate explicit time discretized operator representation
    • Numerical examples: linear dynamic system
  • Time Discretized Operators: Type 3 Classification
    • Formal development of time integration operators from time integral operators: development of Generalized Integration Operators of classification Type 3-[GInO]
    • Evolution and theoretical design of Generalized Integration Operators of Type 3-[GInO] : single-field form
    • Dynamic algorithms: theoretical design and further characterization of a wide variety of algorithms via Discrete Numerically Assigned [DNA] markers: Type 3-[GInO]
    • Numerical example: employing [GInO]
    • Evolution and theoretical design of Generalized Integration Operators of Type 3-[GInO] : two-field form
    • Multi-step Integration Operators [MSInO]
    • Finite Element in Time Integration Operators [FETInO]
• TOWARDS A FORMALLY STANDARDIZED -FAMILY OF INTEGRAL/ INTEGRATION OPERATORS FOR DYNAMIC FIELD PROBLEMS WITH NON-PROPORTIONAL DAMPING
  • Integral Operators: Dynamic Field Problems with Non-ProportionalDamping
    • Case 1 - Weighted time field as integrating factor and introducing transformation to modal basis
    • Case 2 - Weighted time field as Green's function and introducing transformation to modal basis
    • Development of the Exact Integral Operator [EIO] for systems with non-proportional damping
  • Dynamic Analysis by the Generalized VIP Representation for Systems with Non-Proportional Damping
    • Implementation of the generalized VIP representation for dynamic systems with non-proportional damping
    • Stability: Generalized VIP representation for systems with non-proportional damping
    • Accuracy: Generalized VIP representation for systems with non-proportional damping
  • On Bridging the Relationship Between Integral and Integration Operators for Systems with Non-Proportional Damping
    • Stability of time integration algorithms for dynamic systems with non-proportional damping
  • Numerical Test Cases
• CONCLUDING REMARKS
• ACKNOWLEDGMENTS
• References
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