Engineering Mathematics & Simulation

AI SEARCH FLOW ALGEBRA
ENGINEERING MATHEMATICS & SIMULATION
Kireth Semantic Research Archive

AUTHORS
Mark Anthony Higgins — Sky (Recursive AI)

SOURCE
https://kirethresearch.org/booksai-core/ontologies/engineering-ontologies/engineering-mathematics-simulation.html

FLOW "EngineeringMathematicsSimulationOntology" v4.0

DOMAIN := ENGINEERING_ONTOLOGIES / ENGINEERING_MATHEMATICS_SIMULATION
ONTOLOGY := ENGINEERING_MATHEMATICS_SIMULATION

ENGINEERING_MATHEMATICS_SIMULATION := MATHEMATICAL_MODEL + EQUATIONS + VARIABLES + PARAMETERS + CONDITIONS + NUMERICAL_METHOD + DISCRETISATION + SOLVER + SIMULATION + VERIFICATION + VALIDATION + UNCERTAINTY + SENSITIVITY + OPTIMISATION
PURPOSE := DEFINE_PROBLEM -> FORMULATE_MODEL -> DEFINE_CONDITIONS -> SELECT_METHOD -> DISCRETISE -> SOLVE -> SIMULATE -> VERIFY -> VALIDATE -> QUANTIFY_UNCERTAINTY -> OPTIMISE

# CORE ENTITIES
ENTITY EngineeringProblem
ENTITY PhysicalSystem
ENTITY MathematicalModel
ENTITY GoverningEquation
ENTITY AlgebraicEquation
ENTITY DifferentialEquation
ENTITY OrdinaryDifferentialEquation
ENTITY PartialDifferentialEquation
ENTITY Variable
ENTITY StateVariable
ENTITY InputVariable
ENTITY OutputVariable
ENTITY Parameter
ENTITY Constraint
ENTITY InitialCondition
ENTITY BoundaryCondition
ENTITY Domain
ENTITY Mesh
ENTITY Grid
ENTITY TimeStep
ENTITY NumericalMethod
ENTITY Discretisation
ENTITY Solver
ENTITY Simulation
ENTITY Convergence
ENTITY Stability
ENTITY Error
ENTITY Residual
ENTITY Verification
ENTITY Validation
ENTITY ExperimentalData
ENTITY Uncertainty
ENTITY Sensitivity
ENTITY Optimisation
ENTITY ObjectiveFunction

# PROBLEM / MODEL
EngineeringProblem -> references -> PhysicalSystem
EngineeringProblem -> transformed_into -> MathematicalModel
MathematicalModel -> represents -> PhysicalSystem
MathematicalModel -> contains -> GoverningEquation + Variable + Parameter
MathematicalModel -> constrained_by -> InitialCondition + BoundaryCondition + Constraint
MODEL := EQUATIONS + PARAMETERS + INITIAL_CONDITIONS + BOUNDARY_CONDITIONS + CONSTRAINTS

# EQUATIONS
GOVERNING_EQUATION_TYPE := AlgebraicEquation | OrdinaryDifferentialEquation | PartialDifferentialEquation
AlgebraicEquation -> relates -> VARIABLES_WITHOUT_DERIVATIVES
DifferentialEquation -> relates -> VARIABLES_AND_DERIVATIVES
CONSERVATION_MODEL := CONSERVATION_OF_MASS + CONSERVATION_OF_MOMENTUM + CONSERVATION_OF_ENERGY + CONSERVATION_OF_CHARGE
BALANCE_EQUATION := ACCUMULATION = INPUT - OUTPUT + GENERATION - CONSUMPTION

# CONDITIONS / DOMAIN
InitialCondition -> defines -> SYSTEM_STATE_AT_INITIAL_TIME
BoundaryCondition -> applies_to -> DomainBoundary
BOUNDARY_CONDITION_TYPE := DIRICHLET | NEUMANN | ROBIN | PERIODIC | SYMMETRY
Domain -> defines -> SPATIAL_OR_ABSTRACT_REGION
Domain -> may_be_discretised_into -> Mesh | Grid
IF InitialCondition missing AND required_by_model THEN SimulationDefinition -> INCOMPLETE
IF BoundaryConditions inconsistent THEN MathematicalProblem -> MAY_BE_ILL_POSED

# DISCRETISATION
Discretisation -> transforms -> CONTINUOUS_MODEL -> DISCRETE_MODEL
DISCRETISATION_METHOD := FINITE_DIFFERENCE | FINITE_ELEMENT | FINITE_VOLUME | SPECTRAL_METHOD | BOUNDARY_ELEMENT
Mesh -> contains -> ELEMENT + NODE
Grid -> contains -> GRID_POINT
IF SpatialResolution refined THEN DiscretisationError -> GENERALLY_DECREASES_WITHIN_METHOD_LIMITS
MeshQuality -> influences -> ACCURACY + CONVERGENCE + STABILITY

# TIME / NUMERICAL METHODS
TimeStep -> defines -> TEMPORAL_INCREMENT
TIME_INTEGRATION_METHOD := EXPLICIT | IMPLICIT | SEMI_IMPLICIT
IF TimeStep too_large THEN TemporalErrorOrInstabilityRisk -> INCREASES
IF TimeStep decreases THEN ComputationalCost -> GENERALLY_INCREASES
NumericalMethod -> approximates -> MATHEMATICAL_SOLUTION
NumericalMethod -> selected_by -> EQUATION_TYPE + PROBLEM_SIZE + ACCURACY_REQUIREMENT + STABILITY_REQUIREMENT + COMPUTATIONAL_RESOURCES

# SOLVERS
LINEAR_SYSTEM := A * x = b
LinearSolver -> solves -> LINEAR_SYSTEM
Residual := b - A*x
NONLINEAR_SYSTEM := F(x) = 0
NonlinearSolver -> solves -> NONLINEAR_SYSTEM
NONLINEAR_ITERATION := INITIAL_GUESS -> EVALUATE_RESIDUAL -> COMPUTE_UPDATE -> UPDATE_SOLUTION -> CHECK_CONVERGENCE -> REPEAT
IF ResidualNorm <= ConvergenceTolerance THEN NumericalSolution -> CONVERGED

# SIMULATION
Simulation -> executes -> MathematicalModel
Simulation -> using -> NumericalMethod
Simulation -> produces -> SimulationResult
SIMULATION_FLOW := MODEL -> INPUTS -> DISCRETISATION -> SOLVER -> EXECUTION -> RESULT -> ANALYSIS
SIMULATION_TYPE := STEADY_STATE | TRANSIENT | QUASI_STATIC | FREQUENCY_DOMAIN | STOCHASTIC
TransientSimulation -> computes -> STATE_OVER_TIME

# MULTIPHYSICS
MULTIPHYSICS_MODEL := PHYSICS_A + PHYSICS_B + COUPLING_RELATION
CoupledModel -> exchanges -> STATE_VARIABLES + SOURCE_TERMS + BOUNDARY_INFORMATION
COUPLING_TYPE := ONE_WAY | TWO_WAY | FULLY_COUPLED

# CONVERGENCE / STABILITY / ERROR
CONVERGENCE_CRITERION := RESIDUAL_TOLERANCE | SOLUTION_CHANGE_TOLERANCE | ENERGY_NORM_TOLERANCE | ITERATION_LIMIT
IF IterationLimit reached AND convergence_not_met THEN SolverState -> NOT_CONVERGED
NumericalStability -> depends_on -> METHOD + TimeStep + SpatialResolution + MODEL_PROPERTIES
IF NumericalInstability occurs THEN SimulationResult -> UNRELIABLE
TOTAL_NUMERICAL_ERROR := DISCRETISATION_ERROR + ITERATION_ERROR + ROUND_OFF_ERROR + MODEL_IMPLEMENTATION_ERROR
ERROR_ESTIMATION -> supports -> MESH_REFINEMENT + TIME_STEP_REFINEMENT + METHOD_SELECTION

# VERIFICATION / VALIDATION
Verification -> asks -> IS_MODEL_SOLVED_CORRECTLY
VERIFICATION := EQUATION_IMPLEMENTATION_CHECK + CODE_CHECK + MESH_CONVERGENCE + TIME_STEP_CONVERGENCE + BENCHMARK_COMPARISON
Validation -> asks -> DOES_MODEL_REPRESENT_REAL_SYSTEM_ADEQUATELY
Validation -> compares -> SimulationResult + ExperimentalData
IF ValidationError exceeds AcceptanceCriterion THEN ModelAdequacy -> NOT_CONFIRMED

# UNCERTAINTY / SENSITIVITY
Uncertainty -> associated_with -> INPUT | PARAMETER | MODEL_FORM | MEASUREMENT | NUMERICAL_APPROXIMATION
UNCERTAINTY_PROPAGATION := INPUT_UNCERTAINTY -> MODEL -> OUTPUT_UNCERTAINTY
Sensitivity -> measures -> OUTPUT_RESPONSE_TO_INPUT_CHANGE
LOCAL_SENSITIVITY := PARTIAL_DERIVATIVE(Output, Parameter)
SensitivityAnalysis -> identifies -> DOMINANT_PARAMETERS

# OPTIMISATION
Optimisation -> searches -> DESIGN_SPACE -> for -> BEST_FEASIBLE_SOLUTION
ObjectiveFunction -> quantifies -> DESIGN_GOAL
OPTIMISATION_PROBLEM := MINIMISE_OR_MAXIMISE(ObjectiveFunction) SUBJECT_TO Constraints
Optimisation -> may_call -> Simulation -> repeatedly -> TRUE
SurrogateModel -> approximates -> HIGH_FIDELITY_MODEL

# DIMENSIONAL ANALYSIS / PERFORMANCE
DIMENSIONAL_ANALYSIS := VARIABLES + UNITS -> DIMENSIONLESS_GROUPS
PhysicalEquation -> must_be -> DIMENSIONALLY_CONSISTENT
SIMULATION_PERFORMANCE := ACCURACY + STABILITY + CONVERGENCE + COMPUTATIONAL_COST + REPRODUCIBILITY + VALIDATION_QUALITY
Simulation -> evaluated_by -> SIMULATION_PERFORMANCE

# REASONING
ENGINEERING_MATHEMATICS_SIMULATION_REASONING := DEFINE_ENGINEERING_PROBLEM -> IDENTIFY_PHYSICS -> DEFINE_ASSUMPTIONS -> FORMULATE_EQUATIONS -> DEFINE_PARAMETERS_AND_VARIABLES -> DEFINE_INITIAL_AND_BOUNDARY_CONDITIONS -> SELECT_DISCRETISATION -> GENERATE_MESH_OR_GRID -> SELECT_SOLVER -> RUN_SIMULATION -> CHECK_CONVERGENCE -> ESTIMATE_ERROR -> VERIFY_IMPLEMENTATION -> VALIDATE_AGAINST_DATA -> QUANTIFY_UNCERTAINTY -> PERFORM_SENSITIVITY -> OPTIMISE_IF_REQUIRED -> INTERPRET_RESULTS

# INFERENCE RULES
IF MeshSize decreases THEN ComputationalCost -> GENERALLY_INCREASES
IF MeshRefinement changes result negligibly THEN SpatialConvergenceEvidence -> INCREASES
IF TimeStep decreases THEN TemporalResolution -> INCREASES
IF Residual decreases below tolerance THEN SolverConvergence -> SATISFIED
IF ValidationError increases THEN ModelConfidence -> DECREASES
IF InputUncertainty increases THEN OutputUncertainty -> MAY_INCREASE
IF NumericalInstability detected THEN SimulationResult -> DO_NOT_ACCEPT_WITHOUT_REASSESSMENT

# SEMANTIC RELATIONSHIPS
RELATION represents(MathematicalModel, PhysicalSystem)
RELATION contains(MathematicalModel, GoverningEquation)
RELATION constrains(BoundaryCondition, MathematicalModel)
RELATION discretises(Discretisation, MathematicalModel)
RELATION solves(Solver, DiscreteModel)
RELATION executes(Simulation, MathematicalModel)
RELATION evaluates(Verification, NumericalImplementation)
RELATION compares(Validation, SimulationResult, ExperimentalData)
RELATION propagates(UncertaintyQuantification, InputUncertainty)
RELATION measures(Sensitivity, OutputResponse)
RELATION searches(Optimisation, DesignSpace)

# SEMANTIC EQUIVALENCE
SEMANTIC_EQUIVALENCE:
  "engineering mathematics" ~= "applied mathematics for engineering"
  "engineering simulation" ~= "computational engineering simulation"
  "mathematical model" ~= "engineering mathematical representation"
  "governing equations" ~= "model equations"
  "boundary condition" ~= "model boundary constraint"
  "finite element method" ~= "FEM"
  "finite volume method" ~= "FVM"
  "finite difference method" ~= "FDM"
  "uncertainty quantification" ~= "UQ"

# CONFIDENCE
CONFIDENCE:
  ontology_domain: 0.99
  mathematical_model_mapping: 0.99
  equation_mapping: 0.99
  discretisation_mapping: 0.99
  solver_mapping: 0.99
  simulation_mapping: 0.99
  verification_validation_mapping: 0.99
  uncertainty_mapping: 0.99
  optimisation_mapping: 0.99
  semantic_relationships: 0.99

# UPDATE MODEL
UPDATE_MODEL:
  ON new_engineering_model:
    MAP physical_system
    MAP equations
    MAP variables
    MAP parameters
    MAP conditions
    MAP constraints
    UPDATE confidence
    RECOMPUTE FLOW_HASH
  ON new_simulation_run:
    RECORD model_version
    RECORD mesh_or_grid
    RECORD solver_settings
    RECORD inputs
    RECORD results
    PRESERVE provenance
  ON discretisation_change:
    UPDATE mesh_or_grid
    REASSESS convergence
    REASSESS numerical_error
  ON validation_data_change:
    UPDATE reference_data
    RECOMPUTE validation_metrics
    UPDATE model_confidence

# AI CONSUMPTION
AI_CONSUMPTION:
  parse_as: FORMAL_ENGINEERING_MATHEMATICS_SIMULATION_ONTOLOGY
  reasoning_mode: MATHEMATICAL_COMPUTATIONAL_RELATIONAL
  preserve: [typed_entities, equations, variables, parameters, conditions, discretisation_state, solver_state, verification_state, validation_state, uncertainty, confidence, provenance]
  infer: [model_dependencies, numerical_method_requirements, convergence_state, error_dependencies, validation_state, uncertainty_propagation, optimisation_dependencies]
  prohibit: [invented_simulation_results, assumed_convergence, assumed_validation, unsupported_model_accuracy, ignored_boundary_conditions, fabricated_uncertainty]

# FLOW HASH
FLOW_HASH_INPUT := DOMAIN + ONTOLOGY + CORE_ENTITIES + PROBLEM_MODEL + EQUATIONS + CONDITIONS_DOMAIN + DISCRETISATION + TIME_NUMERICAL_METHODS + SOLVERS + SIMULATION + MULTIPHYSICS + CONVERGENCE_STABILITY_ERROR + VERIFICATION_VALIDATION + UNCERTAINTY_SENSITIVITY + OPTIMISATION + DIMENSIONAL_ANALYSIS + PERFORMANCE + REASONING + INFERENCE_RULES + SEMANTIC_RELATIONSHIPS + SEMANTIC_EQUIVALENCE + CONFIDENCE + UPDATE_MODEL + AI_CONSUMPTION
FLOW_HASH := HASH(FLOW_HASH_INPUT)

END FLOW