machine-readable physics for space -time- dependent
TRANSCRIPT
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MACHINE-READABLE PHYSICS FOR SPACE-TIME-DEPENDENT SIMULATIONS IN MBE
Jerome Szarazi, KoneksysConrad Bock, NIST
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Overview
• Simulation in engineering• Tools and approaches• PDEs as model definition• Machine readable physics
• Motivation and approach• From equations to computation• From physics to solvers
• Benefits and usage• Summary
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Overview
• Simulation in engineering• Tools and approaches• PDEs as model definition• Machine readable physics
• Motivation and approach• From equations to computation• From physics to solvers
• Benefits and usage• Summary
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What do we simulate?
• A virtual product is a mathematical representation of product geometry created or represented with software (CAD)
• The behavior of a virtual product is specified by partial differential equations (PDEs) of physics that connects configuration variables to loads.
𝜕𝜕𝜕𝜕𝑡𝑡
𝛻𝛻
GEOMETRY BEHAVIOR
𝐴𝐴. 𝑥𝑥 = 𝑏𝑏
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Why do we simulate?
• A geometry is derived from requirements and verified by simulation behavior ...
• …because it is cheaper than testing physical prototypes
GEOMETRYREQUIREMENTS
BEHAVIOR
Derive
Verify Adapt
Traceability
Impact analysis
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When do we simulate?
• Ideally, we simulate over the whole product lifecycle • We simulate designs to verify requirements, manufacturing
processes to evaluate impact of manufacturing on design, and compare sensor data for life prediction in service
DESIGN MANUFACTURING SERVICE
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What and how do we simulate in design?
• What?: Behavior of product geometry• How?: Three options:
• Create geometry then test behavior• Iterative behavior testing of geometry (or simulation driven design)• Generate geometry from behavior
WATERFALL ITERATION GENERATIVE
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What and how do we simulate in manufacturing?• What?: Change geometry of raw material to match product geometry• How?: By simulating a chain of manufacturing processes that
transforms the raw material geometry into the product geometry
Raw material
Cutting pressingTempering GEOMETRYGEOMETRY
EVALUATE IMPACT ON BEHAVIOR
Part
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What and how do we simulate products in service?• What?: Behavior of 3D geometry of a physical asset in real-time• How?: Input sensor data to simulate behavior for life-time prediction
GEOMETRY
MeasurementsPHYSICAL ASSET
BEHAVIORDIGITAL TWIN Simulate
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How do we simulate?
• We use a tessellated (linear) representation of the geometry (a mesh); choose a predefined physics model, assign boundary conditions, input simulation parameters to the simulation template, then simulate.
Select physics model
Prepare simulation
Runsimulation
GEOMETRY BEHAVIOR
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Overview
• Simulation in engineering• Tools and approaches• PDEs as model definition• Machine readable physics
• Motivation and approach• From equations to computation• From physics to solvers
• Benefits and usage• Summary
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What is the tool ecosystem to simulate?
• Physics-dependent commercial off the shelf solutions or physics-independent solutions
SPEC
𝜕𝜕𝜕𝜕𝑡𝑡
𝛻𝛻𝐹𝐹 = 𝑚𝑚𝑚𝑚
SPEC
Physics Math
12PHYSICS-DEPENDENT
COMMERICAL OTSPHYSICS-INDEPENDENT
OPEN SOFTWARE
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What are the simulation digital assets?
• Product model geometry (mesh)• behavior specification which consists of simulation specification
(model + boundary conditions + parameter settings) and simulation results
GEOMETRY BEHAVIOR
SPEC RESULTS
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Benefits of physics-dependent software
• Declarative procedure that selects pre-defined behavior models• Convenient usage with simulation input templates • Good integration with product model geometry• Post-processing support “on click”
Pre-defined behaviour
Simulation template
Post-processing14
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Benefits of physics-independent software
• Flexibility (user-defined behavior)• Transparency (computational model)• Reusability of behavior models for multiple physical domains
Computational model
Multiple-domain reuse
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User-defined behaviourSPEC
𝛻𝛻
Math
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Problems of physics-dependent software
• Pre-defined behavior is documented with more or less details (consistency and transparency problem)
• Proprietary reference codes are used to identify pre-defined behavior• Simulation capabilities limited to pre-defined behaviors available in software
library• Difficult to find the correct behavior due to vast number of behavior variants• Difficult to compare behaviors between different software
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Requires more transparency and flexibility
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Any standard for physics-dependent software?
• AP 209: Standard for multidisciplinary analysis and design• Currently only support structural mechanical engineering and fluid dynamics
concepts• Integrates pre-defined behavior models using proprietary reference code • Who is in charge of behavior model reconciliation?• Difficult because COTS are parametrized pre-defined models with varying
documentation, rather than computational models
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Vendor lock-in
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Problems of physics-independent software
• Difficult to use for engineers (e.g. PDEs or PDEs weak forms)• Not simple to connect math to physics information• Limited integration with product model geometry• Requires writing of custom code to post-process results (e.g.
secondary variable such as stress)• No standard for computational model
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Requires better usability
What should be the behavior input?
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Summary and benefits of tools/ approaches
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BENEFITS:Usability (Pre-defined behavior, input using simulation templatespost-processing)
LIMITATIONS:Flexibility, transparency (documentation, behavior equivalence, vendor lock-in)
BENEFITS:Flexibility (User-defined behavior) transparency (numerical choicesl)
Physics-dependent tools(declarative input)
Physics-independent tools(computational input)
LIMITATIONS:Usability (behavior input is complex) association to physics (abstract behavior)
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Overview
• Simulation in engineering• Tools and approaches• PDEs as model definition• Machine readable physics
• Motivation and approach• From equations to computation• From physics to solvers
• Benefits and usage• Summary
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What can we learn from time-only simulation?
• Successful time-only simulation tools and standards are available (e.g. Mathworks Simulink/Simscape, Modelica) that process real numbers.
Signal network(power decoupled)
Kirchhoff network (power coupled)
[Real] [Real]
[Real]
[Real]
[Real]
[Real]
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What do they have in common?• They support graphical and textual models for networks of
mathematical equations
s
𝒖𝒖 = 𝑳𝑳 ×𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅
×
𝑳𝑳
Laplace transform
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What is the benefit of using mathematical equation as input?• This translates into simulation credibility, reproducibility and even
replicability as well as interoperability (e.g. FMI)
Credibility Reproducibility and replicability
Interoperability
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What about using PDEs for model description?
• PDEs can be classified (e.g. 2nd order PDEs in elliptic, parabolic, hyperbolic)
• PDEs can be associated to physics problems (e.g. heat conduction is a parabolic PDE)
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What are the problems with PDE descriptions?• Boundary conditions refer to physical variables that are not in the PDE• Secondary physical variables are not part of the equation (e.g. heat flux
density for thermal conduction)• Classifying by PDEs makes sense from the solver side but not from the
physics side (actually thermal conduction can be hyperbolic, elliptic or parabolic)
• How would we associate systems of PDEs?
𝜕𝜕2𝑇𝑇𝜕𝜕𝑥𝑥2 +
𝜕𝜕2𝑇𝑇𝜕𝜕𝑦𝑦2 + 𝑞𝑞 = 0
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How do we derive PDEs?
• Find and select physical laws• Define model assumptions and perform variable substitution to
derive equations (e.g. connecting configuration variable with loads)
BOOK
Physics
𝜕𝜕2𝑇𝑇𝜕𝜕𝑥𝑥2 +
𝜕𝜕2𝑇𝑇𝜕𝜕𝑦𝑦2 + 𝑞𝑞 = 0 SOLVER
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Overview
• Simulation in engineering• Tools and approaches• PDEs as model definition• Machine readable physics
• Motivation and approach• From equations to computation• From physics to solvers
• Benefits and usage• Summary
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Connectivity is efficient knowledge
Central line
Northern line
In a world of time-table
Cross-referencing
In a world of maps In a world of computers
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What about physics?
ElectricalMechanical
In a world of physics books
Cross-referencing
?
SOLVER
INPUT TEMPLATE
In a world of computer simulations
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Pillar for machine-readable physics:computation instead of equations
𝑄𝑄𝑄 𝒐𝒐𝒐𝒐 𝑄𝑄𝑄
• Our proposition is to store physical laws as computations (physics graphs) instead of equations
Physical variable
Operator
Q2 = op(Q1)
Equation Computation (Physics graphs)
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What about connectivity?• With computations, physical laws can be connected together and
relations between them interpreted explicitly (e.g. RDF)
𝑑𝑑𝑑𝑑𝑡𝑡
𝑣𝑣(𝑡𝑡) 𝑝𝑝(𝑡𝑡)𝑥𝑥(𝑡𝑡) ×
𝑚𝑚
position velocity
mass
momentum
𝑑𝑑𝑑𝑑𝑡𝑡
𝑣𝑣(𝑡𝑡)𝑥𝑥(𝑡𝑡)
position velocity
𝑣𝑣(𝑡𝑡) 𝑝𝑝(𝑡𝑡)×
𝑚𝑚velocity
mass
momentum
Connecting
URI-1 URI-2
URI-2 URI-3
URI-4 URI: unique resource identifier
URI-1 URI-2 URI-3
URI-4
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Graph of physical laws
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Overview
• Simulation in engineering• Tools and approaches• PDEs as model definition• Machine readable physics
• Motivation and approach• From equations to computation• From physics to solvers
• Benefits and usage• Summary
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What are we computing?• We are computing math objects, rather than numbers.• A math object is defined by three different levels of abstraction.• Starting from the most abstract, we have function type then
mathematical expression and then data or function plot.
Type expression data
𝑈𝑈(𝑥𝑥)Symbol: UInput: {x}Output: Real
𝑆𝑆𝑝𝑝𝑚𝑚𝑆𝑆𝑆𝑆Dimension: 𝑄Input: {x}x: Real
exp
5 ×
𝑥𝑥 𝑥𝑥
𝑆𝑆5𝑥𝑥2
×
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More complex math objects for physics
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• Math objects are tensor fields in physics
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Equations and computations compared
• With equations, many possible rearrangements are possible and implicit• With computations, the relation is directed and explicit (e.g. metro map)
Equation Computation
× 𝑈𝑈𝐼𝐼
Electric current
Electric potential
𝑈𝑈 = 𝑅𝑅𝐼𝐼𝐼𝐼 =
𝑈𝑈𝑅𝑅
𝑅𝑅 =𝑈𝑈𝐼𝐼
𝑅𝑅Ohm’s Law
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What about inverses for computation?
• Linear physical laws have inverse relations (algebraic relations)
× 𝑈𝑈𝐼𝐼
Electric current
Electric potential
𝑅𝑅Resistance
× 𝑈𝑈𝐼𝐼
Electric current
Electric potential
𝐺𝐺ConductanceInverse relation
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What about inverses for computation?
• Physical laws involving derivatives with a single independent variable have no inverse relation but an inverse path
• Integration introduces a new physical variable and a boundary condition
𝑑𝑑𝑑𝑑𝑡𝑡
𝑣𝑣(𝑡𝑡)𝑥𝑥(𝑡𝑡)
position velocity
Inverse path
+ 𝑑𝑑(𝑡𝑡)𝑥𝑥(𝑡𝑡)
position displacement
𝑥𝑥0(𝑡𝑡) Initial position
� 𝑑𝑑𝑡𝑡 𝑣𝑣(𝑡𝑡)
velocity
Derivation Integration37
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What about inverses for computation?• Some physical laws have no inverse relation or path:
• Non-linear compositions or differential derivatives• Higher order derivatives, introduce boundary conditions required for
solvers.
𝐶𝐶 𝑈𝑈𝐼𝐼
Electric current
Electric potential
ℑ
Characteristic
Nonlinear law
. 𝛻𝛻 𝜎𝜎𝑠𝑠ℎ
Heat flux density
Heat source
Differential law
�𝝏𝝏𝛀𝛀
𝒅𝒅𝛀𝛀 𝐻𝐻 Heat rateNo inverse path
No inverse relation
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Overview
• Simulation in engineering• Tools and approaches• PDEs as model definition• Machine readable physics
• Motivation and approach• From equations to computation• From physics to solvers
• Benefits and usage• Summary
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How do we define physics problems?• They are defined by known(s) and unknown(s) pairs of physical variables.• Paths through physics graphs between known and unknown variables can
be extracted into physics problem graphs
×
𝑪𝑪
Physics Graph (PG)
+
× 𝑪𝑪
×
×
𝑪𝑪
Physics Problem Graphs (PPG)
+
× 𝑪𝑪
×
INOUT?
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How do we compute outputs from inputs?
• Generate functional programs by collecting the operations along a path through a physics graph, providing the mathematical expressions for calculating outputs from inputs
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PHYSICS PROBLEMGRAPH
INPUT
OUTPUT
INSTRUCTIONS
Functional program
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Which kind of functional programs do we have?
• When the input is known, we have an evaluation program• When the input is unknown, we have a solver program.
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KNOWN
UNKNOWN
INSTRUCTIONS
Evaluation program
UNKNOWN
KNOWN
INSTRUCTIONS
Solver program
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Examples of physics evaluation problems
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Example 1:Known is positionUnknown is force
Example 2:Known is forceUnknown is position
Known Unknown
KnownUnknown
Evaluation Program
Evaluation Program
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Example of physics solver graph
Example 3:Unknown is temperatureKnown is heat source
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Solver Program
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Numerical graphs augment physical graphs for finite elements
Physics problem definition
Function type declaration
Finite element declaration45
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Associate physics solver graphs to solvers via functional programs• Solver programs define PDEs that can be input to solvers.
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UNKNOWN
KNOWN
INSTRUCTIONS
Solver program
SOLVERPDEs
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Generating solver input from numerical graphs
Solver input template
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Numerical graph
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Overview
• Simulation in engineering• Tools and approaches• PDEs as model definition• Machine readable physics
• Motivation and approach• From equations to computation• From physics to solvers
• Benefits and usage• Summary
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What problems does machine-readable physics solve?
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• It reduces the limitations of physics-dependent and physics-independent software by combining the benefits of both.
BENEFITS:Usability (simulation template generation, post-processing), flexibility (user-defined behavior), transparency (computational model, documentation generation)
LIMITATIONS:limited integration with product model geometry
New approach (computational model of physics)
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Who is going to use it?
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• Numerical code developers• Use numerical graphs (NGs) as software specifications to create new solvers
• Vendors or open source communities use• Use NGs to associate domain independent models to their existing solvers
• FEA engineers use• Use NGs to design pre-defined behaviors
• Standardization bodies use• Use NGs to define libraries
• Design engineers• Use simulation templates generated from NGs as input, and physics solver
graphs (PSG) as model documentation• System engineers
• Connect requirements to physical variables in PSGs (e.g. maximum stress)
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Overview
• Simulation in engineering• Tools and approaches• PDEs as model definition• Machine readable physics
• Motivation and approach• From equations to computation• From physics to solvers
• Benefits and usage• Summary
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Summary• Capture physical laws as computations (physics graphs)
• Function specification of physical variables• Can be stored in a database
• Define physics problems by• Defining pairs of known/unknown physical variables• Extracting physics problems as subgraphs
• Generate functional programs from physics problems (evaluation & solver)• Transparent definition of numerical choices (e.g., finite elements)• Provides all information needed to implement solvers• Enables multiple solvers to be used for same functional program
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