ReferenceAvailable
Piezoelectricity
Coupled linear piezoelectricity.
Reference
| Field | Value | Description |
|---|---|---|
| constraints | [] | — |
| coupled_runtime | {"adapter_module":"telion_backend.v2.runtimes.coupled_fields","adjoint_vjp_owner":"PiezoelectricCoupledAdapter","certification":"Structural Mechanics and Electrostatics operators plus the thermodynamically reciprocal piezoelectric cross coupling.","checkpointing":true,"dynamic_state_schema":false,"execution_backend":"telion-v2-shared-execution","gradient_channels":[],"history_depth":2,"history_jacobian_owner":"PiezoelectricCoupledAdapter","jacobian_owner":"PiezoelectricCoupledAdapter","nonlinear_solver":"CoupledNewtonSession","output_vjp_owner":"PiezoelectricCoupledAdapter","requires_spatial_mesh":true,"residual_owner":"PiezoelectricCoupledAdapter","runtime":"telion-coupled-graph","state_fields":["pze.u","pze.V"],"supported_integrators":["BDF1","BDF2"]} | — |
| differentiable_runtime | {"certification":"PiezoelectricDifferentiableRuntime","coordinate_expression_gradient":true,"coordinate_gradient":true,"design_properties":["*"],"execution_backend":"telion-v2-shared-execution","expression_objectives":true,"material_expression_gradient":true,"material_gradient":true,"objective_rules":[{"entity_dimensions":["global"],"modes":["stationary"],"operators":["value"],"supports_selection":false,"symbols":["pze.compliance"],"value_kind":"scalar-global"},{"entity_dimensions":["global"],"modes":["stationary"],"operators":["value"],"supports_selection":false,"symbols":["pze.mechanicalCompliance"],"value_kind":"scalar-global"},{"entity_dimensions":["global","domain","boundary","point"],"modes":["stationary","transient"],"operators":["norm"],"supports_selection":true,"symbols":["pze.u"],"value_kind":"field"},{"entity_dimensions":["global","domain","boundary","point"],"modes":["stationary","transient"],"operators":["integral","average","minimum","maximum","rms","norm","smooth-maximum","smooth-minimum"],"supports_selection":true,"symbols":["pze.disp"],"value_kind":"field"},{"entity_dimensions":["global","domain","boundary","point"],"modes":["stationary","transient"],"operators":["integral","average","minimum","maximum","rms","norm","smooth-maximum","smooth-minimum"],"supports_selection":true,"symbols":["pze.ux"],"value_kind":"field"},{"entity_dimensions":["global","domain","boundary","point"],"modes":["stationary","transient"],"operators":["integral","average","minimum","maximum","rms","norm","smooth-maximum","smooth-minimum"],"supports_selection":true,"symbols":["pze.uy"],"value_kind":"field"},{"entity_dimensions":["global","domain","boundary","point"],"modes":["stationary","transient"],"operators":["integral","average","minimum","maximum","rms","norm","smooth-maximum","smooth-minimum"],"supports_selection":true,"symbols":["pze.uz"],"value_kind":"field"},{"entity_dimensions":["global","domain","boundary","point"],"modes":["stationary","transient"],"operators":["integral","average","minimum","maximum","rms","norm","smooth-maximum","smooth-minimum"],"supports_selection":true,"symbols":["pze.V"],"value_kind":"field"},{"entity_dimensions":["global","domain"],"modes":["stationary"],"operators":["integral","average","minimum","maximum","rms","norm","smooth-maximum","smooth-minimum"],"supports_selection":true,"symbols":["pze.strain"],"value_kind":"field"},{"entity_dimensions":["global","domain"],"modes":["stationary"],"operators":["integral","average","minimum","maximum","rms","norm","smooth-maximum","smooth-minimum"],"supports_selection":true,"symbols":["pze.stress"],"value_kind":"field"},{"entity_dimensions":["global","domain","boundary","point"],"modes":["stationary"],"operators":["integral","average","minimum","maximum","rms","norm","smooth-maximum","smooth-minimum"],"supports_selection":true,"symbols":["pze.mises"],"value_kind":"field"},{"entity_dimensions":["global","domain"],"modes":["stationary"],"operators":["integral","average","minimum","maximum","rms","norm","smooth-maximum","smooth-minimum"],"supports_selection":true,"symbols":["pze.E"],"value_kind":"field"},{"entity_dimensions":["global","domain"],"modes":["stationary"],"operators":["integral","average","minimum","maximum","rms","norm","smooth-maximum","smooth-minimum"],"supports_selection":true,"symbols":["pze.D"],"value_kind":"field"},{"entity_dimensions":["global"],"modes":["stationary"],"operators":["value"],"supports_selection":false,"symbols":["pze.energy"],"value_kind":"scalar-global"},{"entity_dimensions":["global"],"modes":["stationary"],"operators":["value"],"supports_selection":false,"symbols":["pze.couplingEnergy"],"value_kind":"scalar-global"},{"entity_dimensions":["global"],"modes":["stationary"],"operators":["value"],"supports_selection":false,"symbols":["pze.strainEnergy"],"value_kind":"scalar-global"},{"entity_dimensions":["global"],"modes":["stationary"],"operators":["value"],"supports_selection":false,"symbols":["pze.electricEnergy"],"value_kind":"scalar-global"},{"entity_dimensions":["global","domain","boundary","point"],"modes":["transient"],"operators":["integral","average","minimum","maximum","rms","norm","smooth-maximum","smooth-minimum"],"supports_selection":true,"symbols":["pze.velocity"],"value_kind":"field"},{"entity_dimensions":["global","domain","boundary","point"],"modes":["transient"],"operators":["integral","average","minimum","maximum","rms","norm","smooth-maximum","smooth-minimum"],"supports_selection":true,"symbols":["pze.acceleration"],"value_kind":"field"}],"residual_owner":"Piezoelectricity.element_residual","state_fields":["u","V"],"supported_objective_symbols":{"coordinate":["pze.compliance","pze.mechanicalCompliance","pze.u","pze.disp","pze.ux","pze.uy","pze.uz","pze.V","pze.strain","pze.stress","pze.mises","pze.E","pze.D","pze.energy","pze.couplingEnergy","pze.strainEnergy","pze.electricEnergy"],"material":["pze.compliance","pze.mechanicalCompliance","pze.u","pze.disp","pze.ux","pze.uy","pze.uz","pze.V","pze.strain","pze.stress","pze.mises","pze.E","pze.D","pze.energy","pze.couplingEnergy","pze.strainEnergy","pze.electricEnergy"],"topology":["pze.compliance","pze.mechanicalCompliance","pze.u","pze.disp","pze.ux","pze.uy","pze.uz","pze.V"]},"topology_expression_gradient":true} | — |
| dynamic_outputs | [] | — |
| equation_summary | div(sigma)+F=rho*d2u/dt2; div(D)=rho_q; sigma=C:strain-e^T E; D=e:strain+eps E | — |
| frequency_domain_runtime | null | — |
| metadata | {"releaseSupport":{"certifiedStudies":["stationary","eigenfrequency","parametric-sweep","transient"],"manifest":"docs/september-2026-classroom-physics-matrix.json","manifestSha256":"0a819976ac943e9eedf5bc7f9c80880dd9103cb62bb4c5a915b98c452b135a5c","profile":"september-2026-classroom","reason":"Available.","selectable":true}} | — |
| preferred_material | PZT-5H | — |
| schema_version | 0.1.0 | — |
| stateVariables | [{"adjointDescription":"Coupled piezoelectric adjoint solve supports mechanical displacement contributions to shape, topology, and expression objectives.","adjointSupported":true,"availability":{"topologyMethods":[]},"canonicalSymbol":"pze.u","complexValued":false,"componentDescriptions":["X-direction displacement","Y-direction displacement","Z-direction displacement"],"componentNames":["ux","uy","uz"],"componentSymbols":["pze.ux","pze.uy","pze.uz"],"components":3,"description":"Mechanical displacement field of the piezoelectric solid.","differentiability":"exact-ad","displayName":"Displacement","finiteElementSpace":"H1-P1","initialValue":"0","location":"node","name":"u","namingPolicy":"fixed","quantityId":"pze.u","rank":"vector","role":"state","runtimeEvidence":{"certificationFixture":"tests/python_backend/test_differentiable_physics_contract.py::test_builtin_physics_optimization_contract_gate","provider":"Piezoelectricity.element_residual"},"symbolPattern":null,"unit":"m"},{"adjointDescription":"Coupled piezoelectric adjoint solve supports electric-potential contributions to shape, topology, and expression objectives.","adjointSupported":true,"availability":{"topologyMethods":[]},"canonicalSymbol":"pze.V","complexValued":false,"componentDescriptions":["Electric potential"],"componentNames":["V"],"componentSymbols":["pze.V"],"components":1,"description":"Electrostatic potential coupled to mechanical deformation.","differentiability":"exact-ad","displayName":"Electric potential","finiteElementSpace":"H1-P1","initialValue":"0","location":"node","name":"V","namingPolicy":"fixed","quantityId":"pze.V","rank":"scalar","role":"state","runtimeEvidence":{"certificationFixture":"tests/python_backend/test_differentiable_physics_contract.py::test_builtin_physics_optimization_contract_gate","provider":"Piezoelectricity.element_residual"},"symbolPattern":null,"unit":"V"}] | — |
| studies | [{"certified":true,"differentiability":"exact-ad","kind":"stationary","reason":"september-2026-classroom verified contract: constitutive Jacobian finite-difference agreement and linear displacement scaling with applied voltage"},{"certified":true,"differentiability":"exact-ad","kind":"eigenfrequency","reason":"september-2026-classroom verified contract: coupled mechanical/electrical residuals and monotone spatial refinement"},{"certified":false,"differentiability":"exact-ad","kind":"eigenvalue","reason":"Use Eigenfrequency for physical resonance frequencies."},{"certified":true,"differentiability":"exact-ad","kind":"parametric-sweep","reason":"september-2026-classroom verified contract: two-point body-load sweep publishes distinct linearly scaled coupled fields on one unchanged mesh"},{"certified":false,"differentiability":"exact-ad","kind":"sensitivity","reason":"Sensitivity is unavailable for this physics interface."},{"certified":false,"differentiability":"exact-ad","kind":"optimization","reason":"Optimization is unavailable for this physics interface."},{"certified":true,"differentiability":"exact-ad","kind":"transient","reason":"september-2026-classroom verified contract: second-order coupled modal time convergence with distinct variable-step operators"}] | — |
| transient_runtime | {"capacity_properties":["rho"],"certification":"PiezoelectricDynamicBdfRuntime","checkpointing":true,"design_properties":["cE","e","epsS","rho"],"design_property_conditions":{},"design_vjp_owner":"PiezoelectricTransientTopologyRuntime","expression_objectives":true,"gradient_channels":["coordinates","material","input","initial"],"gradient_vjp_owner":"PiezoelectricTransientTopologyRuntime","history_depth":2,"input_gradient":true,"input_vjp_owner":"PiezoelectricTransientTopologyRuntime","observation_vjp_owner":"PiezoelectricTransientTopologyRuntime","residual_linearization_owner":"PiezoelectricTransientTopologyRuntime","runtime":"piezoelectric-bdf","state_fields":["u","V"],"state_reconstruction_owner":"PiezoelectricTransientTopologyRuntime","supported_integrators":["BDF1","BDF2"],"supported_observation_symbols":["pze.u","pze.ux","pze.uy","pze.uz","pze.disp","pze.V","pze.velocity","pze.acceleration"],"time_derivative_orders":{"V":0,"u":2},"topology_gradient":true,"transpose_solve_owner":"PiezoelectricTransientTopologyRuntime"} | — |
| weak_contributions | [] | — |
| category | Structural Mechanics / AC/DC | — |
| description | Coupled linear piezoelectricity. | — |
| dimensions | [3] | — |
| display_name | Piezoelectricity | — |
| kind | Piezoelectricity | — |
| namespace | pze | — |
| status | certified | — |
| visible | true | — |
Coupled linear piezoelectricity.
Contract
| Property | Value |
|---|---|
| constraints | [] |
| coupled_runtime | {"adapter_module":"telion_backend.v2.runtimes.coupled_fields","adjoint_vjp_owner":"PiezoelectricCoupledAdapter","certification":"Structural Mechanics and Electrostatics operators plus the thermodynamically reciprocal piezoelectric cross coupling.","checkpointing":true,"dynamic_state_schema":false,"execution_backend":"telion-v2-shared-execution","gradient_channels":[],"history_depth":2,"history_jacobian_owner":"PiezoelectricCoupledAdapter","jacobian_owner":"PiezoelectricCoupledAdapter","nonlin... |
| differentiable_runtime | {"certification":"PiezoelectricDifferentiableRuntime","coordinate_expression_gradient":true,"coordinate_gradient":true,"design_properties":["*"],"execution_backend":"telion-v2-shared-execution","expression_objectives":true,"material_expression_gradient":true,"material_gradient":true,"objective_rules":[{"entity_dimensions":["global"],"modes":["stationary"],"operators":["value"],"supports_selection":false,"symbols":["pze.compliance"],"value_kind":"scalar-global"},{"entity_dimensions":["global"]... |
| dynamic_outputs | [] |
| equation_summary | div(sigma)+F=rho*d2u/dt2; div(D)=rho_q; sigma=C:strain-e^T E; D=e:strain+eps E |
| frequency_domain_runtime | null |
| metadata | {"releaseSupport":{"certifiedStudies":["stationary","eigenfrequency","parametric-sweep","transient"],"manifest":"docs/september-2026-classroom-physics-matrix.json","manifestSha256":"0a819976ac943e9eedf5bc7f9c80880dd9103cb62bb4c5a915b98c452b135a5c","profile":"september-2026-classroom","reason":"Available.","selectable":true}} |
| preferred_material | PZT-5H |
| schema_version | 0.1.0 |
| stateVariables | [{"adjointDescription":"Coupled piezoelectric adjoint solve supports mechanical displacement contributions to shape, topology, and expression objectives.","adjointSupported":true,"availability":{"topologyMethods":[]},"canonicalSymbol":"pze.u","complexValued":false,"componentDescriptions":["X-direction displacement","Y-direction displacement","Z-direction displacement"],"componentNames":["ux","uy","uz"],"componentSymbols":["pze.ux","pze.uy","pze.uz"],"components":3,"description":"Mechanical di... |
| studies | [{"certified":true,"differentiability":"exact-ad","kind":"stationary","reason":"september-2026-classroom verified contract: constitutive Jacobian finite-difference agreement and linear displacement scaling with applied voltage"},{"certified":true,"differentiability":"exact-ad","kind":"eigenfrequency","reason":"september-2026-classroom verified contract: coupled mechanical/electrical residuals and monotone spatial refinement"},{"certified":false,"differentiability":"exact-ad","kind":"eigenvalu... |
| transient_runtime | {"capacity_properties":["rho"],"certification":"PiezoelectricDynamicBdfRuntime","checkpointing":true,"design_properties":["cE","e","epsS","rho"],"design_property_conditions":{},"design_vjp_owner":"PiezoelectricTransientTopologyRuntime","expression_objectives":true,"gradient_channels":["coordinates","material","input","initial"],"gradient_vjp_owner":"PiezoelectricTransientTopologyRuntime","history_depth":2,"input_gradient":true,"input_vjp_owner":"PiezoelectricTransientTopologyRuntime","observa... |
| weak_contributions | [] |
| category | Structural Mechanics / AC/DC |
| description | Coupled linear piezoelectricity. |
| dimensions | [3] |
| display_name | Piezoelectricity |
| kind | Piezoelectricity |
| namespace | pze |
| status | certified |
| visible | true |