Peridynamic Field Recovery
Design the strain-gauge layout before the test — recover the full strain, displacement and stress fields, and the elastic constants, after it.
About
Peridynamic Field Recovery (PDRecovery) reconstructs the complete mechanical state of a structural component from strain measured at a small number of discrete points. Strain gauges, fibre-optic sensors and rosettes report a few dozen readings; PDRecovery turns them into continuous strain, displacement and stress fields over the whole part, including around the holes, notches and slits where the gradients matter most.
The engine is the Peridynamic Differential Operator (PDDO) — a non-local, mesh-free framework that integrates over a finite horizon instead of differencing neighbouring points. Because the domain is defined analytically and every sensor family is filtered by line of sight, no data is blended across a cut-out. The result needs no finite element model of the part, no material properties as input, and no expert interpolation of the gauge readings.
PDRecovery now holds two tools that share one model of the part. Sensor Placement runs before the test and answers how many rosettes the test needs and where they go. Field Recovery runs after it and turns the readings into fields. Geometry can be drawn from primitives and polygons or imported straight from a CAD drawing.
Field Recovery, steps 1–4. Domain, peridynamic discretization and the measured sensors, each attached to its nearest point.
Two Tools, One Model
A single project holds the domain, the peridynamic discretization and the boundary conditions. Both tools read them, so the layout you design is optimised on exactly the part you later measure, and the switch between them is one click in the workspace bar.
Sensor Placement — before the test. How many strain rosettes are needed, and where. A linear-elastic plane FE model of the part is solved for a whole family of loads the article may see, and layouts are optimised so that the PDDO reconstruction of those fields is as accurate as possible. The output is an error-against-sensor-count curve, a recommended number of sensors, and a sensor plan to take into the laboratory.
Field Recovery — after the test. The measured strains are imported, the full strain field is recovered at every peridynamic point, the displacement field is constructed by peridynamic least squares, and the elastic constants are characterised by the virtual fields method. Stresses, von Mises, deformed shape and line profiles follow.
Geometry and CAD Import
The domain is built from analytic primitives — rotatable rectangles, circles, ellipses and polygons with any number of vertices — each one a part that adds material or a cut-out that removes it. Polygons are drawn vertex by vertex, edited with a handle per vertex, given new vertices from the midpoint handles, and listed as an editable coordinate table. Sides that would cross are refused. Any shape converts to a polygon, with curves becoming 64 segments.
DXF import brings a 2D CAD drawing in directly: lines, arcs, circles, ellipses, splines, polylines with bulges, and block references. Open curves are joined end to end, curves become segments within a chord tolerance you set, and full circles and ellipses stay exact. Closed outlines are nested automatically — an outline inside an odd number of others becomes a cut-out, so a plate with holes imports as one part and its holes in a single step.
The import panel lists every outline with its role, type, vertex count, area and source layer, shows a canvas preview, reads the drawing unit from the file and scales it to the project unit. Grid and object snapping, spreadsheet-style geometry tables, CSV import and export, live diagnostics for net area, boundary length, regions and holes, and unlimited undo and redo apply throughout.
Step 1. The domain from analytic primitives as parts and cut-outs, with live diagnostics for net area, boundary length, regions and holes.
Sensor Placement — Before the Test
Gauges are expensive, channels are limited, and a layout chosen by intuition wastes both. Sensor Placement turns the question into an optimisation. You mark which boundary edges are supported and which may carry load, then define a load model: normal and shear tractions on every loadable edge, varying along the edge as polynomials up to a chosen degree, from uniform tension through bending to higher orders. Each load is scaled to unit RMS strain, so none dominates.
A plane finite element model of the domain is solved once for the whole family. Every trial layout is then scored with the same PDDO strain recovery that Field Recovery will use after the test: the error is the RMS difference between the recovered and the FE field, per component, over all design loads, with a noise term that penalises ill-conditioned layouts. Candidate positions respect an edge margin for the gauge footprint, a minimum spacing, and exclusion zones at the clamped corners where the strain is singular.
Layouts are optimised across a range of sensor counts. An exchange search moves each sensor to the best of the error hot spots, its neighbours and random trials until nothing improves; a POD and pivoted-QR surrogate places sensors in under a second; and Gauss-Newton polishing lets the sensors leave the grid and take continuous coordinates. Every layout is also scored against held-out loads it was never designed for, and against a space-filling baseline, so an over-fitted layout shows itself.
Optimised placement. Reconstruction error on the plate with a circular hole. The published 72-sensor layout leaves 48.9 % relative error; the optimised 72-sensor layout leaves 9.9 %, and an optimised layout of just 50 sensors leaves 13.0 % — better accuracy from 30 % fewer gauges.
How many sensors? Error against sensor count, with the knee of the curve and the target-error line giving the recommended number. Greedy nested layouts are shown for comparison.
What You Take to the Lab
- An error-against-sensor-count curve for your part and your load cases
- A recommended sensor count: the knee of the curve, or the smallest count that meets a target error you set
- The optimised coordinates of every sensor, listed and drawn on the part
- The error on held-out loads and on a space-filling layout, as an honest check
- The noise amplification of the layout, at the sensor noise level you specify
- A sensor plan exported to CSV, with empty strain columns to fill in after the test and import straight back into Field Recovery
- Optimisation history per sensor count, so you can see when the search converged
- A preview of the FE strain field of any single load in the model
How Field Recovery Works
The part is discretized into peridynamic points carrying cell areas. Each sensor is attached to its nearest point and receives a lumped area from inverse-distance shares of the surrounding cells. The zeroth-order PDDO then estimates all three strain components at every point from the nearest sensors in line of sight, using a quadratic basis with Gaussian weights. Families whose sensors happen to lie on a single ring or curve are enlarged automatically until the estimator is well conditioned, so the reconstruction stays stable wherever the sensors happen to fall.
Displacements follow from a least-squares minimisation: the field whose peridynamic strains best match the measured values at the sensors and the recovered values everywhere else. Since nonlocal derivatives average over a family of neighbours, they barely see a displacement that alternates from point to point, and those near-zero-energy modes would otherwise be excited by measurement noise. A dedicated mode control penalises exactly what the peridynamic family cannot represent — and nothing it can — so the reconstruction comes out smooth without being smoothed.
Step 5. Normal strain εxx recovered at 9,516 points from 68 sensors — the concentration at the hole resolves cleanly.
Step 5. Shear strain γxy over the same domain — the cut-out is never blended across.
Analysis Pipeline
Shared model. Both tools begin from the same three steps.
- Domain: rectangles, circles, ellipses and polygons as parts and cut-outs, drawn or imported from DXF
- Discretization: structured grid, quadtree refinement, body-fitted rings around holes, scattered points, or imported points
- Boundary & material: supported, loadable and free edges, plus the nominal isotropic material and plane hypothesis
Sensor Placement continues with three steps of its own.
- Load model: tractions on the loadable edges, to the polynomial degree the test may produce
- Candidates: where sensors may go — edge margin, minimum spacing, excluded corner zones
- Optimise: layouts across a range of sensor counts, the error curve, and the exported sensor plan
Field Recovery continues with four.
- Sensors & data: import measured strains from CSV or text, in strain, µε or %
- Strain recovery: zeroth-order PDDO with lumped sensor areas and line-of-sight families
- Displacement: peridynamic least-squares minimisation with zero-energy-mode control
- Material characterisation: E and ν by the virtual fields method, then stresses, von Mises, deformed shape, line profiles and export of every field
Step 6. The deformed shape at automatic magnification, coloured by |u| — reconstructed from strain data alone, with no model of the part.
Features
- Two workspaces on one model: Sensor Placement before the test, Field Recovery after it
- Step-by-step graphical user interface with an interactive result canvas
- Works with any sensor layout — no regular grid or minimum count required
- Analytic geometry: rotatable rectangles, circles, ellipses and arbitrary polygons, as parts or cut-outs
- DXF import of 2D CAD drawings, with automatic nesting of outlines into parts and cut-outs
- Grid, quadtree-refined, body-fitted ring, scattered and imported discretizations
- Boundary conditions per edge, surviving moves, resizes and rotations of the shape
- Built-in plane FE solver for the load model, meshed automatically from the domain
- Automatic unit and shear-convention handling (strain, µε, %, tensor or engineering)
- Native C++/OpenMP solver with a NumPy reference implementation as cross-check
- Multi-process layout optimisation with live progress and a stop button
- Interactive pan and zoom, hover read-out of every point's values
- Deformed-shape view with automatic magnification
- Unlimited undo and redo with a history panel, light and dark themes
- Project files that store geometry, points, boundary conditions, sensors, layouts and every result
- Export of the sensor plan, points, strains, displacements and stresses to CSV
- Open Python platform — scriptable, with standalone command-line solvers
Step 6. Horizontal displacement ux, constructed from the strain field by least squares.
Capabilities
- Optimised sensor layouts and a defensible sensor count, before a single gauge is bonded
- Full strain field εxx, εyy, γxy from sparse point measurements
- Continuous displacement field without a finite element model of the part
- Stresses σxx, σyy, τxy and von Mises from the identified law
- Young's modulus E and Poisson's ratio ν from the measured load resultant alone
- Geometry straight from CAD, with polygons for any outline a drawing does not describe
- Geometry-aware families that never reach through a hole, notch or slit
- Zero-energy-mode control for smooth, noise-robust displacement fields
- Body-fitted ring meshes around circular and elliptical cut-outs
- Diagnostics: family size, horizon and noise amplification at every point
- Consistency check that flags an unreliable reconstruction without a reference
- Export of every field for downstream fatigue, fracture or certification work
Material Characterisation — the Virtual Fields Method
With the strain field reconstructed, the elastic constants follow from the principle of virtual work. For every admissible virtual field — zero on the supported edges, where the reactions are unknown, and rigid on the loaded edge, so that only the measured resultant does work — the internal and external virtual work must balance. Each field gives one linear equation in the plane stiffnesses. Nothing but the reconstructed strains, the measured load, the thickness and the plane hypothesis enters the calculation; no reference specimen and no iteration over a finite element model are needed.
Because the virtual strains are PDDO derivatives taken with the same operator that differentiates the reconstruction, the quadratic virtual fields are differentiated exactly. In polynomial mode, six admissible fields are solved by least squares; they would agree perfectly on a field in equilibrium, so their spread is reported as a consistency measure — a warning that the reconstruction is unreliable, available without any reference solution to compare against. In optimised mode, piecewise-bilinear fields on a coarse virtual mesh are built so that each parameter is recovered with the least possible sensitivity to strain noise; the panel reports that sensitivity η directly, and the mode works with any supports that do not touch the loaded edge.
Both plane stress and plane strain are supported, the strains may be taken from the reconstructed displacement field or from the recovered strains, and the identified law immediately gives σxx, σyy, τxy and von Mises stress at every peridynamic point.
Step 7. Von Mises stress from the characterised constants. Here E = 0.9995 and ν = 0.3306 against true values of 1 and 1/3, with the six virtual fields agreeing within 0.61 %.
Step 2. Peridynamic points coloured by cell area; cells are trimmed exactly to the curved cut-out.
Validated Accuracy
The reconstruction is validated against finite element reference solutions for a square plate under tension with a circular hole, an inclined elliptical hole, and a slit. Errors below are the relative RMS over every peridynamic point, from 42 to 82 sensors on grids of 9,500 to 9,900 points.
- Circular cut-out, 68 sensors: strains 2.9 to 6.2 %, displacements 0.8 and 3.8 %
- Elliptical cut-out, 82 sensors: strains 3.6 to 5.6 %, displacements 1.1 and 2.1 %
- Slit, 42 sensors: strains 5.1 to 8.0 %, displacements 4.5 and 4.3 %
- Characterised constants: E within 0.1 to 8.2 % and ν within 0.8 to 12.2 % of the true values
- Native and reference solvers agree to machine precision on every case
Sensor Placement is validated on the same plate with a circular hole, scoring every layout with the identical reconstruction and the identical error measure.
- Published 72-sensor layout: 48.9 % relative L2 strain error
- Optimised 72-sensor layout: 9.9 % — five times more accurate for the same gauge count
- Optimised 50-sensor layout: 13.0 % — more accurate than either reference layout with 30 % fewer gauges
- Near the hole, the optimised 72-sensor layout halves the error of a hand-made layout of comparable size
- Layouts are re-scored on held-out loads they were never designed for, and against a space-filling baseline
Layouts. Space-filling and optimised layouts for the circle, ellipse and slit, on the equivalent-strain field. Optimised layouts crowd the gradients without abandoning the far field, whether or not the load is known in advance.
Who Is It For
- Test engineers — plan the instrumentation, then recover the fields between the gauges on a component or full-scale test article
- Structural health monitoring — site a permanent gauge array where it earns its cost, and turn it into a full-field picture
- Certification and airworthiness — evidence of strain and stress where no gauge could be placed, from a defensible layout
- Materials characterisation — identify elastic constants from a single heterogeneous test
- Digital twins — feed a measured, continuous field into a model instead of point readings
- Fatigue and fracture assessment — measured stress fields at the critical detail
Documentation
Each tool has its own manual, because they serve different jobs at different points in a test campaign.
- Sensor Placement User Manual — the load model, candidate positions, the search methods, reading the error curve, and exporting the sensor plan
- Field Recovery User Manual — importing measured strains, strain recovery, displacement reconstruction, material characterisation, and the result tools
- Validation case: plate with a cut-out — the full comparison against finite element reference solutions
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