SHORTEST WAYS FINDING IN DISCRETE BODIES

Transcription

1 SHORTEST WAYS FINDING IN DISCRETE BODIES Milan CHLADA*, Zdenek PREVOROVSKY Impact and Waves in Solids, Institute of Thermomechanics AS CR, v. v. i.; Prague, Czech Republic *Phone: , Fa: ; NDT in Progress 2015

2 SHORTEST WAYS FINDING IN DISCRETE BODIES POSTER PRESENTATION CONTENTS Definition of the problem Review of possible approaches - differential geometry solution (finding of geodesics on simple and combined surfaces) - graph theory solution (if there eists an interpretation of a body as a graph) New flooding algorithm inspired by Huygens' principle - flooding propagation scheme - a hole over-jumping problem - discrete distance computation Results on samples of discretized real bodies NDT in Progress 2015

3 Definition of the problem The shortest way estimation in bodies - eample on a real aircraft structure part (elastic wave tracing while propagating from point A to B) A? distance estimate B The choice of most suitable method - depends largely on the kind or form of the body shape definition

4 Differential geometry solution Finding geodesics on simple and combined surfaces - the theory of differential geometry provides necessary apparatus for case of mathematically well-describable surfaces. The theory of differential geometry provides apparatus describing geodesics - the system of geodesic equations (It supposes that the surfaces are mathematically described as a conjunction of several simple parametric shapes) source: M.Záveský presentation on EWGAE 2012

5 Graph theory solution The graph theory is capable to solve the problem when there eists an interpretation of a body as a system of mass points (nodes) and their interconnections (edges), i.e. a graph. Shortest ways estimate: the shortest path between two graph vertices (u,v) of a graph. (there may be more than one different shortest paths, all of the same length) Methods: - Breadth-first traversal - Dijkstra's algorithm source: mathworld.wolfram.com

6 Graph theory solution - too complicated for higher number of edges - problem of construction of an adequate graph (holes in material etc.) Horizontal and vertical directions only More general directions allowed B d d d y A d

7 Huygens' principle (model of wave propagation) - Each point on an advancing wave front may be considered to be a new point source generating spherical Huygens' wavelets. - If one knows the location of a wave front at one time, one can determine the location of the wave front at a slightly later time by drawing spherical arcs centered at points along the wave front. Each spherical arc is a Huygens wavelet. (the radius of each arc is given by the distance that sound travels in that small amount of time) SPHERICAL WAVEFRONTS PLANE WAVEFRONTS secondary sources initial source t0 wave propagation tn-1 tn tn-2 tn-1 tn tn+1 tn+1 source:

8 Flooding propagation Another recently designed method was inspired by Huygens' principle as a simple to implement alternative suitable for discrete bodies derived from 2D or 3D bitmap pictures. Such approach also enables the tracing of elastic waves propagating through bodies of complicated shape. Body piels Already reached piels Actual piels (wavefront points) Potencially net piels Net piels Pattern of level 3 Out-of-mass piels Default piel

9 A hole over-jumping problem To avoid over-jumping of a material hole, it is needed to stop propagation in a space angle behind each reached piel out of mass.!! Pattern of level 3!!! Body piels Already reached piels Actual piels (wavefront points) Potencially net piels Net piels Overjumping problem Out-of-mass piels Default piel

10 Eamples of forbidden pattern piels It is needed to eclude all points having with some tolerance similar angular coordinate and simultaneously higher radial coordinate than all out-ofmass points. 8 Level 7 (diameter 14 piels) 1 5 Level 13 (diameter 26 piels) piel coordinates piel coordinates piel coordinates piel coordinates Default piel Out-of-mass piels Forbidden piels Allowed piels

11 Discrete distance computation?? +p -1,3 +p 0,3 +p -2,2 +p -1,2 +p 0,2 +p -1,1 +p 0,1 +p 1,2 +p 1,1 +p 2,1 Body piels Already reached piels??? d +p 1,0 +p 2,0 +p 3,0 Actual piels (wavefront points) Direction of propagation??????? +p 1,-1???? Pattern of level 3 +p 2,-1 +p 3,-1 +p 2,-2 d Potencially net piels Net piels Out-of-mass piels Default piel p i, j = i 2 + j 2 where i and j are from the adapted range of pattern?: if (d+p i,j )<actual value in piel [i,j], then overwrite it by (d+p i,j )

12 Algorithm steps INITIATION: - set zero distance to all mass piels - set initial propagation piel(s) as actual, i.e. wave source point PROPAGATION: - for each actual piels find potential net piels (laying inside arc pattern) find piels to forbid ( out-of-mass or overjumping problem points) set propriate new distance number to all allowed piels of arc pattern set net piels as actual Repeat PROPAGATION until all mass piels are reached...

13 Eamples of results Typical map of distances from one selected AE sensor (point B) to all mass piels. piels A distance: 583 piels distance [piel width] B piels

14 Eamples of results Separation of a structure by first arrival of elastic wave to each AE sensor Y coordinate [piels] S 9 9 S S 3 S 2 S 5 S 10 S distance to sensor [piels] 450 S X coordinate [piels] 0

15 Eamples of results Segmentation of the body by unique chronology of arrivals of elastic waves to sensors. AE SENSORS piels eisting chronologies 6 piels

16 MOVIE SAMPLE...

17 ZONAL LOCATION Badly interpretable results of ANN location (points out of the body). An alternative: algorithm of segment (zonal) location (structure is divided up by the chronology of signal arrivals to sensors) For geometrically sparse structures it is rather difficult to specify particular segments with unique chronology of arrivals of elastic waves to sensors (It is needed to use an algorithm for finding the shortest ways in the structure). Analysis of all possible elastic wave arrival chronologies in the structure: (number of sensor permutations: 13!= ,, there out 235 possible). SEGMENTATION (by first 13 arrivals to sensors)

18 ZONAL LOCALIZATION SEGMENTATION (by first 5 arrivals to sensors) IDENTICAL CHRONOLOGIES SEGMENTATION (by first 3 arrivals to sensors) IDENTICAL CHRONOLOGIES

19 MOVIE SAMPLE...

20 THANK YOU FOR ATTENTION...