BoundaryDiscretizeRegion Plots does not work with ParametricPlot3D

Question

I am trying wit this data

b = 0.042; a = 0.1075;
rh1 = 0.071553571428571;
rh2 = 0.077946428571429;
θ1 = -(0.085497114090722/2);
θ2 = (0.085497114090722)/2;
γ0 = 27.5925876895488499940256588160991668701171875`50.; \
γ1 = 27.6561421691630044961129897274076938629150390625`50.;
γ2 = 29.526641831934380633128967019729316234588623046875`50.;
γ3 = 35.430661411534771332298987545073032379150390625`50.;
γ4 = 44.104585229662689016549848020076751708984375`50.;
γ5 = 53.77374161874728741850049118511378765106201171875`50.;
B0 = -0.258727433320186828158426806112402118742465972900390625`50.;
C0 = -0.087231063588538659825388776880572549998760223388671875`50.;
D0 = -1.54343899231474868116720244870521128177642822265625`50.;
B1 = 10.8721158771190697933661795104853808879852294921875`50.;
C1 = 1.0083854268623009264871370760374702513217926025390625`50.;
D1 = 12.755787693798620097140883444808423519134521484375`50.;
B2 = 0.6119179333464430659006438872893340885639190673828125`50.;
C2 = 0.11419103062758979849622420488231000490486621856689453125`50.;
D2 = 0.56721637438979011225370641113840974867343902587890625`50.;
B3 = 0.1702199985379937718921183886777726002037525177001953125`50.;
C3 = 0.07329541454198429395461289459490217268466949462890625`50.;
D3 = 0.1560232223658888361939034439274109899997711181640625`50.;
B4 = 0.0526358651260851184705558125642710365355014801025390625`50.;
C4 = 0.04176822369822615066414783768777851946651935577392578125`50.;
D4 = 0.051253182800385883866045588774795760400593280792236328125`50.;
B5 = 0.01679753231958447390326227832701988518238067626953125`50.;
C5 = 0.022207085726827675842276477169434656389057636260986328125`50.;
D5 = 0.017742295388893665475382732665821094997227191925048828125`50.;
H1 = {{BesselJ[0, γ0*r] + B0*BesselY[0, γ0*r] + 
     C0*BesselI[0, γ0*r] + 
     D0*BesselK[0, γ0*r]}, {BesselJ[1, γ1*r] + 
      B1*BesselY[1, γ1*r] + C1*BesselI[1, γ1*r] + 
      D1*BesselK[1, γ1*r]}*(Cos[θ]), {BesselJ[
       2, γ2*r] + B2*BesselY[2, γ2*r] + 
      C2*BesselI[2, γ2*r] + D2*BesselK[2, γ2*r]}*(Cos[
      2*θ]), {BesselJ[3, γ3*r] + 
      B3*BesselY[3, γ3*r] + C3*BesselI[3, γ3*r] + 
      D3*BesselK[3, γ3*r]}*(Cos[
      3*θ]), {BesselJ[4, γ4*r] + 
      B4*BesselY[4, γ4*r] + C4*BesselI[4, γ4*r] + 
      D4*BesselK[4, γ4*r]}*(Cos[
      4*θ]), {BesselJ[5, γ5*r] + 
      B5*BesselY[5, γ5*r] + C5*BesselI[5, γ5*r] + 
      D5*BesselK[5, γ5*r]}*(Cos[5*θ])};
H2 = {{BesselJ[1, γ1*r] + B1*BesselY[1, γ1*r] + 
      C1*BesselI[1, γ1*r] + 
      D1*BesselK[1, γ1*r]}*(Sin[θ]), {BesselJ[
       2, γ2*r] + B2*BesselY[2, γ2*r] + 
      C2*BesselI[2, γ2*r] + D2*BesselK[2, γ2*r]}*(Sin[
      2*θ]), {BesselJ[3, γ3*r] + 
      B3*BesselY[3, γ3*r] + C3*BesselI[3, γ3*r] + 
      D3*BesselK[3, γ3*r]}*(Sin[
      3*θ]), {BesselJ[4, γ4*r] + 
      B4*BesselY[4, γ4*r] + C4*BesselI[4, γ4*r] + 
      D4*BesselK[4, γ4*r]}*(Sin[
      4*θ]), {BesselJ[5, γ5*r] + 
      B5*BesselY[5, γ5*r] + C5*BesselI[5, γ5*r] + 
      D5*BesselK[5, γ5*r]}*(Sin[5*θ])};
H = Join[H1, H2];
Eigvectors = \
{{-0.00032910702592332067748330550715383735242784158137713845786365866\
618131449249`50., 
    0.0000619394300120392780844012821989786217898198231051515592104294\
1747885144002`50., 
    0.0005547827300665782271433910475035989813436308122253333513903714\
1848307835336`50., 
    0.0005231988120040556552360013802091916709121452211867352600736746\
8527124122845`50., 
    0.0001838703756141756214053904725815494830601516198880360538176908\
4779577600227`50., 1.`50., 
    2.535318998719537730027615020797307965593938101494039361034438`50.\
*^-17, -2.\
1199319771545881344744738949843672854921570996787640418875656`50.*^-\
16, 1.34543416580296032974037527983559183583191075512792285969355953`\
50.*^-15, 
    1.45414196658736704189109420869275005922675219240791977060328612`\
50.*^-15, \
-2.917097488216128637398507230661172533566547614650862244192105116`50.\
*^-14}, {4.\
1393906816958560506978253749717613270207895244841348699741100140013322\
45713633`50.*^-18, \
-9.14021353961675242207947861212444698545448700280830102418224`50.*^-\
18, -2.909029886480655906567397243598961616405208425684555840862853`\
50.*^-16, 
    9.3614156404083407742753906079531560632438588700394338002564902`\
50.*^-16, 
    1.4128637809158172162529083345653765060314564044217952827602257`\
50.*^-15, 
    2.90765574314389495791231923027726231261113923608199467872172016`\
50.*^-14, \
-0.0000102487513305948084431870771189154082388487279146178514491923798\
1836808159`50., \
-0.0002240825961280424559068758588142077514355999816542655770043371938\
7236316922`50., \
-0.0005143836872790891355266039041046145928385094397241827389040327182\
2346420488`50., \
-0.0012074595447196376182807727974712165509603740267021568252014645476\
4359518489`50., 
    1.`50.}, \
{0.0004608033513713831258375485171807049504541716271843716959516879997\
7954873463`50., \
-0.0000856174018654443306072450212002276545270315224574867444593157249\
0976293488`50., \
-0.0007497256113880201907096291212615010208187874984455863008128208092\
33944644`50., \
-0.0007045369820657110223994646630313088602108232772246448816068537928\
9026644997`50., -1.`50., \
-0.0007292709021946642586591303431807983205254318563468210388878191979\
3230329422`50., 
    1.612465340701469773140111230767500956284428913412180015912364`50.\
*^-17, 2.2718176145494184590612295330354512566068520834977601089972492\
`50.*^-16, 
    1.53374998822348400734702009686123657274307302110011805725316381`\
50.*^-15, \
-6.6476932216263309515467886755298524111379493050733579115455005892`\
50.*^-13, \
-3.4210817258822349903667884338340901589397780953005271635150603`50.*^\
-16}, {3.9215110591163238921985484848432872986726068018374286243968374\
`50.*^-16, \
-7.060293531811999168219332314229657629906880338977727849738489`50.*^-\
17, -4.2203271327201239511388284905277631283833695511007867566252656`\
50.*^-16, \
-1.10492079661353270253589364762728014029228771216039031875811473`50.*^\
-15, -6.64447150372026487128378926830272110962921937953808918595995809\
56`50.*^-13, \
-1.1039589080359062320096115641314423281295291969769716184211785`50.*^\
-15, -0.00002193153145134803755895737369104039031458486962738630602584\
085765083799767`50., \
-0.0004952446540505775859824195160651590958951137819892825972857427871\
1851131332`50., \
-0.0013867066231735361951413009403097147160753784730872362437262964511\
962098162`50., 1.`50., 
    0.0015242107960386576553082305181457021160941979625148388414183400\
1581319880917`50.}, \
{-0.000721695041384639970379347844909717422462164250717250107153752199\
92263667019`50., 
    0.0001341141851614028158479228936924702597153616176660938614165098\
2407034412274`50., 
    0.0013214363877639745778778738053876766822419495682974748092042064\
8012566633748`50., 1.`50., 
    0.0002029831726643346866436351146244634360890031726167254591278677\
6861797776094`50., 
    0.0002977781820516691385431807618954168637544970611384250279080029\
5039049646433`50., 
    1.4344546788155352577996314770057287699130055037444734390653782420\
93035081086896`50.*^-18, \
-5.0168005290211820231440331061843788947295286084580853880737910939370\
33563677789`50.*^-18, 
    1.297712171501468983410624722875930248222442436629399653403748406`\
50.*^-14, 
    4.8581438122295662451507703426348866793651927344204790733055165`\
50.*^-16, \
-4.0552917052426496603290162438647799902278269979897741182127621`50.*^\
-16}, {-8.\
235323830670512169881118021458569738049510303663170791226304`50.*^-17,
     9.23655360963475521664004862478283204162384908333810889082003`50.\
*^-18, -7.\
872020760977684814881212429986455457253066516284795301803299`50.*^-17,\
 -1.296154100788077082551258188600238507468231933102631878490866991`\
50.*^-14, 
    2.4385168962471148278190479611897767936639927505803644059180493`\
50.*^-16, \
-2.0450432993050091825511873734756232069796967020622757330467207`50.*^\
-16, -0.00006122399474276091388487955110005694087188105331930580248885\
845825566637149`50., \
-0.0015861652476044066717707566467004430446023792304980359386936773699\
7786889289`50., 1.`50., 
    0.0018686818872298847181351525985422925101326444010139838017798752\
0983537635227`50., 
    0.0008791955018964318404070731849445697041615663396599332953553779\
5866495493148`50.}, \
{-0.001580744502339414896539179840369942320380239654112722162971839684\
154962524`50., 
    0.0003289875243886837776528933204461343854758217073002874974616918\
7514762146841`50., 
    1.`50., -0.\
0009728175599309765319619183142830433297403137404732184238142773216224\
4252881`50., \
-0.0002522599989090151053284432215487695457841690756867243606866855825\
535958794`50., \
-0.0000785975687305026448545819167358022930218343550901059664321527559\
3472854784`50., \
-2.9859160441888262092813465351342666016122408696689304854323547603282\
08137509329`50.*^-18, \
-5.116058689661065049487818532294625230857992745437944041028724491`50.\
*^-14, -1.\
1755737298383730580520312731546830494148662411819392897719667`50.*^-\
16, -1.2655011041710821870181758690178517933359278620149434248094979`\
50.*^-16, 
    6.640564452018317633907939020558928290820710031845098527571613`50.\
*^-17}, {-1.\
1057830913544361669733951539486951152289493830247499150239963`50.*^-\
16, -2.363081701143659722879942836864166518743190830905134202347195`\
50.*^-17, \
-5.108359693912843973911294522726808232669380070122321435568296463`50.\
*^-14, 2.434711327966822362738131425038585790958945426419939438167281`\
50.*^-17, 
    6.549004864251973982474258354586105422930739954128607600271419`50.\
*^-17, -1.\
1366687348301205975623936541232952034656131011204965921899129`50.*^-\
16, 0.0003198544659064150541532486941729514891847165226407919351488775\
9515285897949`50., -1.`50., \
-0.0029515001275693582121083534832353832974200229953610048870468754845\
2592675859`50., \
-0.0012494216883185347043268372138726866888759617328994661540689855925\
6588513197`50., \
-0.0007163867701357530554207893962970338177896660067841435430743099698\
060169262`50.}, \
{-0.001687689286197699785669067051406701055823325566989708591650799350\
28546454916`50., -1.`50., 
    0.0261069627679070011954391841099060896056894575306742869798623776\
4444362989925`50., 
    0.0128124775781892929209661336591617074048585937965968486882684081\
890850667266`50., 
    0.0072183327550258495931317314872393677087586332960887847130624619\
1648616608986`50., 
    0.0045179985220927684774607925446066132752416709361615753462512216\
825761743851`50., \
-5.962728273654544734495044158633264287143624080341275776310369434`50.\
*^-14, -1.\
22894867235371551095475331666109739751325961807560023423961793`50.*^-\
15, 1.004785594164117355971128123493380117697773246297865098442237`50.\
*^-17, -1.\
13919807651009683115123456760653707110054408494576589209130413`50.*^-\
15, -1.2079747110409626402292720978386539116276001400584523234139492`\
50.*^-16}, \
{1.808526475002847594801973463269620688084868519645429305669613339`50.\
*^-14, -5.\
969185435205016837823427245246054469633834080993506438006781939`50.*^-\
14, 3.08805526287246500425428402264016519031368661706228586545755311`\
50.*^-15, 
    2.0586426135603362137833902198326297647002956589535961785063012`\
50.*^-16, 
    7.9307944168456240437402614205078006959588119302476261811764927`\
50.*^-16, \
-3.6782662894364863892475509209929013190151919350085249236678966`50.*^\
-16, 1.`50., 
    0.0374050766055095015288445953609994506168469420062098350710965886\
1876610146802`50., 
    0.0134600109022510329650372430591675355512145691096044172006040052\
9863454521428`50., 
    0.0065283257970991131672355961829056296484904283757364644938075274\
1590137059805`50., 
    0.0038649320987731052336756224552645495376819790980291023724402847\
9247623006113`50.}, {1.`50., \
-0.0001484441797251027373835513255741263776559023318831909023630893804\
590932471`50., 
    0.0024592397194208468772068156707164262284976757372885669910269049\
2606679975748`50., 
    0.0015005312438479287455417456039735845815224520313779003073659163\
1202244087766`50., 
    0.0009088942139718065447307708431438307251551632444673489694336995\
7998167921769`50., 
    0.0005905464817700233861262550023893767570233648988605261846297219\
8980340796028`50., \
-3.690391194295661686797560288286279485225137860116469033333616`50.*^-\
16, -4.0386376430758020245668911832126660425875450686723454203692188`\
50.*^-16, 
    1.2172850312878280837979078959830368288265953286580952089571358`\
50.*^-16, \
-8.608407088469488809355424962462519273138095681824263875753515`50.*^-\
17, -2.281642952879382399672649393634184780094337182121603198326391`\
50.*^-17}};
Dcomplete = Annulus[{0, 0}, {b, a}];
Dhole = 
  Annulus[{0, 0}, {rh1, rh2}, {θ1, θ2}];
D = 
  RegionDifference[Dcomplete, Dhole];
BoundaryDiscretizeRegion[D, PrecisionGoal -> 5]

which produces this discretized region

I want to produce a ParametricPlot3D like this

but with the hole created in the region. I already tried with this

ParametricPlot3D[{r Sin[θ], r Cos[θ], 
  Eigvectors[[All, 1]].H[[All, 1]]}, {θ, 
   r} ∈ D, 
 Mesh -> {Range[0, 2 Pi, 2 Pi/50], 20}, Boxed -> False, Axes -> False,
  BoxRatios -> 1, BoundaryStyle -> Black]

which produces this

it is clear that it does not have resolution, something wrong it is going; even increasing the values of PrecisionGoal and including AccuracyGoal in the region.

Answer 1

You could use a Finite Element Mesh which seems to track the boundaries better. Note that I prepend the "D" terms with an "S" because "D" is usually used for differentiation.

Needs["NDSolve`FEM`"]
SDcomplete = Annulus[{0, 0}, {b, a}];
SDhole = Annulus[{0, 0}, {rh1, rh2}, {\[Theta]1, \[Theta]2}];
SD = RegionDifference[SDcomplete, SDhole];
m = ToElementMesh[SD, 
  "RegionHoles" -> {{0, 0}, Mean@{rh1, rh2} {1, 0}}]
ParametricPlot3D[{x, 
  y, (Eigvectors[[All, 1]].H[[All, 1]]) /. {r -> Sqrt[
     x^2 + y^2], \[Theta] -> ArcTan[x, y]}}, {x, y} \[Element] m, 
 Mesh -> {Range[0, 2 Pi, 2 Pi/50], 20}, Boxed -> False, Axes -> False,
  BoxRatios -> 1, BoundaryStyle -> Black]

With this approach, you will need to use MeshFunctions to get the desired mesh lines.

ParametricPlot3D[{x, 
  y, (Eigvectors[[All, 1]].H[[All, 1]]) /. {r -> Sqrt[
     x^2 + y^2], \[Theta] -> ArcTan[x, y]}}, {x, y} \[Element] m, 
 Mesh -> {Range[b, a, (a - b)/10], Range[0, 2 Pi, 2 Pi/50]}, 
 MeshFunctions -> {Function[{x, y, z, \[Phi], \[Theta]}, Sqrt[
    x^2 + y^2]], 
   Function[{x, y, z, \[Phi], \[Theta]}, 
    PlanarAngle[{0, 0} -> {{-1, 1}, {x, y}}, "Counterclockwise"]]}, 
 Boxed -> False, Axes -> False, BoxRatios -> 1, 
 BoundaryStyle -> Black, PlotRange -> All]