3D Doublet Panel Method

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1 3D Doublet Pal Method V

2 An Issue With Doublet Pals Conside a ut though the body sufae no flow Pals appea as a ies of dite voties with stengths equal to the diffee between adjaent pal stengths. At the ontol points we e alost no of the tangential veloity geated by aby voties. This is wong. It is as though the sufae wee a ontinuous votex sheet, and we alulated the tangential veloity at the.p. ignoing the dintinuous jup between the inside and outside. This jup, equal to half the loal sheet stength is alled the pinipal value and ust be added to the tangent veloity alulated at a ontol point V=V tang V=0 V=V tang V<V tang V<<V tang This ust be added to the tangential veloity alulated at any ontol point The pinipal value fo a doublet pal an be estiated as -½ loal gadient of the pal stength on the sufae.

3 3D Doublet Pal Code Non-lifting bodies Handling vetos in Matlab Speifying a 3D body Speifying Pal Geoety Pal Influe lving fo the pal stengths Getting the sufae pessue

4 %3D doublet pal ethod fo ayli flow aound 3D bodies. lea all; vinf=[1;0;0]; %fee stea veloity %Speify body of evolution geoety th=-pi/2:pi/20:pi/2;xp=sin(th;yp=os(th;n=20; [,,,,,,no,we,,ea]=bodyofrevolution(xp,yp,n; % detei sufae aea and noal vetos at ontol points (assues ounte lokwi aound o a=0.5*v_oss((:,-(:,,(:,-(:,;n=a./v_ag(a; %detei influe oeffiient atix oeff npals=length((1,:;oef=zeos(npals; fo n=1:npals n=f((:,n,(:,,(:,+f((:,n,(:,,(:,+f((:,n,(:,,(:,+f oef(n,:=n(1,n*n(1,:+n(2,n*n(2,:+n(3,n*n(3,:; end %detei esult veto and lve atix fo aent stengths =(-n(1,:*vinf(1-n(2,:*vinf(2-n(3,:*vinf(3'; oef(end+1,:=1;(end+1=0; %pevents singula atix - su of pal stengths on lod body is z ga=oef\; %Detei veloity and pessue at ontol points ga=epat(ga',[3 1]; fo n=1:npals %Detei veloity at eah.p. without pinipal value n=f((:,n,(:,,(:,+f((:,n,(:,,(:,+f((:,n,(:,,(:,+f v(:,n=vinf+su(ga.*n,2; end %Detei piniple value of veloity at eah.p., -gad(ga/2 gg=v_oss(((:,we-(:,no.*(ga(:,we+ga(:,no+((:,-(:,we.*(ga(:,+ga(:,we+((:, v=v-gg/2; %veloity veto p=1-su(v.^2/(vinf'*vinf; %pessue %plotting of pessue distibution and veloity vetos

5 Vetos In Matlab U aays with 3 ows, o fo eah opont. E.g. V Easy to ake funtions fo veto opeations like dot and oss poduts, and agnitude With the it is now easie to ake oe oplex funtions E.g. f (, s, e %3D doublet pal ethod fo flow aound 3D bodies. lea all; vinf=[1;0;0]; %fee stea veloity funtion =v_dot(a,b; =zeos(size(a; (1,:=a(1,:.*b(1,:+a(2,:.*b(2,:+a(3,:.*b(3,:; (2,:=(1,:;(3,:=(1,:; funtion =v_oss(a,b; =zeos(size(a; (1,:=(a(2,:.*b(3,:-a(3,:.*b(2,:; (2,:=(a(3,:.*b(1,:-a(1,:.*b(3,:; (3,:=(a(1,:.*b(2,:-a(2,:.*b(1,:; funtion q=f(,s,e; 1(1,:=s(1,:-(1,:; 1(2,:=s(2,:-(2,:; 1(3,:=s(3,:-(3,:; 2(1,:=e(1,:-(1,:; 2(2,:=e(2,:-(2,:; 2(3,:=e(3,:-(3,:; 0(1,:=1(1,:-2(1,:;0(2,:=1(2,:-2(2,:;0(3,:=1(3,:-2(3,:; =v_oss(1,2; 2=v_dot(,; q=./2.*v_dot(0,1./v_ag(1-2./v_ag(2/4/pi;

6 Speifying a 3D Body Fist we ust hoo shape. E.g. Body of evolution: %Speify body of evolution geoety th=-pi/2:pi/20:pi/2;xp=sin(th;yp=os(th;n=20; [,,,,,,no,we,,ea]=bodyofrevolution(xp,yp,n Then we geate the oodinates, in veto fo, of all the points on the body. The points (in a etangula aay, beoe the pal o points (nube of iufeential points funtion [,,,,,,no,we,,ea]=bodyofrevolution(xp,yp,n %Defi veties of pals x=epat(xp,[n+1 1]; %gid of x points al=[0:n]/(n*2*pi; %veto of iufeential angles (about x axis y=os(al'*yp; z=sin(al'*yp; =zeos([3 size(x];(1,:=x(:;(2,:=y(:;(3,:=z(:; %position veto of veties i=eshape(1:pod(size(x,size(x; %index of veties

7 no Pal Geoety ea Contol point loations have loations Pal veties have loations we n 4 1 Defining the indies, al defis how the body los =zeos([3 size(x];(1,:=x(:;(2,:=y(:;(3,:=z(:; %position veto of veties i=eshape(1:pod(size(x,size(x; %index of veties %pal i,j has os i,j i+1,j i+1,j+1 i,j+1 =i(1:end-1,1:end-1;=i(2:end,1:end-1; %indies of uppe and lowe left os =i(1:end-1,2:end;=i(2:end,2:end; %indies of uppe and lowe ight os % detei pal entes (ontol points and indies =((:,+(:,+(:,+(:,/4;=eshape(,[3 size(]; i=eshape(1:pod(size(,size(; % detei indies of pals bodeing eah ontol point no=i([end 1:end-1],:;=i([2:end 1],:; %Pals above and below. we=i(:,[1 1:end-1];ea=i(:,[2:end end]; %Pals to left and ight.

8 no Pal Geoety ea Contol point loations have loations Pal veties have loations we n 4 1 Defining the indies, al defis how the body los Sea 2nd index iasing =zeos([3 size(x];(1,:=x(:;(2,:=y(:;(3,:=z(:; %position veto of veties i=eshape(1:pod(size(x,size(x; %index of veties n e %pal i,j has os i,j i+1,j i+1,j+1 i,j+1 w s =i(1:end-1,1:end-1;=i(2:end,1:end-1; %indies of uppe and lowe left os =i(1:end-1,2:end;=i(2:end,2:end; %indies of uppe and lowe ight os % detei pal entes (ontol points and indies =((:,+(:,+(:,+(:,/4;=eshape(,[3 size(]; i=eshape(1:pod(size(,size(; % detei indies of pals bodeing eah ontol point no=i([end 1:end-1],:;=i([2:end 1],:; %Pals above and below. we=i(:,[1 1:end-1];ea=i(:,[2:end end]; %Pals to left and ight. 1st index iasing

9 Pal Geoety - Detei aea veto and outwad noal of eah pal by oss podut - n a n 1 2 a ( / a ( [,,,,,,no,we,,ea]=bodyofrevolution(xp,yp,n; % detei sufae aea and noal vetos at ontol points (assues ounte lokwi aound opass by RH ule points out of sufae a=0.5*v_oss((:,-(:,,(:,-(:,;n=a./v_ag(a;

10 Pal Influe Veloity due to pal at ontol point n: V( f (,, f (,, f (,, f (,, ( (n Pal o V( C Noal opont is Sued veloity at ontol point n is thus: V ( V C V(. n 0 V n C. n So, to get the ( s we ed to lve the siultaous equations: (w/o tangential veloity due to pinipal value Contol point n Nx1 atix of feestea oponts V n C.n NxN oeffiient atix Nx1 atix of pal stengths

11 Pal Influe ( (n Pal Contol point n Veloity due to pal at ontol point n: V( o V( f f ( ( C,,,, f f %detei influe oeffiient atix oef npals=length((1,:;oef=zeos(npals; fo n=1:npals V( V C (w/o tangential veloity n=f((:,n,(:,,(:,+f((:,n,(:,,(:, due to piniple value +f((:,n,(:,,(:,+f((:,n,(:,,(:,; oef(n,:=n(1,n*n(1,:+n(2,n*n(2,:+n(3,n*n(3,:; Noal opont is end ( ( Total veloity at ontol point n is thus:,, V(. n 0 V n C. n %detei esult veto and lve atix fo aent stengths =(-n(1,:*vinf(1-n(2,:*vinf(2-n(3,:*vinf(3'; oef(end+1,:=1;(end+1=0; So, to get the ( %pevents singula atix s we ed to lve the ga=oef\; siultaous equations: Nx1 atix of feestea oponts V n C.n NxN oeffiient atix,, Nx1 atix of pal stengths

12 Deteining Sufae Pessue Total veloity at ontol point we no n ea V( V C Tangential veloity due to pinipal value at Using the gadient theoe and values of at ighboing ontol points we an show that ( we no ( we no ( we ( ( ea ( ea ( no ea ( ( ( no 1 2 we Tangential veloity due to pinipal value at Evaluated at ea no we n ea We an then u Benoulli V( C ( 1 to opute the pessue p 2 V 2

13 Deteining Sufae Pessue Veloity at ontol point we no n ea V( V C Tangential veloity due to pinipal value at %Detei veloity and pessue at ontol points fo n=1:npals %Get veloity at eah.p. w/o pinipal value n=f((:,n,(:,,(:,+f((:,n,(:,,(:, +f((:,n,(:,,(:,+f((:,n,(:,,(:,; v(:,n=vinf+su(ga.*n,2; end ( %Detei we no ( piniple we no value, ( -gad(ga/2 we ( we gg=v_oss(((:,we-(:,no.*(ga(:,we+ga(:,no+((:,-... ( ea ( ea ( no ea ( no ea v=v-gg/2; %veloity veto ( no ( we ea p=1-su(v.^2/(vinf'*vinf; %pessue 1 2 Using the gadient theoe and values of at ighboing ontol points we an show that Tangential veloity due to pinipal value at Evaluated at n We an then u Benoulli V( C ( 1 to opute the pessue p 2 V 2

14 Deteining Sufae Pessue we no %Detei veloity and pessue at ontol points fo n=1:npals %Get veloity at eah.p. w/o pinipal value n=f((:,n,(:,,(:,+f((:,n,(:,,(:, Tangential veloity +f((:,n,(:,,(:,+f((:,n,(:,,(:,; V( V C due to piniple v(:,n=vinf+su(ga.*n,2; value at end %Detei piniple value, -gad(ga/2 gg=v_oss(((:,we-(:,no.*(ga(:,we+ga(:,no+((:,-... ea Tangential veloity v=v-gg/2; %veloity veto p=1-su(v.^2/(vinf'*vinf; n due to piniple %pessue value at Veloity at ontol point 1 2 Using the gadient theoe and values of at ighboing ontol points we an show that ( we no ( we no ( we ( ( ea ( ea ( no ea ( ( ( no we Evaluated at ea no we n ea We an then u Benoulli V( C ( 1 to opute the pessue p 2 V 2

15 Using the Code Plotting the pessue, stealis Defoing the body shape Changing the shape Moe than o body

Determine the Stress Calculating Mode of Sliding Failure of Soil Mass under the Push-Extend Multi-under-Reamed Pile

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