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Maple helps in simulating multiscale-multidomain...

by: Dr. Venkat Subramanian 431 Maple
battery pde numeric
October 06 2020
1  0

We recently published a paper on multiscale-multidomain simulation of battery models.

https://iopscience.iop.org/article/10.1149/1945-7111/abb37b

Some challenges are listed at 

https://twitter.com/UT_MAPLE/status/1311356957941522438

Maple and symbolic math can play a critical role in solving many challenging problems. For example, consider a seemingly-trivial problem

uxx+uyy = 0

x = 0 and x = 1, ux = 0 for all y

y = 1, u = 0, for all x

y = 0, u =1, 0<=x<=0.5

y = 0, uy = 0, 0.5<x<=1

There is a singularity at (0.5,0) and most numerical methods will have trouble there. In these equations, uxx means the second derivative of u with respect to x.

Maple can help solve this problem with conformal mapping to achieve arbitrary precision. As of today, machine precision is not possible with any numerical method even with millions of Degrees of Freedom. The Maple code is given below. A FEM code is given below as well. Models like this can benefit from Maple adding parallel sparse direct and iterative solvers.


 

> 

 

> 

restart;
> 

Digits:=15;

Digits := 15

 (1)

 The domain is tranformed from Z = X+IY to w. The points tranformed are

> 

Zdomain:=[[0,0],[1,0],[1,1],[0,1]];

Zdomain := [[0, 0], [1, 0], [1, 1], [0, 1]]

 (2)
> 

wdomain:=[0,1,1+a,a+2];

wdomain := [0, 1, 1+a, a+2]

 (3)
> 

eq1:=diff(Z(w),w)=-I*K1/sqrt(w)/sqrt(w-1)/sqrt(w-a-1)/sqrt(w-a-2);

eq1 := diff(Z(w), w) = -I*K1/(w^(1/2)*(w-1)^(1/2)*(w-a-1)^(1/2)*(w-a-2)^(1/2))

 (4)

 a is not known apriori. The value of a should be found to make sure [1,1] in the Z coordinate is transformed to [1+a] in the w coordinate.

> 

a:=sqrt(2)-1;

a := 2^(1/2)-1

 (5)

 Value of K1 is found using the transformation of [1,0] to 1 in the w coordinate

> 

eq11:=1=int(rhs(eq1),w=0..1);

eq11 := 1 = -K1*2^(1/2)*EllipticK(2^(1/2)/2)

 (6)
> 

K1:=solve(eq11,K1);

K1 := -(1/2)*(2^(1/2)/EllipticK(2^(1/2)/2))

 (7)

 The height Y in the Z coordinate is found by integrating from 1 to 1+a in the w coordinate.

> 

simplify(int(rhs(eq1),w=1..1+a));

(2*I)*2^(1/2)*EllipticK((2^(1/2)-1)/(2^(1/2)+1))/(EllipticK(2^(1/2)/2)*(2^(1/2)+1))

 (8)
> 

evalf(%);

1.00000000000000*I

 (9)

 The choice of a = sqrt(2)-1 gives the height of 1 for Z coordinate.

> 

eval(simplify(int(rhs(eq1),w=0..1+a)));

(2^(1/2)*EllipticK(2^(1/2)/2)+EllipticK(2^(1/2)/2)+(2*I)*2^(1/2)*EllipticK((2^(1/2)-1)/(2^(1/2)+1)))/(EllipticK(2^(1/2)/2)*(2^(1/2)+1))

 (10)
> 

evalf(%);

1.00000000000000+1.00000000000001*I

 (11)

 integration from 0 to 1+a in the w coordinate gives 1,1 in the Z coordinate.

> 

simplify(int(rhs(eq1),w=0..1/sqrt(2)));

EllipticF(2^(1/2)/(2+2^(1/2))^(1/2), 2^(1/2)/2)/EllipticK(2^(1/2)/2)

 (12)
> 

evalf(%);

.500000000000001

 (13)

 Integrating w from 0 to wmid =1/sqrt(2) gives the point 0.5,0 in the Z coordinate

> 

wmid:=1/sqrt(2);

wmid := 2^(1/2)/2

 (14)

 Next w domain is transformed to Z2 domain Z2 = X2+IY2

> 

Z2domain:=[[0,0],[1,0],[1,H],[0,H]];

Z2domain := [[0, 0], [1, 0], [1, H], [0, H]]

 (15)
> 

wdomain2:=[0,1/sqrt(2),1+a,a+2];

wdomain2 := [0, 2^(1/2)/2, 2^(1/2), 2^(1/2)+1]

 (16)
> 

eq2:=diff(Z2(w),w)=-I*K2/sqrt(w)/sqrt(w-wmid)/sqrt(w-1-a)/sqrt(w-a-2);

eq2 := diff(Z2(w), w) = -(2*I)*K2/(w^(1/2)*(4*w-2*2^(1/2))^(1/2)*(w-2^(1/2))^(1/2)*(w-2^(1/2)-1)^(1/2))

 (17)

 K2 is found based on the transformation of wmid to [1,0] in Z2 coordinate.

> 

eq21:=1=int(rhs(eq2),w=0..wmid);

eq21 := 1 = -2*K2*EllipticK(1/(((2+2^(1/2))^(1/2))))/(2^(1/2)+1)^(1/2)

 (18)
> 

K2:=solve(eq21,K2);

K2 := -(1/2)*((2^(1/2)+1)^(1/2)/EllipticK(1/(((2+2^(1/2))^(1/2)))))

 (19)

 The corner 0,1 in the Z coordinate is mapped by integrating eq2 from 0 to 1 in the w coordinate

 

 
> 

int(rhs(eq2),w=0..1);

1+EllipticF((2-2^(1/2))^(1/2), ((2^(1/2)+1)/(2+2^(1/2)))^(1/2))*I/EllipticK(1/(((2+2^(1/2))^(1/2))))

 (20)
> 

corner:=evalf(%);

corner := 1.+.559419351518322*I

 (21)

 This is the point 1,0 in the original coordinate.

 The height in the Z2 coorinate is found by integrating eq2 from wmid to 1+a.

> 

ytot:=int(rhs(eq2),w=wmid..1+a);

ytot := EllipticK(((2^(1/2)+1)/(2+2^(1/2)))^(1/2))*I/EllipticK(1/(((2+2^(1/2))^(1/2))))

 (22)
> 

evalf(%);

1.22004159128347*I

 (23)

 The magnitude in the Y direction is given by the coefficient of I, the imaginary number

> 

Ytot:=coeff(ytot,I);

Ytot := EllipticK(((2^(1/2)+1)/(2+2^(1/2)))^(1/2))/EllipticK(1/(((2+2^(1/2))^(1/2))))

 (24)

 The analytial solution in the Z2 corordinate is a line in Y2 to satisfy simple zero flux conditions at X2 = 0 and X2 = 1.

> 

phianal:=1+b*Y2;

phianal := 1+b*Y2

 (25)

 The value of phi is zero at Y2 = Ytot (originally the cathode domain in the Z domain)

> 

bc:=subs(Y2=Ytot,phianal)=0;

bc := 1+b*EllipticK(((2^(1/2)+1)/(2+2^(1/2)))^(1/2))/EllipticK(1/(((2+2^(1/2))^(1/2)))) = 0

 (26)
> 

b:=solve(bc,b);

b := -EllipticK(1/(((2+2^(1/2))^(1/2))))/EllipticK(((2^(1/2)+1)/(2+2^(1/2)))^(1/2))

 (27)

 The analytical solution is given by

> 

phianal;

1-EllipticK(1/(((2+2^(1/2))^(1/2))))*Y2/EllipticK(((2^(1/2)+1)/(2+2^(1/2)))^(1/2))

 (28)
> 

evalf(phianal);

1.-.819644188480505*Y2

 (29)

 The potential at the corner is given by substituting the imaginary value of corner for Y2 in phinanal)

> 

phicorner:=subs(Y2=Im(corner),phianal);

phicorner := 1-.559419351518322*EllipticK(1/(((2+2^(1/2))^(1/2))))/EllipticK(((2^(1/2)+1)/(2+2^(1/2)))^(1/2))

 (30)
> 

evalf(phicorner);

.541475179604475

 (31)

local flux/current density calculation, written in terms of w is

> 

curr:=b*rhs(eq2)/rhs(eq1);

curr := -(2^(1/2)+1)^(1/2)*2^(1/2)*EllipticK(2^(1/2)/2)*(w-1)^(1/2)/(EllipticK(((2^(1/2)+1)/(2+2^(1/2)))^(1/2))*(4*w-2*2^(1/2))^(1/2))

 (32)

average flux/current density calculation for the anode

> 

currave:=int((curr),w=0..wmid)/wmid;

currave := -(1/2)*((2^(1/2)+1)^(1/2)*(2^(1/2)-1)*EllipticK(2^(1/2)/2)*(2^(3/4)+2^(1/4)+arctanh(2^(3/4)/2))*2^(1/2)/EllipticK(2^(3/4)/2))

 (33)
> 

Digits:=25:

 The average current density at Y =0, local current density at X = 0,Y=0 and potential at X=1,Y=0 (Corner) can be used to study convergence of FEM and other numerical methods

> 

evalf(currave),evalf(subs(w=0,curr)),evalf(phicorner);

-1.656507648777793388522396, -1.161311530233258689567781, .5414751796044734741869534

 (34)
> 

 


 

Download Conformalmapping.mws


 

This FEM code is for solving Laplace's equation with primary current distribution considered in Model 1.
This code is based on FEM weak-form. Biquadratic Lagrange shape functions (9nodes in an element) are used.

> 

restart;
> 

with(LinearAlgebra):
> 

Lx:=1: #length in X
> 

Ly:=1: #length in Y
> 

nx:=10: #number of elements in X (even numbers only)
> 

ny:=10: #number of elements in Y, to be kept same as nx in this version
> 

hx:=Lx/nx: #element size x
> 

hy:=Ly/ny: #element size y

Procedure to perform numerical integration on shape functions to obtain local matrices (can be replaced with analytical integration for this particular problem)
  -Shape functions are also used as weight functions in applying weak formulation. Numerical integration is done using Simpson's rule.
  -Local cartesian coordinates x,y are converted to natural coordinates zeta and eta. This transformation is not required for this simple geomerty but useful in general. zeta and eta are obtained by scaling x and y with hx/2 and hy/2, respectively, in this code.

 

> 

localmatrices:=proc(a1,a2,a3,b1,b2,b3,q1,q2)
global Kx,Ky,Nx,Ny,zeta,eta,c;
local A,dAdzeta,dAdeta,y,x,J,terms,i,j,k,l,dx,dy,fx,fy,fxy,fyy,dzeta,deta,J1,J2;
> 

A:=[(1-zeta)*zeta*(1-eta)*eta/4,-(1-(zeta)^2)*(1-eta)*eta/2,-(1+zeta)*zeta*(1-eta)*eta/4,(1-(eta)^2)*(1+zeta)*zeta/2,(1+zeta)*zeta*(1+eta)*eta/4,(1-(zeta)^2)*(1+eta)*eta/2,-(1-zeta)*zeta*(1+eta)*eta/4,-(1-(eta)^2)*(1-zeta)*(zeta)/2,(1-(zeta)^2)*(1-(eta)^2)]; #bi quadratic langrange shape functions
> 

dAdzeta:=diff(A,zeta);
> 

dAdeta:=diff(A,eta);
> 

y:=[-a1,-a2,-a3,0,a3,a2,a1,0,0];x:=[-b1,0,b1,b2,b3,0,-b3,-b2,0];
> 

J:=simplify(add(dAdzeta[i]*x[i],i=1..9)*add(dAdeta[i]*y[i],i=1..9)-add(dAdzeta[i]*y[i],i=1..9)*add(dAdeta[i]*x[i],i=1..9));
> 

Nx:=[seq((simplify(add(dAdeta[i]*y[i],i=1..9))*dAdzeta[j]-simplify(add(dAdzeta[i]*y[i],i=1..9))*dAdeta[j])/simplify(J),j=1..9)];
> 

Ny:=[seq((-dAdzeta[j]*simplify(add(dAdeta[i]*x[i],i=1..9))+dAdeta[j]*simplify(add(dAdzeta[i]*x[i],i=1..9)))/simplify(J),j=1..9)];
> 

Kx:=Matrix(nops(A),nops(A),datatype=float[8]):
> 

Ky:=Matrix(nops(A),nops(A),datatype=float[8]):
c:=Vector(nops(A),datatype=float[8]):
> 

terms:=20:#number of terms for numerical integration
dzeta:=2/terms:
deta:=2/terms:
> 

for i from 1 to nops(Nx) do #loop to obtain local matrices      

for j from 1 to nops(Ny) do
Kx[i,j]:=0;
Ky[i,j]:=0;

for k from 0 to terms do #outer loop double integration, integration in zeta

if k = 0 then fx[k]:= subs(zeta=-1,Nx[i]*Nx[j]*J); fy[k]:= subs(zeta=-1,Ny[i]*Ny[j]*J);  
elif k = terms then
fx[k]:= subs(zeta=-1+(k*dzeta),Nx[i]*Nx[j]*J);
fy[k]:= subs(zeta=-1+(k*dzeta),Ny[i]*Ny[j]*J);  
elif irem(k,2) = 0 then
fx[k]:= 2*subs(zeta=-1+(k*dzeta),Nx[i]*Nx[j]*J);
fy[k]:=     2*subs(zeta=-1+(k*dzeta),Ny[i]*Ny[j]*J);
else fx[k]:= 4*subs(zeta=-1+(k*dzeta),Nx[i]*Nx[j]*J);
fy[k]:=     4*subs(zeta=-1+(k*dzeta),Ny[i]*Ny[j]*J);  end if;

for l from 0 to terms do #inner loop double integration, integration in eta

if l = 0 then fxy[l]:= subs(eta=-1,fx[k]); fyy[l]:= subs(eta=-1,fy[k]);
elif l = terms then fxy[l]:= subs(eta=-1+(l*deta),fx[k]); fyy[l]:= subs(eta=-1+(l*deta),fy[k]);
elif irem(l,2) = 0 then fxy[l]:= 2*subs(eta=-1+(l*deta),fx[k]); fyy[l]:= 2*subs(eta=-1+(l*deta),fy[k]);
else fxy[l]:=4*subs(eta=-1+(l*deta),fx[k]); fyy[l]:=4*subs(eta=-1+(l*deta),fy[k]); end if;
Kx[i,j]:=Kx[i,j]+fxy[l];
Ky[i,j]:=Ky[i,j]+fyy[l];

end do;
    
end do;
Kx[i,j]:=Kx[i,j]*dzeta*deta/9;
Ky[i,j]:=Ky[i,j]*dzeta*deta/9;
end do;
end do:
> 

end proc:
> 

n:=nx*ny; #total number of elements

n := 100

 (1)
> 

Nx1:=nx*2+1: #number of nodes in x in one row
> 

N:=Nx1*(2*ny+1); # total number of nodes/equations

N := 441

 (2)
> 

K:=Matrix(N,N,storage=sparse): # global K matrix
> 

C:=Vector(N,storage=sparse): # global c matrix
> 

L2G:=Matrix(n,9):  #mapping matrix - each row has node numbers for each element
> 

l:=1:k:=1:
localmatrices(hy/2,hy/2,hy/2,hx/2,hx/2,hx/2,0,0): kx:=copy(Kx):ky:=copy(Ky):c0:=copy(c):
> 

for i from 1 to n do #modifying,adding and assembling matrices to get global matrix
 
if i<=nx/2 then  
a1:=copy(kx); a2:=copy(ky); a3:=0; a1[1..3,1..9]:=IdentityMatrix(3,9); a2[1..3,1..9]:=Matrix(3,9,shape=zero); a4:=a1+a2; c:=copy(c0); c[1..3]:=1.0;

elif i=nx/2+1 then  a1:=copy(kx); a2:=copy(ky); a3:=0; a1[1,1..9]:=IdentityMatrix(1,9); a2[1,1..9]:=Matrix(1,9,shape=zero); a4:=a1+a2; c:=copy(c0); c[1]:=1.0;

elif i>nx*(ny-1) then a1:=copy(kx); a2:=copy(ky); a3:=0; a1[5..7,5..7]:=IdentityMatrix(3,3);
a1[5..7,1..4]:=ZeroMatrix(3,4);
a1[5..7,8..9]:=ZeroMatrix(3,2); a2[5..7,1..9]:=Matrix(3,9,shape=zero); a4:=a1+a2; c:=copy(c0); c[5..7]:=0;

else a1:=kx; a2:=ky; a3:=0;a4:=a1+a2; c:=c0;  end if;

L2G[i,1..9]:=Matrix([l,l+1,l+2,l+2+Nx1,l+2+Nx1*2,l+1+Nx1*2,l+Nx1*2,l+Nx1,l+1+Nx1]):

k:=k+1:
 
if k>nx then k:=1; l:=l+Nx1+3;

else l:=l+2; end if:

indx2:=L2G[i,1..9]:
indx2:=convert(indx2,list):

C[indx2]:=C[indx2]+c[1..9];
c[1..9];

for i1 from 1 to 9 do
indx1:=L2G[i,i1]:
K[indx1,indx2]:=K[indx1,indx2]+a4[i1,1..9]:
end do:

end do:
> 

phi:=LinearSolve(K,C,method=SparseLU): #linear set of equations solved using Sparse LU solver
> 

phi_at(1,0):=phi[Nx1];
 

phi_at(1, 0) := .546587799122513

 (3)
> 

dNdy:=copy(Ny):
> 

dNdy_bottom_left:=subs(eta=-1,zeta=-1,dNdy):
> 

current_at(0,0):=add(dNdy_bottom_left[i]*phi[L2G[1,i]],i=1..nops(Ny));

current_at(0, 0) := -1.15773815354626

 (4)
> 

if irem(nx/2,2)=0
then current_at(0,0.25):=add(dNdy_bottom_left[i]*phi[L2G[nx/4+1,i]],i=1..nops(Ny));
else
dNdy_bottom_center:=subs(eta=-1,zeta=0,dNdy):
current_at(0,0.25):=add(dNdy_bottom_center[i]*phi[L2G[(nx/2+1)/2,i]],i=1..nops(Ny));
end if;

dNdy_bottom_center := [0, -30, 0, 0, 0, -10, 0, 0, 40]

current_at(0, .25) := -1.26989097821724

 (5)
> 

 

> 

 


 

Download FEM_2D.mws

what are the main things you like to see improved...

by: nm 11218 Maple
August 28 2020
5  43

What are the things you most like to see improved/add to next version of Maple? 

This is my list for a starter:

1.  Improve the debugger. Debugger is very useful but needs more work. At least be able to see code listing in larger view as one steps in for example. See Matlab debugger for inspiration.

2.  Improve Latex. It still does not do fractions well. Posted about this before.

3. Eliminate hangs when using timelimit(). On long runs, random hangs happen when timelimit() do not expire as requested. Posted about this before.

 

 

Maple's user-interface deters me from doing math

by: DoingMath2018 35
August 10 2020
1  17

I don't stand by all I said here, so I'm deleting. I think there was valid criticism to be made, but I didn't make it any valid way.

Several contentious issues in the treatment of...

by: mmcdara 7721
discrete randomvariable statistics
December 18 2019
1  3

Hi, 

This is more of an open discussion than a real question. Maybe it would gain to be displaced in the post section?

Working with discrete random variables I found several inconsistencies or errors.
In no particular order: 

  • The support of a discrete RV is not defined correctly (a real range instead of a countable set)
  • The plot of the probability function (which, in my opinion, would gain to be renamed "Probability Mass Function, see https://en.wikipedia.org/wiki/Probability_mass_function) is not correct.
  • The  ProbabiliytFunction of a discrte rv of EmpiricalDistribution can be computed at any point, but its formal expression doesn't exist (or at least is not accessible).
  • Defining the discrete rv "toss of a fair dice"  with EmpiricalDistribution and DiscreteUniform gives different results.


The details are given in the attached file and I do hope that the companion text is clear enough to point the issues.
I believe there is no major issues here, but that Maple suffers of some lack of consistencies in the treatment of discrete (at least some) rvs. Nothing that could easily be fixed.


As I said above, if some think this question has no place here and ought to me moved to the post section, please feel free to do it.

Thanks for your attention.


 

> 

restart:
> 

with(Statistics):


Two alternate ways to define a discrete random variable on a finite set
of equally likely outcomes.

> 

Universe    := [$1..6]:
toss_1_dice := RandomVariable(EmpiricalDistribution(Universe));
TOSS_1_DICE := RandomVariable(DiscreteUniform(1, 6));

_R

  

_R0

 (1)


Let's look to the ProbabilityFunction of each RV

> 

ProbabilityFunction(toss_1_dice, x);
ProbabilityFunction(TOSS_1_DICE, x);

"_ProbabilityFunction[Typesetting:-mi("x",italic = "true",mathvariant = "italic")]"

  

piecewise(x < 1, 0, x <= 6, 1/6, 6 < x, 0)

 (2)


It looks like the procedure ProbabilityFunction is not an attribute of RV with EmpiticalDistribution.
Let's verify

> 

law := [attributes(toss_1_dice)][3]:
lprint(exports(law))

Conditions, ParentName, Parameters, CDF, DiscreteValueMap, Mean, Median, Mode, ProbabilityFunction, Quantile, Specialize, Support, RandomSample, RandomVariate

  


Clearly ProbabilityFunction is an attribute of toss_1_dice.

In fact it appears the explanation of the difference of behaviours relies upon different definitions
of the set of outcomes of toss_1_dice and TOSS_1_DICE

> 

LAW := [attributes(TOSS_1_DICE)][3]:
exports(LAW):

law:-Conditions;
LAW:-Conditions;

[(Vector(6, {(1) = 1, (2) = 2, (3) = 3, (4) = 4, (5) = 5, (6) = 6}))::rtable]

  

[1 < 6]

 (3)


From :-Conditions one can see that toss_1_dice is realy a discrete RV defined on a countable set of outcomes,
but that nothing is said about the set over which TOSS_1_DICE is defined.

The truly discrete definition of toss_1_dice is confirmed here :
(the second result is correct

ProbabilityFinction(toss_1_dice, x) = {0 if x < 1, 0 if x > 6, 1/6 if x::integer, 0 otherwise

> 

ProbabilityFunction~(toss_1_dice, Universe);
ProbabilityFunction~(toss_1_dice, [seq(0..7, 1/2)]);

[1/6, 1/6, 1/6, 1/6, 1/6, 1/6]

  

[0, 0, 1/6, 0, 1/6, 0, 1/6, 0, 1/6, 0, 1/6, 0, 1/6, 0, 0]

 (4)


One can also see that the Support of both of these RVs are wrong

(see for instance https://en.wikipedia.org/wiki/Discrete_uniform_distribution)

There should be {1, 2, 3, 4, 5, 6}, not a RealRange.

> 

Support(toss_1_dice);
Support(TOSS_1_DICE);

RealRange(1, 6)

  

RealRange(1, 6)

 (5)
> 

 

0

  

{1, 2, 3, 4, 5, 6}

  

 


Now this is the surprising ProbabilityFunction of TOSS_1_DICE.
This obviously wrong result probably linked to the weak definition of the conditions for this RB.

> 

# plot(ProbabilityFunction(TOSS_1_DICE, x), x=0..7);
plot(ProbabilityFunction(TOSS_1_DICE, x), x=0..7, discont=true)

 


These differences of treatments raise a lot of questions :
    -  Why is a DiscreteUniform RV not defined on a countable set?
    -  Why does the ProbabilityFunction of an EmpiricalDistribution return no result
        if its second parameter is not set to one  its outcomes.

 All this without even mentioning the wrong plot shown above.
 

I believe something which would work like the module below would be much better than what is done

right now

 

> 

EmpiricalRV := module()
export MassDensityFunction, PlotMassDensityFunction, Support:

MassDensityFunction := proc(rv, x)
  local u, v, N:
  u := [attributes(rv)][3]:
  if u:-ParentName = EmpiricalDistribution then
    v := op([1, 1], u:-Conditions);
    N := numelems(v):
    return piecewise(op(op~([seq([x=v[n], 1/N], n=1..N)])), 0)
  else
    error "The random variable does not have an EmpiricalDistribution"
  end if
end proc:

PlotMassDensityFunction := proc(rv, x1, x2)
  local u, v, a, b:
  u := [attributes(rv)][3]:
  if u:-ParentName = EmpiricalDistribution then
    v := op([1, 1], u:-Conditions);
    a := select[flatten](`>=`, v, x1);
    b := select[flatten](`<=`, a, x2);
    PLOT(seq(CURVES([[n, 0], [n, 1/numelems(v)]], COLOR(RGB, 0, 0, 1), THICKNESS(3)), n in b), VIEW(x1..x2, default))
  else
    error "The random variable does not have an EmpiricalDistribution"
  end if
end proc:

Support := proc(rv, x1, x2)
  local u, v, a, b:
  u := [attributes(rv)][3]:
  if u:-ParentName = EmpiricalDistribution then
    v := op([1, 1], u:-Conditions);
    return {entries(v, nolist)}
  else
    error "The random variable does not have an EmpiricalDistribution"
  end if
end proc:

end module:
 
> 

EmpiricalRV:-MassDensityFunction(toss_1_dice, x);
 

piecewise(x = 1, 1/6, x = 2, 1/6, x = 3, 1/6, x = 4, 1/6, x = 5, 1/6, x = 6, 1/6, 0)

 (6)
> 

f := unapply(EmpiricalRV:-MassDensityFunction(toss_1_dice, x), x):
f(2);
f(5/2);
 

1/6

  

0

 (7)
> 

EmpiricalRV:-PlotMassDensityFunction(toss_1_dice, 0, 7);

 
> 

 


 

Download Discrete_RV.mw

 

 

Wait for 2019.3 - 2019.2 flawed - Maplesoft please...

by: Christopher2222 6005 Maple 2019
November 19 2019
5  2

I'm only just hearing (haven't experienced) about some serious issues with the 2019.2 updates.  I would recommend waiting for Maplesoft to release an emergency 2019.3 fix update - Maplesoft can NOT leave the last update of 2019 in this state.

Maple May benefit from starting a Stack Exchange...

by: Adam Ledger 360 Maple
stackexchange
October 12 2019
6  7

I have noticed that there exists a Stack Exchange site for mathematica, and not for maple. My discussions with the part of Stack Exchange that handle the creation of a new Stack Exchange community have said that I must accrue a certain level of interest in the subject in order for it to be approved, and so I thought I would begin here to see if there is suffice level of interest.

This would not diminish the use of the Maple Primes forum, and an additional proposal, in consideration of the years of dedication that have gone into this domain, be to pool the data between the two, make reputation points the same on both, perhaps even user profiles and questions answered already linkable, and all of the questions already addressed here showing up in the search on both domains.

I am proposing this simply because I want to encourage the use of maple, and have noted that Stack Exchange is very popular. 

So I am posting this to get overall feedback from other Maple users, as to what their opinion is regarding this proposal, and advice on whether it should and how it ought to be pursued.

We’d Like Your Input! - Updates to FAQs

by: TechnicalSupport 750 Maple , MapleNet , Maple Toolboxes , MapleSim , MaplePrimes , MapleSim Add-Ons
August 02 2019
6  2

We are currently in the process of updating the support FAQs at https://faq.maplesoft.com. We’ve been working on updating the existing content for clarity, and have added several new articles already.

 

The majority of our FAQs are from questions people ask us in Technical Support by support request form, but we’d also like to like to add content from other sources.

Since we have such a great community here at MaplePrimes, we wanted to reach out and ask if there are any articles or questions that you'd like to see added to our FAQ.

 

We look forward to hearing your feedback!

2D contour plot and legend

by: acer 32091 Maple
contourplot contour legend
February 12 2019
15  1

There have been several posts, over the years, related to visual cues about the values associated with particular 2D contours in a plot.

Some people ask or post about color-bars [1]. Some people ask or post about inlined labelling of the curves [1, 2, 3, 4, 5, 6, 7]. And some post about mouse popup/hover-over functionality [1]., which got added as general new 2D plot annotation functionality in Maple 2017 and is available for the plots:-contourplot command via its contourlabels option.

Another possibility consists of a legend for 2D contour plots, with distinct entries for each contour value. That is not currently available from the plots:-contourplot command as documented. This post is about obtaining such a legend.

Aside from the method used below, a similar effect may be possible (possibly with a little effort) using contour-plotting approaches based on individual plots:-implicitplot calls for each contour level. Eg. using Kitonum's procedure, or an undocumented, alternate internal driver for plots:-contourplot.

Since I like the functionality provided by the contourlabels option I thought that I'd highjack that (and the _HOVERCONTENT plotting substructure that plot-annotations now generate) and get a relatively convenient way to get a color-key via the 2D plotting legend.  This is not supposed to be super-efficient.

Here below are some examples. I hope that it illustrates some useful functionality that could be added to the contourplot command. It can also be used to get a color-key for use with densityplot.

> 

restart;
> 

contplot:=proc(ee, rng1, rng2)
  local clabels, clegend, i, ncrvs, newP, otherdat, others, tcrvs, tempP;
  (clegend,others):=selectremove(type,[_rest],identical(:-legend)=anything);
  (clabels,others):= selectremove(type,others,identical(:-contourlabels)=anything);
  if nops(clegend)>0 then
    tempP:=:-plots:-contourplot(ee,rng1,rng2,others[],
                                ':-contourlabels'=rhs(clegend[-1]));
    tempP:=subsindets(tempP,'specfunc(:-_HOVERCONTENT)',
                      u->`if`(has(u,"null"),NULL,':-LEGEND'(op(u))));
    if nops(clabels)>0 then
      newP:=plots:-contourplot(ee,rng1,rng2,others[],
                              ':-contourlabels'=rhs(clabels[-1]));
      tcrvs:=select(type,[op(tempP)],'specfunc(CURVES)');
      (ncrvs,otherdat):=selectremove(type,[op(newP)],'specfunc(CURVES)');
      return ':-PLOT'(seq(':-CURVES'(op(ncrvs[i]),op(indets(tcrvs[i],'specfunc(:-LEGEND)'))),
                          i=1..nops(ncrvs)),
                      op(otherdat));
    else
      return tempP;
    end if;
  elif nops(clabels)>0 then
    return plots:-contourplot(ee,rng1,rng2,others[],
                              ':-contourlabels'=rhs(clabels[-1]));
  else
    return plots:-contourplot(ee,rng1,rng2,others[]);
  end if;
end proc:
 
> 

contplot(x^2+y^2, x=-2..2, y=-2..2,
      coloring=["Yellow","Blue"],
      contours = 9,
      size=[500,400],
      legendstyle = [location = right],
      legend=true,
      contourlabels=true,
      view=[-2.1..2.1,-2.1..2.1]
);

> 

contplot(x^2+y^2, x=-2..2, y=-2..2,
      coloring=["Yellow","Blue"],
      contours = 17,
      size=[500,400],
      legendstyle = [location = right],
      legend=['contourvalue',$("null",7),'contourvalue',$("null",7),'contourvalue'],
      contourlabels=true,
      view=[-2.1..2.1,-2.1..2.1]
);

> 

# Apparently legend items must be unique, to persist on document re-open.

contplot(x^2+y^2, x=-2..2, y=-2..2,
      coloring=["Yellow","Blue"],
      contours = 11,
      size=[500,400],
      legendstyle = [location = right],
      legend=['contourvalue',seq(cat($(` `,i)),i=2..5),
              'contourvalue',seq(cat($(` `,i)),i=6..9),
              'contourvalue'],
      contourlabels=true,
      view=[-2.1..2.1,-2.1..2.1]
);

> 

contplot(x^2+y^2, x=-2..2, y=-2..2,
      coloring=["Green","Red"],
      contours = 8,
      size=[400,450],
      legend=true,
      contourlabels=true
);

> 

contplot(x^2+y^2, x=-2..2, y=-2..2,
      coloring=["Yellow","Blue"],
      contours = 13,
      legend=['contourvalue',$("null",5),'contourvalue',$("null",5),'contourvalue'],
      contourlabels=true
);

> 

(low,high,N):=0.1,7.6,23:
conts:=[seq(low..high*1.01, (high-low)/(N-1))]:
contplot(x^2+y^2, x=-2..2, y=-2..2,
      coloring=["Yellow","Blue"],
      contours = conts,
      legend=['contourvalue',$("null",floor((N-3)/2)),'contourvalue',$("null",ceil((N-3)/2)),'contourvalue'],
      contourlabels=true
);

> 

plots:-display(
  subsindets(contplot((x^2+y^2)^(1/2), x=-2..2, y=-2..2,
                      coloring=["Yellow","Blue"],
                      contours = 7,
                      filledregions),
             specfunc(CURVES),u->NULL),
  contplot((x^2+y^2)^(1/2), x=-2..2, y=-2..2,
      coloring=["Yellow","Blue"],
      contours = 7, #grid=[50,50],
      thickness=0,
      legendstyle = [location=right],
      legend=true),
  size=[600,500],
  view=[-2.1..2.1,-2.1..2.1]
);
> 

 

> 

plots:-display(
  contplot(x^2+y^2, x=-2..2, y=-2..2,
      coloring=["Yellow","Blue"],
      contours = 5,
      thickness=0, filledregions),
  contplot(x^2+y^2, x=-2..2, y=-2..2,
      coloring=["Yellow","Blue"],
      contours = 5,
      thickness=3,
      legendstyle = [location=right],
      legend=typeset("<=",contourvalue)),
  size=[700,600],
  view=[-2.1..2.1,-2.1..2.1]
);

> 

N:=11:
plots:-display(
  contplot(sin(x)*y, x=-2*Pi..2*Pi, y=-1..1,
      coloring=["Yellow","Blue"],
      contours = [seq(-1+(i-1)*(1-(-1))/(N-1),i=1..N)],
      thickness=3,
      legendstyle = [location=right],
      legend=true),
   plots:-densityplot(sin(x)*y, x=-2*Pi..2*Pi, y=-1..1,
      colorscheme=["zgradient",["Yellow","Blue"],colorspace="RGB"],
      grid=[100,100],
      style=surface, restricttoranges),
   plottools:-line([-2*Pi,-1],[-2*Pi,1],thickness=3,color=white),
   plottools:-line([2*Pi,-1],[2*Pi,1],thickness=3,color=white),
   plottools:-line([-2*Pi,1],[2*Pi,1],thickness=3,color=white),
   plottools:-line([-2*Pi,-1],[2*Pi,-1],thickness=3,color=white),
   size=[600,500]
);

> 

N:=13:
plots:-display(
  contplot(sin(x)*y, x=-2*Pi..2*Pi, y=-1..1,
      coloring=["Yellow","Blue"],
      contours = [seq(-1+(i-1)*(1-(-1))/(N-1),i=1..N)],
      thickness=6,
      legendstyle = [location=right],
      legend=['contourvalue',seq(cat($(` `,i)),i=2..3),
              'contourvalue',seq(cat($(` `,i)),i=5..6),
              'contourvalue',seq(cat($(` `,i)),i=8..9),
              'contourvalue',seq(cat($(` `,i)),i=11..12),
              'contourvalue']),
   plots:-densityplot(sin(x)*y, x=-2*Pi..2*Pi, y=-1..1,
      colorscheme=["zgradient",["Yellow","Blue"],colorspace="RGB"],
      grid=[100,100],
      style=surface, restricttoranges),
   plottools:-line([-2*Pi,-1],[-2*Pi,1],thickness=6,color=white),
   plottools:-line([2*Pi,-1],[2*Pi,1],thickness=6,color=white),
   plottools:-line([-2*Pi,1],[2*Pi,1],thickness=6,color=white),
   plottools:-line([-2*Pi,-1],[2*Pi,-1],thickness=6,color=white),
  size=[600,500]
);

> 

 

Download contour_legend_post.mw

 

 

 

MapleSim 2018 Bug: Simulink Component Block Genera...

by: wmcnally 30 MapleSim 2018
December 17 2018
0  1

Error when running 'm_model.m' to generate simulink function block when using 'Group all parameters into nested structure'. No errors when using 'Do not group parameters'.

Error using m_fullSwing (line 125)
Invalid setting in 'MapleSim_fullSwing/MapleSim_fullSwing/MapleSimParameters' for parameter 'Value'.
Caused by:
    Error using m_fullSwing (line 125)
    Expression '[Param.Address.ArmPlane, Param.Address.ClubRelHand,  ... ]' for
    parameter 'Value' in 'MapleSim_fullSwing/MapleSim_fullSwing/MapleSimParameters' cannot be evaluated.
        Error using m_fullSwing (line 125)
        Error: Invalid expression. When calling a function or indexing a variable, use parentheses. Otherwise, check for
        mismatched delimiters.
 

Maple 2019 wishlist

by: Christopher2222 6005
October 11 2018
6  33

Starting the Maple 2019 wishlist.  Anything in particular users would like to see added or upgraded to maple can add it here. 

 

Bug in inequal

by: Markiyan Hirnyk 7809 Maple 2018
gui inequal
September 24 2018
1  2

Let us consider

plots:-inequal(max(1, min(x, 2))+max(1, min(y, 2)) <= 3, x = -4 .. 4, y = -4 .. 4);


and compare it with

plots:-implicitplot(max(1, min(x, 2))+max(1, min(y, 2)) = 3, x = -4 .. 4, y = -4 .. 4, gridrefine = 2);

The latter plot must be a subset of the former plot, but it isn't so. bug_in_inequal.mw

GitHub support for Maple file extensions and syntax...

by: SGJanssens 125 Maple
August 27 2018
6  2

Hello,

At the moment there is no support on GitHub for language recognition and syntax highlighting for Maple. I think better support for Maple on GitHub would be a good thing:

  1. It makes Maple more recognizable, for example in language searches such as this one for one of its competitors.
     
  2. The list of currently supported languages is long and even contains many obscure entries. Maple is not obscure and deserves to be there.

So, why am I posting this here, as it concerns GitHub more than Maple? The reason is that adding support for new languages is often done by GitHub users themselves, using the Linguist library mentioned on this help page. The process does not seem very difficult to me, but it requires a few careful steps.

Referring to those steps:

  1. I think it would be nice to add support for the extensions , and .
     
  2. Perhaps this project by @Daniel Skoog could be used for syntax highlighting? However, currently it does not have a license.
     
  3. Does anybody know of a body of Maple source code that is representative of the modern Maple language as a whole? Probably it should be available under a permissive open source license such as MIT or BSD. (I am not sure whether GPLed work would qualify.)
     
  4. I know of a few Maple projects by others that are maintained on GitHub, such as the ParametricMatrixTools package and some packages by Daniel Skoog. If there are other Maple projects being maintained on GitHub, it would be good to know about them in the comments.

So, in summary, could you perhaps help me with the above Steps 2, 3 and 5 or, if you prefer, could you take care of them yourself and open a pull request, so Maple on GitHub can literally get the recognition it deserves?

My best wishes,

Sebastiaan Janssens.

 

improving Latex output of Maple for expressions

by: nm 11218 Maple 2018
June 11 2018
8  3

There are many questions that complain about Latex conversion in Maple.

I'd like to again request that Maplesoft improves Latex output of its expressions. If Maple can just fix how it generates fractions, that will good enough for now.

I am willing to send Maplesoft a personal check of the amount of one month salary for one of your developers to do this fix if you are willing to do it. It should not take more than one month to do this simple fix in your code. It might even take one day if someone knows the code.

The problem comes when there is a fraction in the expression. the Latex output instead of using proper latex code using "\frac{}{}", it instead uses "/" which makes the output terrible.

Another case, where Maple generate (expression)^{-1} instead of \frac{1}{expression}.

It can't be that hard to fix these 2 issues, which can go a long way towards making the latex generated by Maple much better. Here is an example

eq:=-(1/2)*1/y = (1/3)*x^3+z:
sol:=solve(eq,y);

latex(sol);
-3/2\, \left( {x}^{3}+3\,z \right) ^{-1}

Which renders as

Which is terrible. The screen output is much better.

Compare this to Mathematica

eq = -(1/2)*(1/y) == (1/3)*x^3 + z;
sol = y /. First@Solve[eq, y];
TeXForm[sol]

   -\frac{3}{2 \left(x^3+3 z\right)}

Which renders in Latex as

If Maplesoft does not think Latex is improtant, then they are completely wrong. CAS support in Latex is very important. Ignoring Latex means you will lose customers who want good Latex support of the math output of Maple. After all, Math and Latex go togother. And Maple is supposed to be all about Mathematics.

Any chance of Maplesoft taking some time to fix these issues in Latex? Maple has not had any improvement in Latex for years and years. I keep buying Maple each year, and nothing changes in its Latex export.

thank you

Math Container Embedded Component

by: Adam Ledger 360 Maple
May 26 2018
0  2

In as much as the embedded component suite is a brilliant tool for the custom design of online educational programs, student tests, etc, my purposes are orientated around encouraging and assisting of self directed investigation, with the utilization of the packages of maple, but in a manner that allows the user to neglect the requirement to have any knowledge of maple code itself, allowing them to focus entirely on their discipline of choice. 

So because the content the user will enter into the interface I am designing is naturally going to be quite variant from individual to individual, one of the  necessary properties that does not exist is for the math containers to have the option of being resizeable at the discretion of the user.

At the moment, I have added buttons that allow the user to resize the window by pressing one of four buttons entitled "increase height". "decrease height", "increase width" and "decrease height". This will suffice for my first working prototype but i just feel that it would be much neater if it were possible to resize the math component directly, and some option of the neighbouring embedded components to either shift their position accordingly, or maintain rectilinear alignment with the greater proportion of other components by all embedded components shifting accordingly when one math container is resized.

 

Maple splash screen

by: Christopher2222 6005 Maple
April 07 2018
2  4

The Maple splash screen needs a makeover, it's not too exciting so looking at the maplesoft website the opening screen has an image that would have been rather fitting for the Maple 2018 splash screen.  Here's the image I'm talking about.