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ENH: improve analytic eigen for small off-diagonals
This commit is contained in:
Kutalmis Bercin
committed by
Andrew Heather
Andrew Heather
parent
b476dd92e6
commit
6576397e81
@ -62,9 +62,24 @@ unsigned nFail_ = 0;
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// Create a random symmTensor
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symmTensor makeRandomContainer(Random& rnd)
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{
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symmTensor A(Zero);
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std::generate(A.begin(), A.end(), [&]{ return rnd.GaussNormal<scalar>(); });
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return A;
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symmTensor T(Zero);
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std::generate(T.begin(), T.end(), [&]{ return rnd.GaussNormal<scalar>(); });
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return T;
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}
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// Create a symmTensor based on a given value
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template<class Type>
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typename std::enable_if
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<
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std::is_same<floatScalar, Type>::value ||
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std::is_same<doubleScalar, Type>::value,
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symmTensor
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>::type makeContainer(const Type val)
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{
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symmTensor T(Zero);
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std::fill(T.begin(), T.end(), val);
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return T;
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}
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@ -83,11 +98,11 @@ typename std::enable_if
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const word& msg,
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const Type& x,
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const Type& y,
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const scalar relTol = 1e-8, //<! are values the same within 8 decimals
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const scalar absTol = 0 //<! useful for cmps near zero
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const scalar absTol = 0, //<! useful for cmps near zero
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const scalar relTol = 1e-8 //<! are values the same within 8 decimals
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)
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{
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Info<< msg << x << endl;
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Info<< msg << x << "?=" << y << endl;
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unsigned nFail = 0;
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@ -121,11 +136,11 @@ typename std::enable_if
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const word& msg,
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const Type& x,
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const Type& y,
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const scalar relTol = 1e-8,
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const scalar absTol = 0
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const scalar absTol = 0,
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const scalar relTol = 1e-8
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)
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{
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Info<< msg << x << endl;
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Info<< msg << x << "?=" << y << endl;
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unsigned nFail = 0;
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@ -338,7 +353,6 @@ void test_global_funcs(Type)
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Type(1), Type(0),
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Type(-0.05555556)
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),
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1e-8,
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1e-8
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);
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cmp
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@ -351,7 +365,6 @@ void test_global_funcs(Type)
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Type(1), Type(0),
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Type(-0.05555556)
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),
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1e-8,
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1e-8
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);
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cmp(" First invariant = ", invariantI(sT), Type(-3));
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@ -514,21 +527,21 @@ void test_global_opers(Type)
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// Return false if given eigenvalues fail to satisy eigenvalue relations
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// Relations: (Beauregard & Fraleigh (1973), ISBN 0-395-14017-X, p. 307)
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void test_eigenvalues(const symmTensor& sT, const vector& EVals)
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void test_eigenvalues(const symmTensor& T, const vector& EVals)
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{
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{
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const scalar determinant = det(sT);
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const scalar determinant = det(T);
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const scalar EValsProd = EVals.x()*EVals.y()*EVals.z();
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cmp("# Product of eigenvalues = det(sT):", EValsProd, determinant);
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cmp("# Product of eigenvalues = det(T):", EValsProd, determinant, 1e-6);
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}
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{
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const scalar trace = tr(sT);
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const scalar trace = tr(T);
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scalar EValsSum = 0.0;
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for (const auto& val : EVals)
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{
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EValsSum += val;
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}
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cmp("# Sum of eigenvalues = trace(sT):", EValsSum, trace);
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cmp("# Sum of eigenvalues = trace(T):", EValsSum, trace);
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}
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}
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@ -537,7 +550,7 @@ void test_eigenvalues(const symmTensor& sT, const vector& EVals)
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// fails to satisfy the characteristic equation
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void test_characteristic_equation
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(
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const symmTensor& sT,
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const symmTensor& T,
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const vector& EVals,
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const tensor& EVecs
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)
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@ -547,39 +560,39 @@ void test_characteristic_equation
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{
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Info<< "EVal = " << EVals[dir] << nl
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<< "EVec = " << EVecs.row(dir) << endl;
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const vector leftSide(sT & EVecs.row(dir));
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const vector leftSide(T & EVecs.row(dir));
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const vector rightSide(EVals[dir]*EVecs.row(dir));
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const vector X(leftSide - rightSide);
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for (const auto x : X)
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{
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cmp(" (sT & EVec - EVal*EVec) = 0:", x, 0.0, 1e-8, 1e-6);
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cmp(" (T & EVec - EVal*EVec) = 0:", x, 0.0, 1e-5);
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}
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}
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}
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// Return false if the eigen functions fail to satisfy relations
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void test_eigen_funcs(const symmTensor& sT)
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void test_eigen_funcs(const symmTensor& T)
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{
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Info<< "# Operand:" << nl
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<< " symmTensor = " << sT << nl;
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<< " symmTensor = " << T << nl;
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Info<< "# Return eigenvalues of a given symmTensor:" << nl;
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const vector EVals(eigenValues(sT));
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const vector EVals(eigenValues(T));
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Info<< EVals << endl;
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test_eigenvalues(sT, EVals);
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test_eigenvalues(T, EVals);
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Info<< "# Return eigenvectors of a given symmTensor corresponding to"
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<< " given eigenvalues:" << nl;
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const tensor EVecs0(eigenVectors(sT, EVals));
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const tensor EVecs0(eigenVectors(T, EVals));
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Info<< EVecs0 << endl;
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test_characteristic_equation(sT, EVals, EVecs0);
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test_characteristic_equation(T, EVals, EVecs0);
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Info<< "# Return eigenvectors of a given symmTensor by computing"
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<< " the eigenvalues of the symmTensor in the background:" << nl;
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const tensor EVecs1(eigenVectors(sT));
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const tensor EVecs1(eigenVectors(T));
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Info<< EVecs1 << endl;
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}
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@ -629,7 +642,7 @@ int main()
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run_tests(types, typeID);
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Info<< nl << " ## Test SymmTensor<scalar> eigen functions: ##" << nl;
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Info<< nl << " ## Test symmTensor eigen functions: ##" << nl;
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const label numberOfTests = 10000;
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Random rndGen(1234);
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@ -640,14 +653,13 @@ int main()
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}
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{
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Info<< nl << " ## Test eigen functions by a zero tensor: ##"<< nl;
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Info<< nl << " ## A zero symmTensor: ##"<< nl;
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const symmTensor zeroT(Zero);
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test_eigen_funcs(zeroT);
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}
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{
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Info<< nl
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<< " ## Test eigen functions by a tensor consisting of"
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<< " 2 repeated eigenvalues: ##"
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<< " ## A symmTensor with 2 repeated eigenvalues: ##"
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<< nl;
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const symmTensor T
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(
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@ -659,8 +671,7 @@ int main()
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}
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{
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Info<< nl
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<< " ## Test eigen functions by a tensor consisting of"
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<< " 3 repeated eigenvalues: ##"
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<< " ## A symmTensor with 3 repeated eigenvalues: ##"
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<< nl;
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const symmTensor T
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(
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@ -671,22 +682,7 @@ int main()
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test_eigen_funcs(T);
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}
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{
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Info<< nl
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<< " ## Test eigen functions by a tensor consisting of"
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<< " SMALL off-diagonal elements: ##"
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<< nl;
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for (label i = 0; i < numberOfTests; ++i)
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{
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symmTensor T(makeRandomContainer(rndGen));
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T.xy() = SMALL*rndGen.GaussNormal<scalar>();
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T.xz() = SMALL*rndGen.GaussNormal<scalar>();
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T.yz() = SMALL*rndGen.GaussNormal<scalar>();
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test_eigen_funcs(T);
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}
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}
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{
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Info<< nl << " ## Test eigen functions by a stiff tensor: ##" << nl;
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Info<< nl << " ## A stiff symmTensor: ##" << nl;
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const symmTensor stiff
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(
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pow(10.0, 10), pow(10.0, 8), pow(10.0, -8),
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@ -695,6 +691,165 @@ int main()
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);
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test_eigen_funcs(stiff);
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}
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{
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Info<< nl
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<< " ## Random symmTensors with tiny off-diag elements: ##"
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<< nl;
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const List<scalar> epsilons
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({
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0, SMALL, Foam::sqrt(SMALL), sqr(SMALL), Foam::cbrt(SMALL),
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-SMALL, -Foam::sqrt(SMALL), -sqr(SMALL), -Foam::cbrt(SMALL)
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});
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for (label i = 0; i < numberOfTests; ++i)
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{
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for (const auto& eps : epsilons)
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{
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{
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symmTensor T(makeRandomContainer(rndGen));
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T.xy() = eps*rndGen.GaussNormal<scalar>();
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test_eigen_funcs(T);
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}
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{
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symmTensor T(makeRandomContainer(rndGen));
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T.xz() = eps*rndGen.GaussNormal<scalar>();
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test_eigen_funcs(T);
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}
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{
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symmTensor T(makeRandomContainer(rndGen));
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T.yz() = eps*rndGen.GaussNormal<scalar>();
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test_eigen_funcs(T);
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}
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{
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symmTensor T(makeRandomContainer(rndGen));
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T.xy() = eps*rndGen.GaussNormal<scalar>();
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T.xz() = eps*rndGen.GaussNormal<scalar>();
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test_eigen_funcs(T);
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}
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{
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symmTensor T(makeRandomContainer(rndGen));
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T.xy() = eps*rndGen.GaussNormal<scalar>();
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T.yz() = eps*rndGen.GaussNormal<scalar>();
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test_eigen_funcs(T);
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}
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{
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symmTensor T(makeRandomContainer(rndGen));
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T.xz() = eps*rndGen.GaussNormal<scalar>();
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T.yz() = eps*rndGen.GaussNormal<scalar>();
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test_eigen_funcs(T);
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}
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{
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symmTensor T(makeRandomContainer(rndGen));
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T.xy() = eps*rndGen.GaussNormal<scalar>();
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T.xz() = eps*rndGen.GaussNormal<scalar>();
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T.yz() = eps*rndGen.GaussNormal<scalar>();
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test_eigen_funcs(T);
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}
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{
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symmTensor T(makeRandomContainer(rndGen));
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T.xy() = eps;
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T.xz() = eps;
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T.yz() = eps;
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test_eigen_funcs(T);
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}
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{
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symmTensor T(makeRandomContainer(rndGen));
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T.xy() = eps;
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T.xz() = eps;
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T.yz() = eps;
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T.zz() = eps;
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test_eigen_funcs(T);
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}
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{
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symmTensor T(makeRandomContainer(rndGen));
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T.xy() = 0;
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T.xz() = eps*rndGen.GaussNormal<scalar>();
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T.yz() = 0;
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test_eigen_funcs(T);
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}
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}
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}
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}
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#if 0
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// Numerical diagonalisation of 2x2 or 3x3 matrices with analytic methods
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// are, like the methods currently being used in OpenFOAM, inherently error
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// prone. Despite its speed, the analytic methods may becomes inaccurate or
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// may even fail completely if the matrix entries differ greatly in
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// magnitude, particularly with large off-diagonal elements.
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// The remedy is to use iterative or hybrid analytic/iterative methods
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// such as published here (for 3x3/2x2 matrices):
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// (Kopp, 2008) arXiv.org: physics/0610206
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// mpi-hd.mpg.de/personalhomes/globes/3x3/index.html
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{
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Info<< nl << " ## symmTensors consisting machine epsilons: ##" << nl;
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Info<< " # floatScalar" << nl;
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const List<floatScalar> floatEpsilons
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({
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floatScalarGREAT, floatScalarVGREAT, floatScalarROOTVGREAT,
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floatScalarSMALL, floatScalarVSMALL, floatScalarROOTVSMALL,
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Foam::sqrt(floatScalarSMALL), 0
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});
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for (const auto& eps : floatEpsilons)
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{
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const symmTensor T(makeContainer(eps));
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test_eigen_funcs(T);
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}
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Info<< " # doubleScalar" << nl;
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const List<doubleScalar> doubleEpsilons
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({
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doubleScalarGREAT, doubleScalarROOTVGREAT, // doubleVGREAT fails
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doubleScalarSMALL, doubleScalarVSMALL, doubleScalarROOTVSMALL,
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Foam::sqrt(doubleScalarSMALL), 0
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});
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for (const auto& eps : doubleEpsilons)
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{
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const symmTensor T(makeContainer(eps));
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test_eigen_funcs(T);
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}
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}
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{
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Info<< nl
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<< " ## Random symmTensors with machine eps off-diag elmes: ##"
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<< nl;
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const List<floatScalar> floatEpsilons
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({
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floatScalarGREAT, floatScalarVGREAT, floatScalarROOTVGREAT,
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floatScalarSMALL, floatScalarVSMALL, floatScalarROOTVSMALL
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});
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const List<doubleScalar> doubleEpsilons
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({
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doubleScalarGREAT, doubleScalarVGREAT, doubleScalarROOTVGREAT,
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doubleScalarSMALL, doubleScalarVSMALL, doubleScalarROOTVSMALL
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});
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for (label i = 0; i < numberOfTests; ++i)
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{
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symmTensor T(makeRandomContainer(rndGen));
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for (const auto& eps : floatEpsilons)
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{
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T.xy() = eps;
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T.xz() = eps;
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T.yz() = eps;
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test_eigen_funcs(T);
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}
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for (const auto& eps : doubleEpsilons)
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{
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T.xy() = eps;
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T.xz() = eps;
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T.yz() = eps;
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test_eigen_funcs(T);
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}
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}
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}
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#endif
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if (nFail_)
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