Namespace to group the methods used in testing of the library. More...

Functions
double	getExactIntegrationRefQuad (size_t alpha, size_t beta)
	Computes integration of polynomial exactly over reference quadrangle. More...

double	getExactIntegrationRefTet (size_t alpha, size_t beta, size_t theta)
	Computes integration of polynomial exactly over reference tetrahedral. More...

double	getExactIntegrationRefTri (size_t alpha, size_t beta)
	Computes integration of polynomial exactly over reference triangle. More...

double	getNChooseR (size_t n, size_t r)
	Computes ${n\choose r}$ "n choose r". More...

void	testFracture ()
	Perform test on Fracture class and check if bit-wise functions are working correctly.

void	testLineElem (size_t n)
	Perform test on quadrature points on line elements. More...

void	testQuadElem (size_t n)
	Perform test on quadrature points on quadrangle elements. More...

void	testTetElem (size_t n)
	Perform test on quadrature points on tetrahedral elements. More...

void	testTriElem (size_t n)
	Perform test on quadrature points on triangle elements. More...

void	testTriElemTime (size_t n, size_t N)
	Computes the time needed when quad data for elements are stored and when they are computed as and when needed. More...

Detailed Description

Namespace to group the methods used in testing of the library.

Function Documentation

◆ getExactIntegrationRefQuad()

double test::getExactIntegrationRefQuad	(	size_t	alpha,
		size_t	beta
	)

Computes integration of polynomial exactly over reference quadrangle.

Given $f(s,t) = s^\alpha\, t^\beta$ , the exact integration is given by

$I_{exact} = \int_0^1 \int_0^{1-s} s^\alpha\, t^\beta \, dt\, ds.$

If either $\alpha$ or $\beta$ are odd number then $I_{exact} = 0$ . Otherwise, $I_{exact} = \frac{4}{(\alpha +1) (\beta+1)}$ .

Parameters

alpha	Polynomial order in variable s
beta	Polynomial order in variable t

Returns: I Exact integration of $f(s,t) = s^\alpha\, t^\beta$

Referenced by testQuadElem().

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◆ getExactIntegrationRefTet()

double test::getExactIntegrationRefTet	(	size_t	alpha,
		size_t	beta,
		size_t	theta
	)

Computes integration of polynomial exactly over reference tetrahedral.

Parameters

alpha	Polynomial order in variable s
beta	Polynomial order in variable t
theta	Polynomial order in variable r

Returns: I Exact integration of $f(s,t) = s^\alpha\, t^\beta \, r^\theta$

Referenced by testTetElem().

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◆ getExactIntegrationRefTri()

double test::getExactIntegrationRefTri	(	size_t	alpha,
		size_t	beta
	)

Computes integration of polynomial exactly over reference triangle.

Given $f(s,t) = s^\alpha\, t^\beta$ , the exact integration is given by

$I_{exact} = \int_0^1 \int_0^{1-s} s^\alpha\, t^\beta \, dt\, ds = \sum_{i=0}^{\beta+1} (-1)^i \frac{{{\beta + 1} \choose i}}{(\alpha + i +1) (\beta + 1)},$

where

${a \choose b} = \frac{a (a-1) (a-2) ... (a-b+1)}{1*2*3 ... *b}.$

We have ${a \choose 0} = 1$ so that term for $ i=0$ is not zero. Above formula gives the exact value of integral of $f(s,t) = s^\alpha\, t^\beta$ over reference triangle.

Parameters

alpha	Polynomial order in variable s
beta	Polynomial order in variable t

Returns: I Exact integration of $f(s,t) = s^\alpha\, t^\beta$

Referenced by testTriElem().

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◆ getNChooseR()

double test::getNChooseR	(	size_t	n,
		size_t	r
	)

Computes ${n\choose r}$ "n choose r".

Computes formula

${a \choose b} = \frac{a (a-1) (a-2) ... (a-b+1)}{1*2*3 ... *b}.$

Parameters

n	Number
r	Number which is smaller or equal to n

Returns: Value Value of "n choose r"

Referenced by getExactIntegrationRefTet(), and getExactIntegrationRefTri().

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◆ testLineElem()

void test::testLineElem ( size_t n )

Perform test on quadrature points on line elements.

This function performs accuracy test of the quadrature points for integration over reference line with vertices at {-1, 1}. List of tests are as follows:

Computes quadrature points of the given order, writes them to the file, and checks if the sum of quadrature weights is equal to 0.5 (area of reference triangle).

Also tests the exactness of the integration of the polynomial up to given order. Suppose $ n$ is the order of quadrature point, then we test if the integration of the function $f(s,t) = s^\alpha\, t^\beta$ is exact for $\alpha$ and $\beta$ such that $\alpha+\beta \leq n$ . The exact integration of function $ f$ over reference triangle is

$I_{exact} = \int_0^1 \int_0^{1-s} s^\alpha\, t^\beta \, dt\, ds = \sum_{i=0}^{\beta+1} (-1)^i \frac{{{\beta + 1} \choose i}}{(\alpha + i +1) (\beta + 1)},$

where

${a \choose b} = \frac{a (a-1) (a-2) ... (a-b+1)}{1*2*3 ... *b}.$

We have ${a \choose 0} = 1$ so that term for $ i=0$ is not zero. Above formula gives the exact value of integral of $f(s,t) = s^\alpha\, t^\beta$ over reference triangle. Approximation by quadrature point is as follows

$I_{approx} = \sum_{q=1}^{Q} w_q f(s_q, t_q)$

where $Q$ is the total number of quad points, $ w_q$ and $(s_q, t_q)$ are the $q^{th}$ quad weight and point. In this test, we compare $I_{exact}$ and $I_{approx}$ and report problem if both do not match.

Test the accuracy for simple quadrangle mesh on square domain [0,1]^2. Exact integration of polynomial $f(x,y) = x^\alpha \, y^\beta$ on is given by
$I_{exact} = \frac{1}{(\alpha+1) (\beta+1)}.$
We compare above with the approximation computed from the quadrature points. We consider $\alpha + \beta \leq n$ where is the order of approximation we are testing.

Parameters

n	Order of quadrature point approximation

Todo:: Implement this method!

◆ testQuadElem()

void test::testQuadElem ( size_t n )

Perform test on quadrature points on quadrangle elements.

This function performs accuracy test of the quadrature points for integration over reference triangle with vertices at {(0,0), (1,0), (0,1)}. List of tests are as follows:

Computes quadrature points of the given order, writes them to the file, and checks if the sum of quadrature weights is equal to 2, i.e. area of reference quadrangle element.

Also tests the exactness of the integration of the polynomial up to given order. Suppose $ n$ is the order of quadrature point, then we test if the integration of the function $f(s,t) = s^\alpha\, t^\beta$ is exact for $\alpha$ and $\beta$ such that $\alpha+\beta \leq n$ . The exact integration of function $ f$ over reference triangle is

$I_{exact} = \int_0^1 \int_0^{1-s} s^\alpha\, t^\beta \, dt\, ds = \sum_{i=0}^{\beta+1} (-1)^i \frac{{{\beta + 1} \choose i}}{(\alpha + i +1) (\beta + 1)},$

where

${a \choose b} = \frac{a (a-1) (a-2) ... (a-b+1)}{1*2*3 ... *b}.$

We have ${a \choose 0} = 1$ so that term for $ i=0$ is not zero. Above formula gives the exact value of integral of $f(s,t) = s^\alpha\, t^\beta$ over reference triangle. Approximation by quadrature point is as follows

$I_{approx} = \sum_{q=1}^{Q} w_q f(s_q, t_q)$

where $Q$ is the total number of quad points, $ w_q$ and $(s_q, t_q)$ are the $q^{th}$ quad weight and point. In this test, we compare $I_{exact}$ and $I_{approx}$ and report problem if both do not match.

Test the accuracy for simple quadrangle mesh on square domain [0,1]^2. Exact integration of polynomial $f(x,y) = x^\alpha \, y^\beta$ on is given by
$I_{exact} = \frac{1}{(\alpha+1) (\beta+1)}.$
We compare above with the approximation computed from the quadrature points. We consider $\alpha + \beta \leq n$ where is the order of approximation we are testing.

Parameters

n	Order of quadrature point approximation

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◆ testTetElem()

void test::testTetElem ( size_t n )

Perform test on quadrature points on tetrahedral elements.

Parameters

n	Order of quadrature point approximation

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◆ testTriElem()

void test::testTriElem ( size_t n )

Perform test on quadrature points on triangle elements.

This function performs accuracy test of the quadrature points for integration over reference triangle with vertices at {(0,0), (1,0), (0,1)}. List of tests are as follows:

Computes quadrature points of the given order, writes them to the file, and checks if the sum of quadrature weights is equal to 0.5, i.e. area of reference triangle.

Also tests the exactness of the integration of the polynomial up to given order. Suppose $ n$ is the order of quadrature point, then we test if the integration of the function $f(s,t) = s^\alpha\, t^\beta$ is exact for $\alpha$ and $\beta$ such that $\alpha+\beta \leq n$ . The exact integration of function $ f$ over reference triangle is

$I_{exact} = \int_0^1 \int_0^{1-s} s^\alpha\, t^\beta \, dt\, ds = \sum_{i=0}^{\beta+1} (-1)^i \frac{{{\beta + 1} \choose i}}{(\alpha + i +1) (\beta + 1)},$

where

${a \choose b} = \frac{a (a-1) (a-2) ... (a-b+1)}{1*2*3 ... *b}.$

We have ${a \choose 0} = 1$ so that term for $ i=0$ is not zero. Above formula gives the exact value of integral of $f(s,t) = s^\alpha\, t^\beta$ over reference triangle. Approximation by quadrature point is as follows

$I_{approx} = \sum_{q=1}^{Q} w_q f(s_q, t_q)$

where $Q$ is the total number of quad points, $ w_q$ and $(s_q, t_q)$ are the $q^{th}$ quad weight and point. In this test, we compare $I_{exact}$ and $I_{approx}$ and report problem if both do not match.

Test the accuracy for simple triangular mesh on square domain [0,1]^2. Exact integration of polynomial $f(x,y) = x^\alpha \, y^\beta$ on is given by
$I_{exact} = \frac{1}{(\alpha+1) (\beta+1)}.$
We compare above with the approximation computed from the quadrature points. We consider $\alpha + \beta \leq n$ where is the order of approximation we are testing.

Parameters

n	Order of quadrature point approximation

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◆ testTriElemTime()

void test::testTriElemTime	(	size_t	n,
		size_t	N
	)

Computes the time needed when quad data for elements are stored and when they are computed as and when needed.

This function allocates dummy elements and test how much time it is required to do computation when the quad data are stored for each element and when the quad data are computed.

Parameters

n	Order of quadrature point approximation
N	Number of elements on which this test is performed

Functions

Detailed Description

Function Documentation

◆ getExactIntegrationRefQuad()

◆ getExactIntegrationRefTet()

◆ getExactIntegrationRefTri()

◆ getNChooseR()

◆ testLineElem()

◆ testQuadElem()

◆ testTetElem()

◆ testTriElem()

◆ testTriElemTime()