mbd_formulas Module

Common formulas used at multiple places.

Contents

Functions

omega_qho alpha_dyn_qho C6_from_alpha sigma_selfint scale_with_ratio

Functions

public function omega_qho(C6, alpha, domega, grad) result(omega)

$$\omega=\frac{4C_6}{3\alpha_{0}^2},\qquad \partial\omega=\omega\left( \frac{\partial C_6}{C_6}-\frac{2\partial\alpha_0}{\alpha_0} \right)$$

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=dp),	intent(in)			::	C6(:)
real(kind=dp),	intent(in)			::	alpha(:)
type(grad_t),	intent(out),	optional		::	domega
type(grad_request_t),	intent(in),	optional		::	grad

Return Value real(kind=dp) (size(C6))

public function alpha_dyn_qho(alpha_0, omega, freq, dalpha, grad) result(alpha)

$$\alpha(\mathrm iu)=\frac{\alpha_0}{1+u^2/\omega^2},\qquad \partial\alpha(\mathrm iu)=\alpha(\mathrm iu)\left( \frac{\partial\alpha_0}{\alpha_0}+ \frac2\omega\frac{\partial\omega}{1+\omega^2/u^2} \right)$$

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=dp),	intent(in)			::	alpha_0(:)
real(kind=dp),	intent(in)			::	omega(:)
type(quad_pt_t),	intent(in)			::	freq(0:)
type(grad_t),	intent(out),		allocatable	::	dalpha(:)
type(grad_request_t),	intent(in)			::	grad

Return Value real(kind=dp) (size(alpha_0),0:ubound(freq,1))

public function C6_from_alpha(alpha, freq, dC6_dalpha, grad) result(C6)

$$\bar C_6=\frac3\pi\int_0^\infty\mathrm du\,\bar\alpha(u)^2,\qquad \partial\bar C_6=\frac6\pi\int_0^\infty\mathrm du \bar\alpha(u)\partial\bar\alpha(u)$$

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=dp),	intent(in)			::	alpha(:,0:)
type(quad_pt_t),	intent(in)			::	freq(0:)
real(kind=dp),	intent(out),	optional	allocatable	::	dC6_dalpha(:,:)
logical,	intent(in),	optional		::	grad

Return Value real(kind=dp) (size(alpha,1))

public function sigma_selfint(alpha, dsigma_dalpha, grad) result(sigma)

$$\begin{gathered} \sigma_i(u)=\left(\frac13\sqrt{\frac2\pi}\alpha_i(u)\right)^{\frac13},\qquad \partial\sigma_i=\sigma_i\frac{\partial\alpha_i}{3\alpha_i} \\ \sigma_{ij}(u)=\sqrt{\sigma_i(u)^2+\sigma_j(u)^2},\qquad \partial\sigma_{ij}= \frac{\sigma_i\partial\sigma_i+\sigma_j\partial\sigma_j}{\sigma_{ij}} \end{gathered}$$

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=dp),	intent(in)			::	alpha(:)
real(kind=dp),	intent(out),	optional	allocatable	::	dsigma_dalpha(:)
logical,	intent(in),	optional		::	grad

Return Value real(kind=dp) (size(alpha))

public function scale_with_ratio(x, yp, y, q, dx, grad) result(xp)

$$x'=x\left(\frac{y'}y\right)^q,\qquad \partial x'=x\left( \frac{\partial x}x+ q\frac{\partial y'}{y'}- q\frac{\partial y}{y} \right)$$

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=dp),	intent(in)			::	x(:)
real(kind=dp),	intent(in)			::	yp(:)
real(kind=dp),	intent(in)			::	y(:)
real(kind=dp),	intent(in)			::	q
type(grad_t),	intent(out),	optional		::	dx
type(grad_request_t),	intent(in),	optional		::	grad

Return Value real(kind=dp) (size(x))