From da69a7da026b730a8989267ef39c0c74d785cd26 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Sat, 11 Jan 2025 10:01:40 +0000 Subject: [PATCH] build based on 13686e9 --- dev/.documenter-siteinfo.json | 2 +- dev/index.html | 8 ++++---- dev/tuto/index.html | 2 +- 3 files changed, 6 insertions(+), 6 deletions(-) diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index 8531e75..638aff2 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2025-01-11T09:56:39","documenter_version":"1.8.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2025-01-11T10:01:37","documenter_version":"1.8.0"}} \ No newline at end of file diff --git a/dev/index.html b/dev/index.html index e323238..894579a 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,16 +1,16 @@ -Home · BloodFlowTrixi.jl
GitHub

BloodFlowTrixi

Documentation for BloodFlowTrixi.

BloodFlowTrixi.BloodFlowTrixi — Module
Package BloodFlowTrixi v0.0.1

This package implements 1D and 2D blood flow models for arterial circulation using Trixi.jl, enabling efficient numerical simulation and analysis.

Docs under https://yolhan83.github.io/BloodFlowTrixi.jl

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BloodFlowTrixi.BloodFlowEquations1D — Type
BloodFlowEquations1D(;h,rho=1.0,xi=0.25,nu=0.04)

Blood Flow equations in one space dimension. This model describes the dynamics of blood flow along a compliant artery using one-dimensional equations derived from the Navier-Stokes equations. The equations account for conservation of mass and momentum, incorporating the effect of arterial compliance and frictional losses.

The governing equations are given by

\[\left\{\begin{aligned} +Home · BloodFlowTrixi.jl

GitHub

BloodFlowTrixi

Documentation for BloodFlowTrixi.

BloodFlowTrixi.BloodFlowTrixi — Module
Package BloodFlowTrixi v0.0.1

This package implements 1D and 2D blood flow models for arterial circulation using Trixi.jl, enabling efficient numerical simulation and analysis.

Docs under https://yolhan83.github.io/BloodFlowTrixi.jl

source
BloodFlowTrixi.BloodFlowEquations1D — Type
BloodFlowEquations1D(;h,rho=1.0,xi=0.25,nu=0.04)

Blood Flow equations in one space dimension. This model describes the dynamics of blood flow along a compliant artery using one-dimensional equations derived from the Navier-Stokes equations. The equations account for conservation of mass and momentum, incorporating the effect of arterial compliance and frictional losses.

The governing equations are given by

\[\left\{\begin{aligned} \frac{\partial a}{\partial t} + \frac{\partial}{\partial x}(Q) &= 0 \\ \frac{\partial Q}{\partial t} + \frac{\partial}{\partial x}\left(\frac{Q^2}{A} + A P(a)\right) &= P(a) \frac{\partial A}{\partial x} - 2 \pi R k \frac Q {A}\\ P(a) &= P_{ext} + \frac{Eh\sqrt{\pi}}{1-\xi^2}\frac{\sqrt{A} - \sqrt{A_0}}{A_0} \\ R &= \sqrt{\frac{A}{\pi}} -\end{aligned}\right.\]

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BloodFlowTrixi.BloodFlowEquations2D — Type
BloodFlowEquations2D(;h,rho=1.0,xi=0.25)

Defines the two-dimensional blood flow equations derived from the Navier-Stokes equations in curvilinear coordinates under the thin-artery assumption. This model describes the dynamics of blood flow along a compliant artery in two spatial dimensions (s, θ).

Parameters

  • h::T: Wall thickness of the artery.
  • rho::T: Fluid density (default 1.0).
  • xi::T: Poisson's ratio (default 0.25).
  • nu::T: Viscosity coefficient.

The governing equations account for conservation of mass and momentum, incorporating the effects of arterial compliance, curvature, and frictional losses.

\[\left\{\begin{aligned} +\end{aligned}\right.\]

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BloodFlowTrixi.BloodFlowEquations2D — Type
BloodFlowEquations2D(;h,rho=1.0,xi=0.25)

Defines the two-dimensional blood flow equations derived from the Navier-Stokes equations in curvilinear coordinates under the thin-artery assumption. This model describes the dynamics of blood flow along a compliant artery in two spatial dimensions (s, θ).

Parameters

  • h::T: Wall thickness of the artery.
  • rho::T: Fluid density (default 1.0).
  • xi::T: Poisson's ratio (default 0.25).
  • nu::T: Viscosity coefficient.

The governing equations account for conservation of mass and momentum, incorporating the effects of arterial compliance, curvature, and frictional losses.

\[\left\{\begin{aligned} \frac{\partial a}{\partial t} + \frac{\partial}{\partial \theta}\left( \frac{Q_{R\theta}}{A} \right) + \frac{\partial}{\partial s}(Q_s) &= 0 \\ \frac{\partial Q_{R\theta}}{\partial t} + \frac{\partial}{\partial \theta}\left(\frac{Q_{R\theta}^2}{2A^2} + A P(a)\right) + \frac{\partial}{\partial s}\left( \frac{Q_{R\theta}Q_s}{A} \right) &= P(a) \frac{\partial A}{\partial \theta} - 2 R k \frac{Q_{R\theta}}{A} + \frac{2R}{3} \mathcal{C}\sin \theta \frac{Q_s^2}{A} \\ \frac{\partial Q_{s}}{\partial t} + \frac{\partial}{\partial \theta}\left(\frac{Q_{R\theta} Q_s}{A^2} \right) + \frac{\partial}{\partial s}\left( \frac{Q_s^2}{A} - \frac{Q_{R\theta}^2}{2A^2} + A P(a) \right) &= P(a) \frac{\partial A}{\partial s} - R k \frac{Q_s}{A} - \frac{2R}{3} \mathcal{C}\sin \theta \frac{Q_s Q_{R\theta}}{A^2} \\ P(a) &= P_{ext} + \frac{Eh}{\sqrt{2}\left(1-\xi^2\right)}\frac{\sqrt{A} - \sqrt{A_0}}{A_0} \\ R &= \sqrt{2A} -\end{aligned}\right.\]

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Trixi.DissipationLocalLaxFriedrichs — Method
(dissipation::Trixi.DissipationLocalLaxFriedrichs)(u_ll, u_rr, orientation_or_normal_direction, eq::BloodFlowEquations1D)

Calculates the dissipation term using the Local Lax-Friedrichs method.

Parameters

  • u_ll: Left state vector.
  • u_rr: Right state vector.
  • orientation_or_normal_direction: Orientation or normal direction.
  • eq: Instance of BloodFlowEquations1D.

Returns

Dissipation vector.

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BloodFlowTrixi.boundary_condition_outflow — Method
boundary_condition_outflow(u_inner, orientation_or_normal, direction, x, t, surface_flux_function, eq::BloodFlowEquations1D)

Implements the outflow boundary condition, assuming that there is no reflection at the boundary.

Parameters

  • u_inner: State vector inside the domain near the boundary.
  • orientation_or_normal: Normal orientation of the boundary.
  • direction: Integer indicating the direction of the boundary.
  • x: Position vector.
  • t: Time.
  • surface_flux_function: Function to compute flux at the boundary.
  • eq: Instance of BloodFlowEquations1D.

Returns

Computed boundary flux.

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BloodFlowTrixi.boundary_condition_pressure_in — Method
boundary_condition_pressure_in(u_inner, orientation_or_normal, direction, x, t, surface_flux_function, eq::BloodFlowEquations1D)

Implements a pressure inflow boundary condition where the inflow pressure varies with time.

Parameters

  • u_inner: State vector inside the domain near the boundary.
  • orientation_or_normal: Normal orientation of the boundary.
  • direction: Integer indicating the boundary direction.
  • x: Position vector.
  • t: Time scalar.
  • surface_flux_function: Function to compute flux at the boundary.
  • eq: Instance of BloodFlowEquations1D.

Returns

Computed boundary flux with inflow pressure specified by:

\[P_{in} = \begin{cases} +\end{aligned}\right.\]

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Trixi.DissipationLocalLaxFriedrichs — Method
(dissipation::Trixi.DissipationLocalLaxFriedrichs)(u_ll, u_rr, orientation_or_normal_direction, eq::BloodFlowEquations1D)

Calculates the dissipation term using the Local Lax-Friedrichs method.

Parameters

  • u_ll: Left state vector.
  • u_rr: Right state vector.
  • orientation_or_normal_direction: Orientation or normal direction.
  • eq: Instance of BloodFlowEquations1D.

Returns

Dissipation vector.

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BloodFlowTrixi.boundary_condition_outflow — Method
boundary_condition_outflow(u_inner, orientation_or_normal, direction, x, t, surface_flux_function, eq::BloodFlowEquations1D)

Implements the outflow boundary condition, assuming that there is no reflection at the boundary.

Parameters

  • u_inner: State vector inside the domain near the boundary.
  • orientation_or_normal: Normal orientation of the boundary.
  • direction: Integer indicating the direction of the boundary.
  • x: Position vector.
  • t: Time.
  • surface_flux_function: Function to compute flux at the boundary.
  • eq: Instance of BloodFlowEquations1D.

Returns

Computed boundary flux.

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BloodFlowTrixi.boundary_condition_pressure_in — Method
boundary_condition_pressure_in(u_inner, orientation_or_normal, direction, x, t, surface_flux_function, eq::BloodFlowEquations1D)

Implements a pressure inflow boundary condition where the inflow pressure varies with time.

Parameters

  • u_inner: State vector inside the domain near the boundary.
  • orientation_or_normal: Normal orientation of the boundary.
  • direction: Integer indicating the boundary direction.
  • x: Position vector.
  • t: Time scalar.
  • surface_flux_function: Function to compute flux at the boundary.
  • eq: Instance of BloodFlowEquations1D.

Returns

Computed boundary flux with inflow pressure specified by:

\[P_{in} = \begin{cases} 2 \times 10^4 \sin^2(\pi t / 0.125) & \text{if } t < 0.125 \\ 0 & \text{otherwise} -\end{cases}\]

The corresponding inflow area A_{in} is computed using the inverse pressure relation, and the boundary state is constructed accordingly.

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BloodFlowTrixi.boundary_condition_slip_wall — Method
boundary_condition_slip_wall(u_inner, orientation_or_normal, direction, x, t, surface_flux_function, eq::BloodFlowEquations1D)

Implements a slip wall boundary condition where the normal component of velocity is reflected.

Parameters

  • u_inner: State vector inside the domain near the boundary.
  • orientation_or_normal: Normal orientation of the boundary.
  • direction: Integer indicating the direction of the boundary.
  • x: Position vector.
  • t: Time.
  • surface_flux_function: Function to compute flux at the boundary.
  • eq: Instance of BloodFlowEquations1D.

Returns

Computed boundary flux at the slip wall.

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BloodFlowTrixi.flux_nonconservative — Method
flux_nonconservative(u_ll, u_rr, orientation::Integer, eq::BloodFlowEquations1D)

Computes the non-conservative flux for the model, used for handling discontinuities in pressure.

Parameters

  • u_ll: Left state vector.
  • u_rr: Right state vector.
  • orientation::Integer: Orientation index.
  • eq: Instance of BloodFlowEquations1D.

Returns

Non-conservative flux vector.

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BloodFlowTrixi.friction — Method
friction(u, x, eq::BloodFlowEquations1D)

Calculates the friction term for the blood flow equations, which represents viscous resistance to flow along the artery wall.

Parameters

  • u: State vector containing cross-sectional area and flow rate.
  • x: Position along the artery.
  • eq::BloodFlowEquations1D: Instance of the blood flow model.

Returns

Friction coefficient as a scalar.

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BloodFlowTrixi.initial_condition_simple — Method
initial_condition_simple(x, t, eq::BloodFlowEquations1D; R0=2.0)

Generates a simple initial condition with a specified initial radius R0.

Parameters

  • x: Position vector.
  • t: Time scalar.
  • eq::BloodFlowEquations1D: Instance of the blood flow model.
  • R0: Initial radius (default: 2.0).

Returns

State vector with zero initial area perturbation, zero flow rate, constant elasticity modulus, and reference area computed as A_0 = \pi R_0^2.

This initial condition is suitable for basic tests without complex dynamics.

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BloodFlowTrixi.inv_pressure — Method
inv_pressure(p, u, eq::BloodFlowEquations1D)

Computes the inverse relation of pressure to cross-sectional area.

Parameters

  • p: Pressure.
  • u: State vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Cross-sectional area corresponding to the given pressure.

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BloodFlowTrixi.pressure — Method
pressure(u, eq::BloodFlowEquations1D)

Computes the pressure given the state vector based on the compliance of the artery.

Parameters

  • u: State vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Pressure as a scalar.

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BloodFlowTrixi.pressure_der — Method
pressure_der(u, eq::BloodFlowEquations1D)

Computes the derivative of pressure with respect to cross-sectional area.

Parameters

  • u: State vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Derivative of pressure.

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BloodFlowTrixi.radius — Method
radius(u, eq::BloodFlowEquations1D)

Computes the radius of the artery based on the cross-sectional area.

Parameters

  • u: State vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Radius as a scalar.

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BloodFlowTrixi.source_term_simple — Method
source_term_simple(u, x, t, eq::BloodFlowEquations1D)

Computes a simple source term for the blood flow model, focusing on frictional effects.

Parameters

  • u: State vector containing area perturbation, flow rate, elasticity modulus, and reference area.
  • x: Position vector.
  • t: Time scalar.
  • eq::BloodFlowEquations1D: Instance of the blood flow model.

Returns

Source terms vector where:

  • s_1 = 0 (no source for area perturbation).
  • s_2 represents the friction term given by s_2 = \frac{2 \pi k Q}{R A}.

Friction coefficient k is computed using the friction function, and the radius R is obtained using the radius function.

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Trixi.cons2prim — Method
Trixi.cons2prim(u, eq::BloodFlowEquations1D)

Converts the conserved variables to primitive variables.

Parameters

  • u: State vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Primitive variable vector.

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Trixi.flux — Method
Trixi.flux(u, orientation::Integer, eq::BloodFlowEquations1D)

Computes the flux vector for the conservation laws of the blood flow model.

Parameters

  • u: State vector.
  • orientation::Integer: Orientation index for flux computation.
  • eq: Instance of BloodFlowEquations1D.

Returns

Flux vector as an SVector.

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Trixi.initial_condition_convergence_test — Method
initial_condition_convergence_test(x, t, eq::BloodFlowEquations1D)

Generates a smooth initial condition for convergence tests of the blood flow equations.

Parameters

  • x: Position vector.
  • t: Time scalar.
  • eq::BloodFlowEquations1D: Instance of the blood flow model.

Returns

Initial condition state vector with zero initial area perturbation, sinusoidal flow rate, a constant elasticity modulus, and reference area.

Details

The returned initial condition has:

  • Zero perturbation in area (a = 0).
  • A sinusoidal flow rate given by Q = sin(\pi x t).
  • A constant elasticity modulus E.
  • A reference cross-sectional area A_0 = \pi R_0^2 for R_0 = 1.

This initial condition can be used to verify the accuracy and stability of numerical solvers.

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Trixi.max_abs_speed_naive — Method
Trixi.max_abs_speed_naive(u_ll, u_rr, orientation::Integer, eq::BloodFlowEquations1D)

Calculates the maximum absolute speed for wave propagation in the blood flow model using a naive approach.

Parameters

  • u_ll: Left state vector.
  • u_rr: Right state vector.
  • orientation::Integer: Orientation index.
  • eq: Instance of BloodFlowEquations1D.

Returns

Maximum absolute speed.

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Trixi.prim2cons — Method
Trixi.prim2cons(u, eq::BloodFlowEquations1D)

Converts the primitive variables to conserved variables.

Parameters

  • u: Primitive variable vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Conserved variable vector.

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Trixi.source_terms_convergence_test — Method
source_terms_convergence_test(u, x, t, eq::BloodFlowEquations1D)

Computes the source terms for convergence tests of the blood flow equations.

Parameters

  • u: State vector containing area perturbation, flow rate, elasticity modulus, and reference area.
  • x: Position vector.
  • t: Time scalar.
  • eq::BloodFlowEquations1D: Instance of the blood flow model.

Returns

Source terms vector.

Details

The source terms are derived based on the smooth initial condition and friction effects:

  • s_1 represents the source term for area perturbation and is given by s_1 = \pi t \cos(\pi x t).
  • s_2 represents the source term for the flow rate and includes contributions from spatial and temporal variations as well as friction effects.

The radius R is computed using the radius function, and the friction coefficient k is obtained using the friction function.

This function is useful for evaluating the correctness of source term handling in numerical solvers.

source
+\end{cases}\]

The corresponding inflow area A_{in} is computed using the inverse pressure relation, and the boundary state is constructed accordingly.

source
BloodFlowTrixi.boundary_condition_slip_wall — Method
boundary_condition_slip_wall(u_inner, orientation_or_normal, direction, x, t, surface_flux_function, eq::BloodFlowEquations1D)

Implements a slip wall boundary condition where the normal component of velocity is reflected.

Parameters

  • u_inner: State vector inside the domain near the boundary.
  • orientation_or_normal: Normal orientation of the boundary.
  • direction: Integer indicating the direction of the boundary.
  • x: Position vector.
  • t: Time.
  • surface_flux_function: Function to compute flux at the boundary.
  • eq: Instance of BloodFlowEquations1D.

Returns

Computed boundary flux at the slip wall.

source
BloodFlowTrixi.flux_nonconservative — Method
flux_nonconservative(u_ll, u_rr, orientation::Integer, eq::BloodFlowEquations1D)

Computes the non-conservative flux for the model, used for handling discontinuities in pressure.

Parameters

  • u_ll: Left state vector.
  • u_rr: Right state vector.
  • orientation::Integer: Orientation index.
  • eq: Instance of BloodFlowEquations1D.

Returns

Non-conservative flux vector.

source
BloodFlowTrixi.friction — Method
friction(u, x, eq::BloodFlowEquations1D)

Calculates the friction term for the blood flow equations, which represents viscous resistance to flow along the artery wall.

Parameters

  • u: State vector containing cross-sectional area and flow rate.
  • x: Position along the artery.
  • eq::BloodFlowEquations1D: Instance of the blood flow model.

Returns

Friction coefficient as a scalar.

source
BloodFlowTrixi.initial_condition_simple — Method
initial_condition_simple(x, t, eq::BloodFlowEquations1D; R0=2.0)

Generates a simple initial condition with a specified initial radius R0.

Parameters

  • x: Position vector.
  • t: Time scalar.
  • eq::BloodFlowEquations1D: Instance of the blood flow model.
  • R0: Initial radius (default: 2.0).

Returns

State vector with zero initial area perturbation, zero flow rate, constant elasticity modulus, and reference area computed as A_0 = \pi R_0^2.

This initial condition is suitable for basic tests without complex dynamics.

source
BloodFlowTrixi.inv_pressure — Method
inv_pressure(p, u, eq::BloodFlowEquations1D)

Computes the inverse relation of pressure to cross-sectional area.

Parameters

  • p: Pressure.
  • u: State vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Cross-sectional area corresponding to the given pressure.

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BloodFlowTrixi.pressure — Method
pressure(u, eq::BloodFlowEquations1D)

Computes the pressure given the state vector based on the compliance of the artery.

Parameters

  • u: State vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Pressure as a scalar.

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BloodFlowTrixi.pressure_der — Method
pressure_der(u, eq::BloodFlowEquations1D)

Computes the derivative of pressure with respect to cross-sectional area.

Parameters

  • u: State vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Derivative of pressure.

source
BloodFlowTrixi.radius — Method
radius(u, eq::BloodFlowEquations1D)

Computes the radius of the artery based on the cross-sectional area.

Parameters

  • u: State vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Radius as a scalar.

source
BloodFlowTrixi.source_term_simple — Method
source_term_simple(u, x, t, eq::BloodFlowEquations1D)

Computes a simple source term for the blood flow model, focusing on frictional effects.

Parameters

  • u: State vector containing area perturbation, flow rate, elasticity modulus, and reference area.
  • x: Position vector.
  • t: Time scalar.
  • eq::BloodFlowEquations1D: Instance of the blood flow model.

Returns

Source terms vector where:

  • s_1 = 0 (no source for area perturbation).
  • s_2 represents the friction term given by s_2 = \frac{2 \pi k Q}{R A}.

Friction coefficient k is computed using the friction function, and the radius R is obtained using the radius function.

source
Trixi.cons2prim — Method
Trixi.cons2prim(u, eq::BloodFlowEquations1D)

Converts the conserved variables to primitive variables.

Parameters

  • u: State vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Primitive variable vector.

source
Trixi.flux — Method
Trixi.flux(u, orientation::Integer, eq::BloodFlowEquations1D)

Computes the flux vector for the conservation laws of the blood flow model.

Parameters

  • u: State vector.
  • orientation::Integer: Orientation index for flux computation.
  • eq: Instance of BloodFlowEquations1D.

Returns

Flux vector as an SVector.

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Trixi.initial_condition_convergence_test — Method
initial_condition_convergence_test(x, t, eq::BloodFlowEquations1D)

Generates a smooth initial condition for convergence tests of the blood flow equations.

Parameters

  • x: Position vector.
  • t: Time scalar.
  • eq::BloodFlowEquations1D: Instance of the blood flow model.

Returns

Initial condition state vector with zero initial area perturbation, sinusoidal flow rate, a constant elasticity modulus, and reference area.

Details

The returned initial condition has:

  • Zero perturbation in area (a = 0).
  • A sinusoidal flow rate given by Q = sin(\pi x t).
  • A constant elasticity modulus E.
  • A reference cross-sectional area A_0 = \pi R_0^2 for R_0 = 1.

This initial condition can be used to verify the accuracy and stability of numerical solvers.

source
Trixi.max_abs_speed_naive — Method
Trixi.max_abs_speed_naive(u_ll, u_rr, orientation::Integer, eq::BloodFlowEquations1D)

Calculates the maximum absolute speed for wave propagation in the blood flow model using a naive approach.

Parameters

  • u_ll: Left state vector.
  • u_rr: Right state vector.
  • orientation::Integer: Orientation index.
  • eq: Instance of BloodFlowEquations1D.

Returns

Maximum absolute speed.

source
Trixi.prim2cons — Method
Trixi.prim2cons(u, eq::BloodFlowEquations1D)

Converts the primitive variables to conserved variables.

Parameters

  • u: Primitive variable vector.
  • eq: Instance of BloodFlowEquations1D.

Returns

Conserved variable vector.

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Trixi.source_terms_convergence_test — Method
source_terms_convergence_test(u, x, t, eq::BloodFlowEquations1D)

Computes the source terms for convergence tests of the blood flow equations.

Parameters

  • u: State vector containing area perturbation, flow rate, elasticity modulus, and reference area.
  • x: Position vector.
  • t: Time scalar.
  • eq::BloodFlowEquations1D: Instance of the blood flow model.

Returns

Source terms vector.

Details

The source terms are derived based on the smooth initial condition and friction effects:

  • s_1 represents the source term for area perturbation and is given by s_1 = \pi t \cos(\pi x t).
  • s_2 represents the source term for the flow rate and includes contributions from spatial and temporal variations as well as friction effects.

The radius R is computed using the radius function, and the friction coefficient k is obtained using the friction function.

This function is useful for evaluating the correctness of source term handling in numerical solvers.

source
diff --git a/dev/tuto/index.html b/dev/tuto/index.html index 4aea07b..24c1c0a 100644 --- a/dev/tuto/index.html +++ b/dev/tuto/index.html @@ -150,4 +150,4 @@ plt2 = Plots.plot(pd["QRθ"]; title = "$(sol.t[i])",aspect_ratio=0.2,clim=(-10,20),label="QRθ") plt3 = Plots.plot(pd["Qs"]; title = "$(sol.t[i])",aspect_ratio=0.2,clim=(-10,90),label="Qs") plot(plt1,plt2,plt3,layout=(1,3)) -end

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+end

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