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FrictionQPotSpringBlock::Line2d::System_Cuspy_Laplace Class Reference
Identical to Line1d::System_Cuspy_Laplace() but with '2d' interactions. More...
#include <FrictionQPotSpringBlock/Line2d.h>
Inheritance diagram for FrictionQPotSpringBlock::Line2d::System_Cuspy_Laplace:
Public Member Functions | |
| System_Cuspy_Laplace (double m, double eta, double mu, double k_interactions, double k_frame, double dt, const std::array< size_t, 2 > &shape, uint64_t seed, const std::string &distribution, const std::vector< double > ¶meters, double offset=-100.0, size_t nchunk=5000) | |
| Standard system with a cuspy potential energy landscape and short range interactions. | |
| const auto & | chunk () const |
| Chunk of (cumulative sum of) random numbers. | |
| auto | size () const |
| Number of particles. | |
| const auto & | shape () const |
| Shape of the system. | |
| auto | dt () const |
| The time step (parameter). | |
| auto | mu () const |
| The curvature of each well (parameter). | |
| auto | eta () const |
| The damping coefficient (parameter). | |
| auto | m () const |
| The mass of each particle (parameter). | |
| auto | k_frame () const |
| The stiffness of the loading frame (parameter). | |
| const auto & | external () const |
| Class that generates and external force that is add to the residual force. | |
| void | set_t (double arg) |
| Set time. | |
| void | set_inc (ptrdiff_t arg) |
| Set increment number. | |
| void | set_u_frame (double arg) |
| Set position of the load frame. | |
| double | u_frame () const |
| Position of the load frame. | |
| void | set_u (const array_type::tensor< double, rank > &arg) |
| Set the slip ('position') of each particle. | |
| void | set_v (const array_type::tensor< double, rank > &arg) |
| Set the velocity of each particle (the first time derivative of the slip). | |
| void | set_a (const array_type::tensor< double, rank > &arg) |
| Set the acceleration of each particle (the second time derivative of the slip). | |
| void | refresh () |
| Recompute all forces. | |
| const auto & | u () const |
| Slip ('position') of each particle. | |
| const auto & | v () const |
| Velocity of each particle. | |
| const auto & | a () const |
| Acceleration of each particle. | |
| const auto & | f () const |
| Resultant force acting on each particle. | |
| const auto & | f_potential () const |
| Force associated to potentials acting on each particle. | |
| const auto & | f_frame () const |
| Force associated to the load frame acting on each particle. | |
| const auto & | f_interactions () const |
| Force associated to interactions between particles. | |
| const auto & | f_damping () const |
| Force associated to damping on each particle. | |
| auto | t () const |
| The time. | |
| auto | inc () const |
| The increment number. | |
| double | temperature () const |
| The instantaneous temperature. | |
| double | residual () const |
| Residual. | |
| void | quench () |
| Set velocities and accelerations to zero. | |
| void | timeStep () |
| Effectuate one time step using the velocity Verlet algorithm. | |
| void | timeSteps (size_t n) |
| Make a number of time steps, see timeStep(). | |
| size_t | timeStepsUntilEvent (double tol=1e-5, size_t niter_tol=10, size_t max_iter=1e9) |
| Perform a series of time-steps until the next plastic event, or equilibrium. | |
| void | flowSteps (size_t n, double v_frame) |
| Make a number of steps with the frame moving at a constant velocity. | |
| size_t | minimise (double tol=1e-5, size_t niter_tol=10, size_t max_iter=1e9, bool time_activity=false, bool max_iter_is_error=true) |
| Minimise energy: run timeStep() until a mechanical equilibrium has been reached. | |
| size_t | quasistaticActivityFirst () const |
| Increment with the first plastic event. | |
| size_t | quasistaticActivityLast () const |
| Increment with the last plastic event. | |
| size_t | minimise_truncate (array_type::tensor< ptrdiff_t, rank > i_n, size_t A_truncate=0, size_t S_truncate=0, double tol=1e-5, size_t niter_tol=10, size_t max_iter=1e9, bool time_activity=true, bool max_iter_is_error=true) |
| Minimise energy: run timeStep() until a mechanical equilibrium has been reached. | |
| double | maxUniformDisplacement (int direction) |
| Find maximum particle displacement for which the system is linear and uniform. | |
| double | eventDrivenStep (double eps, bool kick, int direction=1) |
| Make event driven step. | |
| void | trigger (size_t p, double eps, int direction=1) |
| Trigger a specific particle. | |
| void | advanceToFixedForce (double f_frame, bool allow_plastic=false) |
| Change the position of the particles and of the loading frame such that the mean of f_frame() is equal to a target value, and mechanical equilibrium is maintained. | |
Protected Member Functions | |
| void | initSystem (double m, double eta, double k_frame, double mu, double dt, detail::Cuspy< Generator > *potential, Generator *chunk, detail::Laplace2d *interactions=nullptr, void *external=nullptr) |
| Initialise the system. | |
| void | computeForce () |
| Compute residual force. | |
| void | computeForcePotential () |
| Compute force due to the potential energy. | |
| void | computeForceInteractions () |
| Compute force due to interactions between particles. | |
| void | computeForceFrame () |
| Compute force due to the loading frame. | |
| void | computeForceDamping () |
| Compute force due to damping. | |
| void | updated_inc () |
| Update forces that depend on time. | |
| void | updated_u () |
| Update forces that depend on slip. | |
| void | updated_v () |
| Update forces that depend on velocity. | |
| double | advanceUniformly (double du, bool input_is_frame=true) |
| Advance the system uniformly. | |
Protected Attributes | |
| Generator | m_gen |
| Pointer to chunk of yield 'positions' (automatically updated if needed) | |
| detail::Cuspy< Generator > | m_pot |
| Class to get the forces from the local potential energy landscape. | |
| detail::Laplace2d | m_int |
| Class to get the forces from particle interaction. | |
| size_type | m_N |
| Number of particles. | |
| array_type::tensor< double, rank > | m_f |
| Resultant force acting on each particle. | |
| array_type::tensor< double, rank > | m_f_thermal |
| Force due to thermal fluctuations. | |
| array_type::tensor< double, rank > | m_f_potential |
| Force associated to potentials acting on each particle. | |
| array_type::tensor< double, rank > | m_f_interactions |
| Force associated to interactions between particles. | |
| array_type::tensor< double, rank > | m_f_frame |
| Force associated to the load frame acting on each particle. | |
| array_type::tensor< double, rank > | m_f_damping |
| Force associated to damping on each particle. | |
| array_type::tensor< double, rank > | m_u |
| Slip ('position') of each particle. | |
| array_type::tensor< double, rank > | m_v |
| Velocity of each particle. | |
| array_type::tensor< double, rank > | m_a |
| Acceleration of each particle. | |
| array_type::tensor< double, rank > | m_v_n |
| Temporary for integration. | |
| array_type::tensor< double, rank > | m_a_n |
| Temporary for integration. | |
| ptrdiff_t | m_inc |
| The increment number. | |
| ptrdiff_t | m_qs_inc_first |
| Increment with the first plastic event. | |
| ptrdiff_t | m_qs_inc_last |
| Increment with the last plastic event. | |
| double | m_dt |
| Time step. | |
| double | m_eta |
| Damping constant (same for all particles). | |
| double | m_m |
| Mass (same for all particles). | |
| double | m_inv_m |
== 1 / m_m | |
| double | m_mu |
| Curvature of the potentials. | |
| double | m_k_frame |
| Stiffness of the load fame (same for all particles). | |
| double | m_u_frame |
| Position of the load frame. | |
| detail::Cuspy< Generator > * | m_potential |
| Class to get the forces from the local potential energy landscape. | |
| Generator * | m_chunk |
| Pointer to chunk of yield 'positions' (automatically updated if needed) | |
| detail::Laplace2d * | m_interactions |
| Class to get the forces from particle interaction. | |
| void * | m_external |
| Add an (time dependent) externally defined force to the residual. | |
Detailed Description
Identical to Line1d::System_Cuspy_Laplace() but with '2d' interactions.
Short range interactions based on the Laplacian \( \Delta u_{i, j} \).
Constructor & Destructor Documentation
◆ System_Cuspy_Laplace()
|
inline |
Standard system with a cuspy potential energy landscape and short range interactions.
Physics
The system consists of particles of mass \( m \). The principle degree of freedom is the slip ('position') \( u_i \) of each particle \( i \). Each particle follows Newtonian dynamics \( m \ddot{u}_i = f_i \), with the residual force \( f_i \) given by the sum of:
A piece-wise quadratic local potential energy.
The resulting force is piece-wise linear: \( f_\mathrm{pot}^{(i)} = \mu (u_{\min}^{(i)} - u_i) \). where \( \mu \) is the curvature of the quadratic potentials, and \( u_{\min}^{(i)} \) the position of the current local minimum. The landscape is given by a cumulative sum of random numbers.
Short range elastic interactions with other particles.
The interactions are the short-range Laplacian \( \Delta u_i \): \( f_\mathrm{inter}^{(i)} = k (u_{i - 1} - 2 u_i + u_{i + 1}) \).
A drive though a spring attached to a load frame.
This corresponds to a parabolic potential energy for each particle, with the minimum at a set \( u_\text{frame} \): \( f_\text{frame}^{(i)} = k_\text{frame} (u_\text{frame} - u_i) \).
Damping.
\( f_\text{damping}^{(i)} = - \eta v_i \), with the particle's velocity \( v_i \equiv \dot{u}_i \equiv \partial_t u_i \).
Parameters
The following scaling argument holds to set \( k_\text{frame} \). Because of the competition between disorder and elastic interactions, the interface gets a roughness \( \delta u \sim \ell^\zeta \). This corresponds to fluctuations of interactions \( \delta f_\text{inter} \sim k_\text{inter} \partial^2 \delta u \sim k_\text{inter} \ell^{\zeta - 2} \). The fluctuations of the drive are \( \delta f_\text{frame} \sim k_\text{frame} \delta u \sim k_\text{frame} \ell^\zeta \). For the disorder to be relevant \( \delta f_\text{inter} \sim \delta f_\text{frame} \). Such that \( k_\text{frame} / k_\text{inter} \sim 1 / \ell^2 \). With \( L = 1 / N \) the linear size of the system \( k_\text{frame} / k_\text{inter} \sim 1 / L^2 \)
Dynamics
The dynamics follow \( m a_i = f_i \), with the particle's acceleration \( a_i \equiv \ddot{u}_i \equiv \partial_t^2 u_i \), and the residual force \( f_i = f_\text{damping}^{(i)} + f_\text{pot}^{(i)} + f_\text{inter}^{(i)} + f_\text{frame}^{(i)} \). This differential equation is integrated using Velocity-Verlet in timeStep() Note that this function updates the time t (or in fact the increment number inc), and that this is the only function that does so.
The implementation is is such that all forces are updated (if needed) every time the positions and velocities are updated, either internally in timeStep(), or externally when calling set_u() and set_v(). In absolute need refresh() can be called to force re-computation of forces, but this should normally not be needed.
Member Function Documentation
◆ a()
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inlineinherited |
◆ advanceToFixedForce()
|
inlineinherited |
Change the position of the particles and of the loading frame such that the mean of f_frame() is equal to a target value, and mechanical equilibrium is maintained.
- Warning
- Assumes mechanical equilibrium. No assertions are made on this.
- Parameters
-
f_frame Target value for the mean of f_frame(). allow_plastic Allow plastic events during the advance.
◆ advanceUniformly()
|
inlineprotectedinherited |
Advance the system uniformly.
- All particle move by the same amount.
- The frame moves proportionally such that equilibrium is maintained (the change in
f_potentialis compensated by the change inf_frame).
Thereby:
\( \Delta f^\mathrm{potential}_i = \mu \Delta u_i \)
\( \Delta f^\mathrm{frame} = k (\Delta u^\mathrm{frame} - \Delta u_i) \)
Setting \( \Delta f^\mathrm{potential}_i = \Delta f^\mathrm{frame} \) gives:
- Given \( \Delta u_i \) one finds: \( \Delta u^\mathrm{frame} = (k + \mu) / k \Delta u_i \).
- Given \( \Delta u^\mathrm{frame} \) one finds: \( \Delta u_i = k / (k + \mu) \Delta u^\mathrm{frame} \).
- Parameters
-
du Displacement. input_is_frame If true:du\( = \Delta u^\mathrm{frame} \), iffalse:du\( = \Delta u_i \),
- Returns
dufor the particles ininput_is_frame == true, otherwiseduof the frame.
◆ chunk()
|
inlineinherited |
◆ computeForce()
|
inlineprotectedinherited |
◆ computeForceDamping()
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inlineprotectedinherited |
◆ computeForceFrame()
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inlineprotectedinherited |
◆ computeForceInteractions()
|
inlineprotectedinherited |
◆ computeForcePotential()
|
inlineprotectedinherited |
◆ dt()
|
inlineinherited |
◆ eta()
|
inlineinherited |
◆ eventDrivenStep()
|
inlineinherited |
Make event driven step.
kick = false: Increment the position of the load-frame and that of the particles to a new mechanical equilibrium just before yielding (ifdirection = 1, the new position for particlepclosest to yielding to the right isx[p] = y[p] - eps / 2). This assumes incrementing the load-frame infinitely slowly such that, because there is no yielding, the equilibrium configuration for a new position of the load frame is known.kick = true: Advance the system uniformly (the particles and the frame are moved proportionally) such that the particle closest to yielding is brought just past yielding (ifdirection = 1, the new position for particlepclosest to yielding right isx[p] = y[p] + eps / 2).
- Parameters
-
eps Margin to keep to the position to the closest yield position. kick If false, the increment is elastic (no minimisation has to be applied after). Iftrue, the increment leads to a state out of mechanical equilibrium.direction If +1: move right. If-1move left.
- Returns
- Position increment of the frame.
◆ external()
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inlineinherited |
◆ f()
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inlineinherited |
◆ f_damping()
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inlineinherited |
◆ f_frame()
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inlineinherited |
◆ f_interactions()
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inlineinherited |
◆ f_potential()
|
inlineinherited |
◆ flowSteps()
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inlineinherited |
Make a number of steps with the frame moving at a constant velocity.
Make a number of steps with the following protocol.
- Add a step \( v_\text{frame} \Delta t \) to the frame.
- Make a timeStep().
- Parameters
-
n Number of steps to make. v_frame Velocity of the frame.
◆ inc()
|
inlineinherited |
◆ initSystem()
|
inlineprotectedinherited |
Initialise the system.
- Parameters
-
m Mass (same for all particles). eta Damping constant (same for all particles). k_frame Stiffness of the load fame (same for all particles). mu Curvature of the potentials. dt Time step. potential Class to get the forces from the local potential energy landscape. chunk Pointer to chunk of yield 'positions' (automatically updated if needed) interactions Class to get the forces from particle interaction. external Add an (time dependent) externally defined force to the residual.
◆ k_frame()
|
inlineinherited |
◆ m()
|
inlineinherited |
◆ maxUniformDisplacement()
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inlineinherited |
◆ minimise()
|
inlineinherited |
Minimise energy: run timeStep() until a mechanical equilibrium has been reached.
- Parameters
-
tol Relative force tolerance for equilibrium. See residual() for definition. niter_tol Enforce the residual check for niter_tolconsecutive increments.max_iter Maximum number of time-steps. Throws std::runtime_errorotherwise.time_activity If trueplastic activity is timed. After this function you can find:- quasistaticActivityFirst() : Increment with the first plastic event.
- quasistaticActivityLast() : Increment with the last plastic event. Attention: if you are changing the chunk of yield positions during the minimisation you should copy quasistaticActivityFirst() after the first (relevant) call of minimise(): each time you call minimise(), quasistaticActivityFirst() is reset.
max_iter_is_error If truean error is thrown when the maximum number of time-steps is reached. Iffalsethe function simply returnsmax_iter.
- Returns
0: if stopped when the residual is reached (and number of steps< max_iter).max_iter: if no residual was reached, andmax_iter_is_error = false.
◆ minimise_truncate()
|
inlineinherited |
Minimise energy: run timeStep() until a mechanical equilibrium has been reached.
- Parameters
-
tol Relative force tolerance for equilibrium. See residual() for definition. niter_tol Enforce the residual check for niter_tolconsecutive increments.max_iter Maximum number of time-steps. Throws std::runtime_errorotherwise.time_activity If trueplastic activity is timed. After this function you can find:- quasistaticActivityFirst() : Increment with the first plastic event.
- quasistaticActivityLast() : Increment with the last plastic event. Attention: if you are changing the chunk of yield positions during the minimisation you should copy quasistaticActivityFirst() after the first (relevant) call of minimise(): each time you call minimise(), quasistaticActivityFirst() is reset.
max_iter_is_error If truean error is thrown when the maximum number of time-steps is reached. Iffalsethe function simply returnsmax_iter.
- Returns
0: if stopped when the residual is reached (and number of steps< max_iter).max_iter: if no residual was reached, andmax_iter_is_error = false.
- Parameters
-
A_truncate Truncate if A_truncateblocks have yielded at least once (return > 0). Only considered ifA_truncate > 0.S_truncate Truncate if the number of times blocks yielded is equal to S_truncate(return > 0). Only considered ifS_truncate > 0.i_n Reference potential index of the first integration point.
◆ mu()
|
inlineinherited |
◆ quasistaticActivityFirst()
|
inlineinherited |
Increment with the first plastic event.
This value is only relevant if time_activity = true was used in the last call of minimise().
- Returns
- Increment.
◆ quasistaticActivityLast()
|
inlineinherited |
Increment with the last plastic event.
This value is only relevant if time_activity = true was used in the last call of minimise().
- Returns
- Increment.
◆ quench()
|
inlineinherited |
Set velocities and accelerations to zero.
Call this function after an energy minimisation (taken care of in minimise()).
◆ refresh()
|
inlineinherited |
◆ residual()
|
inlineinherited |
◆ set_a()
|
inlineinherited |
◆ set_inc()
|
inlineinherited |
◆ set_t()
|
inlineinherited |
◆ set_u()
|
inlineinherited |
◆ set_u_frame()
|
inlineinherited |
◆ set_v()
|
inlineinherited |
◆ shape()
|
inlineinherited |
◆ size()
|
inlineinherited |
◆ t()
|
inlineinherited |
◆ temperature()
|
inlineinherited |
◆ timeStep()
|
inlineinherited |
◆ timeSteps()
|
inlineinherited |
Make a number of time steps, see timeStep().
- Parameters
-
n Number of steps to make.
◆ timeStepsUntilEvent()
|
inlineinherited |
Perform a series of time-steps until the next plastic event, or equilibrium.
- Parameters
-
tol Relative force tolerance for equilibrium. See residual() for definition. niter_tol Enforce the residual check for niter_tolconsecutive increments.max_iter Maximum number of iterations. Throws std::runtime_errorotherwise.
- Returns
- Number of steps.
0if there was no plastic activity and the residual was reached.
◆ trigger()
|
inlineinherited |
◆ u()
|
inlineinherited |
◆ u_frame()
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inlineinherited |
◆ updated_inc()
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inlineprotectedinherited |
◆ updated_u()
|
inlineprotectedinherited |
◆ updated_v()
|
inlineprotectedinherited |
◆ v()
|
inlineinherited |
Member Data Documentation
◆ m_a
|
protectedinherited |
◆ m_a_n
|
protectedinherited |
◆ m_chunk
|
protectedinherited |
◆ m_dt
|
protectedinherited |
◆ m_eta
|
protectedinherited |
◆ m_external
|
protectedinherited |
◆ m_f
|
protectedinherited |
◆ m_f_damping
|
protectedinherited |
◆ m_f_frame
|
protectedinherited |
◆ m_f_interactions
|
protectedinherited |
◆ m_f_potential
|
protectedinherited |
◆ m_f_thermal
|
protectedinherited |
◆ m_gen
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protected |
◆ m_inc
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protectedinherited |
◆ m_int
|
protected |
◆ m_interactions
|
protectedinherited |
◆ m_inv_m
|
protectedinherited |
◆ m_k_frame
|
protectedinherited |
◆ m_m
|
protectedinherited |
◆ m_mu
|
protectedinherited |
◆ m_N
|
protectedinherited |
◆ m_pot
|
protected |
◆ m_potential
|
protectedinherited |
◆ m_qs_inc_first
|
protectedinherited |
◆ m_qs_inc_last
|
protectedinherited |
◆ m_u
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protectedinherited |
◆ m_u_frame
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protectedinherited |
◆ m_v
|
protectedinherited |
◆ m_v_n
|
protectedinherited |
The documentation for this class was generated from the following file:
- FrictionQPotSpringBlock/Line2d.h
