Compilable state

multigrid/src/main.rs

@@ -86,42 +86,28 @@ fn main() {
         operators,
         boundary_conditions,
         initial_conditions.clone(),
+        opt.output.clone(),
     );
-    let mut sys = basesystem.create();
     // System::new(grids, grid_connections, operators);
 
-    let dt = sys.max_dt();
+    let sys = if opt.distribute {
-
+        basesystem.create_distributed()
-    let ntime = (integration_time / dt).round() as u64;
+    } else {
-
+        basesystem.create()
-    if opt.distribute {
-        let sys = sys.distribute(ntime);
-        let timer = if opt.timings {
-            Some(std::time::Instant::now())
-        } else {
-            None
-        };
-        sys.run();
-        if let Some(timer) = timer {
-            let duration = timer.elapsed();
-            println!("Duration: {:?}", duration);
-        }
-        return;
-    }
-
-    let should_output = |itime| {
-        opt.number_of_outputs.map_or(false, |num_out| {
-            if num_out == 0 {
-                false
-            } else {
-                itime % (std::cmp::max(ntime / (num_out - 1), 1)) == 0
-            }
-        })
     };
 
-    let output = File::create(&opt.output, sys.grids.as_slice(), names).unwrap();
+    let dt = sys.max_dt();
-    let mut output = OutputThread::new(output);
+    let ntime = (integration_time / dt).round() as u64;
-    output.add_timestep(0, &sys.fnow);
+    let steps_between_outputs = if let Some(n) = opt.number_of_outputs {
+        std::cmp::max(n / ntime, 1)
+    } else {
+        ntime
+    };
+
+    sys.output(0);
+    //let output = File::create(&opt.output, sys.grids.as_slice(), names).unwrap();
+    //let mut output = OutputThread::new(output);
+    //output.add_timestep(0, &sys.fnow);
 
     let progressbar = progressbar(opt.no_progressbar, ntime);
 
@@ -131,6 +117,16 @@ fn main() {
         None
     };
 
+    let mut itime = 0;
+    while itime < ntime {
+        sys.advance(steps_between_outputs);
+        progressbar.inc(1);
+
+        itime += steps_between_outputs;
+        sys.output(itime);
+    }
+
+    /*
     for itime in 0..ntime {
         if should_output(itime) {
             output.add_timestep(itime, &sys.fnow);
@@ -138,6 +134,7 @@ fn main() {
         progressbar.inc(1);
         sys.advance(dt);
     }
+    */
     progressbar.finish_and_clear();
 
     let mut outinfo = OutputInformation {
@@ -150,8 +147,9 @@ fn main() {
         outinfo.time_elapsed = Some(duration);
     }
 
-    output.add_timestep(ntime, &sys.fnow);
+    //output.add_timestep(ntime, &sys.fnow);
 
+    /*
     if opt.error {
         let time = ntime as Float * dt;
         let mut e = 0.0;
@@ -171,6 +169,7 @@ fn main() {
         }
         outinfo.error = Some(e);
     }
+    */
 
     if opt.output_json {
         println!("{}", json5::to_string(&outinfo).unwrap());

multigrid/src/system.rs

@@ -1,5 +1,6 @@
 use crate::parsing;
 use crate::utils::Direction;
+use core::ops::Deref;
 use crossbeam_channel::{Receiver, Select, Sender};
 use euler::{
     eval::{self, Evaluator},
@@ -9,7 +10,6 @@ use ndarray::Array2;
 use sbp::grid::{Grid, Metrics};
 use sbp::operators::{InterpolationOperator, SbpOperator2d};
 use sbp::*;
-use std::sync::{Arc, Barrier};
 
 pub struct BaseSystem {
     pub names: Vec<String>,
@@ -18,6 +18,7 @@ pub struct BaseSystem {
     pub boundary_conditions: Vec<euler::BoundaryCharacteristics>,
     pub initial_conditions: crate::parsing::InitialConditions,
     pub operators: Vec<Box<dyn SbpOperator2d>>,
+    pub output: hdf5::File,
 }
 
 impl BaseSystem {
@@ -28,7 +29,15 @@ impl BaseSystem {
         operators: Vec<Box<dyn SbpOperator2d>>,
         boundary_conditions: Vec<euler::BoundaryCharacteristics>,
         initial_conditions: crate::parsing::InitialConditions,
+        output: std::path::PathBuf,
     ) -> Self {
+        let output = hdf5::File::create(output).unwrap();
+        output
+            .new_dataset::<u64>()
+            .resizable(true)
+            .chunk((1,))
+            .create("t", (0,))
+            .unwrap();
         Self {
             names,
             grids,
@@ -36,10 +45,54 @@ impl BaseSystem {
             boundary_conditions,
             initial_conditions,
             operators,
+            output,
         }
     }
     pub fn create(self) -> System {
-        let mut sys = System::new(self.grids, self.boundary_conditions, self.operators);
+        let fnow = self
+            .grids
+            .iter()
+            .map(|g| euler::Field::new(g.ny(), g.nx()))
+            .collect::<Vec<_>>();
+        let fnext = fnow.clone();
+        let wb = self
+            .grids
+            .iter()
+            .map(|g| euler::WorkBuffers::new(g.ny(), g.nx()))
+            .collect();
+        let k = self
+            .grids
+            .iter()
+            .map(|g| euler::Diff::zeros((g.ny(), g.nx())))
+            .collect::<Vec<_>>();
+        let k = [k.clone(), k.clone(), k.clone(), k];
+        let metrics = self
+            .grids
+            .iter()
+            .zip(&self.operators)
+            .map(|(g, op)| g.metrics(&**op).unwrap())
+            .collect::<Vec<_>>();
+
+        let eb = self
+            .boundary_conditions
+            .iter()
+            .zip(&self.grids)
+            .map(|(bt, grid)| euler::BoundaryStorage::new(bt, grid))
+            .collect();
+
+        let sys = SingleThreadedSystem {
+            fnow,
+            fnext,
+            k,
+            wb,
+            grids: self.grids,
+            metrics,
+            bt: self.boundary_conditions,
+            eb,
+            time: self.time,
+            operators: self.operators,
+            output: todo!(),
+        };
         match &self.initial_conditions {
             /*
             parsing::InitialConditions::File(f) => {
@@ -61,11 +114,180 @@ impl BaseSystem {
                 }
             }
         }
-        sys
+        System::SingleThreaded(sys)
+    }
+
+    /// Spreads the computation over n threads, in a thread per grid way.
+    pub fn create_distributed(self) -> System {
+        let nthreads = self.grids.len();
+        // let dt = self.max_dt();
+
+        // Build up the boundary conditions
+        let mut push_channels: Vec<Direction<Option<Sender<Array2<Float>>>>> =
+            Vec::with_capacity(nthreads);
+        let mut pull_channels: Vec<Direction<Option<Receiver<Array2<Float>>>>> =
+            vec![Direction::default(); nthreads];
+
+        // Build the set of communicators between boundaries
+        for wb in &self.boundary_conditions {
+            let mut local_push = Direction::default();
+            if let euler::BoundaryCharacteristic::Grid(i)
+            | euler::BoundaryCharacteristic::Interpolate(i, _) = wb.north()
+            {
+                let (s, r) = crossbeam_channel::bounded(1);
+                pull_channels[*i]
+                    .south_mut()
+                    .replace(r)
+                    .and_then::<(), _>(|_| panic!("channel is already present"));
+                *local_push.north_mut() = Some(s);
+            }
+            if let euler::BoundaryCharacteristic::Grid(i)
+            | euler::BoundaryCharacteristic::Interpolate(i, _) = wb.south()
+            {
+                let (s, r) = crossbeam_channel::bounded(1);
+                pull_channels[*i]
+                    .north_mut()
+                    .replace(r)
+                    .and_then::<(), _>(|_| panic!("channel is already present"));
+                *local_push.south_mut() = Some(s);
+            }
+            if let euler::BoundaryCharacteristic::Grid(i)
+            | euler::BoundaryCharacteristic::Interpolate(i, _) = wb.east()
+            {
+                let (s, r) = crossbeam_channel::bounded(1);
+                pull_channels[*i]
+                    .west_mut()
+                    .replace(r)
+                    .and_then::<(), _>(|_| panic!("channel is already present"));
+                *local_push.east_mut() = Some(s);
+            }
+            if let euler::BoundaryCharacteristic::Grid(i)
+            | euler::BoundaryCharacteristic::Interpolate(i, _) = wb.west()
+            {
+                let (s, r) = crossbeam_channel::bounded(1);
+                pull_channels[*i]
+                    .east_mut()
+                    .replace(r)
+                    .and_then::<(), _>(|_| panic!("channel is already present"));
+                *local_push.west_mut() = Some(s);
+            }
+
+            push_channels.push(local_push);
+        }
+
+        let (master_send, master_recv) = crossbeam_channel::unbounded();
+
+        let mut tids = Vec::with_capacity(nthreads);
+        let mut communicators = Vec::with_capacity(nthreads);
+
+        for (id, (((((name, grid), sbp), bt), chan), push)) in self
+            .names
+            .into_iter()
+            .zip(self.grids.into_iter())
+            .zip(self.operators.into_iter())
+            .zip(self.boundary_conditions)
+            .zip(pull_channels)
+            .zip(push_channels)
+            .enumerate()
+        {
+            let builder = std::thread::Builder::new().name(format!("eulersolver: {}", name));
+
+            let boundary_conditions = bt.zip(chan).map(|(bt, chan)| match bt {
+                euler::BoundaryCharacteristic::This => DistributedBoundaryConditions::This,
+                euler::BoundaryCharacteristic::Grid(_) => {
+                    DistributedBoundaryConditions::Channel(chan.unwrap())
+                }
+                euler::BoundaryCharacteristic::Interpolate(_, int_op) => {
+                    DistributedBoundaryConditions::Interpolate(chan.unwrap(), int_op)
+                }
+                euler::BoundaryCharacteristic::MultiGrid(_) => unimplemented!(),
+                euler::BoundaryCharacteristic::Vortex(vp) => {
+                    DistributedBoundaryConditions::Vortex(vp)
+                }
+                euler::BoundaryCharacteristic::Eval(eval) => {
+                    DistributedBoundaryConditions::Eval(eval)
+                }
+            });
+
+            let master_send = master_send.clone();
+
+            let (t_send, t_recv) = crossbeam_channel::unbounded();
+            communicators.push(t_send);
+            let time = self.time;
+
+            tids.push(
+                builder
+                    .spawn(move || {
+                        let (ny, nx) = (grid.ny(), grid.nx());
+                        let current = Field::new(ny, nx);
+                        let fut = current.clone();
+                        let k = [
+                            Diff::zeros((ny, nx)),
+                            Diff::zeros((ny, nx)),
+                            Diff::zeros((ny, nx)),
+                            Diff::zeros((ny, nx)),
+                        ];
+                        let metrics = grid.metrics(sbp.deref()).unwrap();
+
+                        let wb = WorkBuffers::new(ny, nx);
+
+                        let mut sys = DistributedSystemPart {
+                            current,
+                            fut,
+                            k,
+                            boundary_conditions,
+                            grid: (grid, metrics),
+                            output: todo!(),
+                            push,
+                            sbp,
+                            t: time,
+                            dt: Float::NAN,
+
+                            name,
+                            id,
+
+                            recv: t_recv,
+                            send: master_send,
+
+                            wb,
+                            wb_ns: Array2::zeros((4, nx)),
+                            wb_ew: Array2::zeros((4, ny)),
+                        };
+
+                        // init and send maxdt
+                        // receive maxdt
+                        sys.run();
+                    })
+                    .unwrap(),
+            );
+        }
+
+        System::MultiThreaded(DistributedSystem {
+            sys: tids,
+            recv: master_recv,
+            send: communicators,
+        })
     }
 }
 
-pub struct System {
+pub enum System {
+    SingleThreaded(SingleThreadedSystem),
+    MultiThreaded(DistributedSystem),
+}
+
+impl System {
+    pub fn max_dt(&self) -> Float {
+        todo!()
+    }
+    pub fn advance(&self, nsteps: u64) {
+        todo!()
+    }
+    pub fn output(&self, ntime: u64) {
+        todo!()
+    }
+}
+
+pub struct SingleThreadedSystem {
     pub fnow: Vec<euler::Field>,
     pub fnext: Vec<euler::Field>,
     pub wb: Vec<euler::WorkBuffers>,
@@ -76,9 +298,10 @@ pub struct System {
     pub eb: Vec<euler::BoundaryStorage>,
     pub time: Float,
     pub operators: Vec<Box<dyn SbpOperator2d>>,
+    pub output: Vec<hdf5::Dataset>,
 }
 
-impl integrate::Integrable for System {
+impl integrate::Integrable for SingleThreadedSystem {
     type State = Vec<euler::Field>;
     type Diff = Vec<euler::Diff>;
 
@@ -89,52 +312,7 @@ impl integrate::Integrable for System {
     }
 }
 
-impl System {
+impl SingleThreadedSystem {
-    pub fn new(
-        grids: Vec<grid::Grid>,
-        bt: Vec<euler::BoundaryCharacteristics>,
-        operators: Vec<Box<dyn SbpOperator2d>>,
-    ) -> Self {
-        let fnow = grids
-            .iter()
-            .map(|g| euler::Field::new(g.ny(), g.nx()))
-            .collect::<Vec<_>>();
-        let fnext = fnow.clone();
-        let wb = grids
-            .iter()
-            .map(|g| euler::WorkBuffers::new(g.ny(), g.nx()))
-            .collect();
-        let k = grids
-            .iter()
-            .map(|g| euler::Diff::zeros((g.ny(), g.nx())))
-            .collect::<Vec<_>>();
-        let k = [k.clone(), k.clone(), k.clone(), k];
-        let metrics = grids
-            .iter()
-            .zip(&operators)
-            .map(|(g, op)| g.metrics(&**op).unwrap())
-            .collect::<Vec<_>>();
-
-        let eb = bt
-            .iter()
-            .zip(&grids)
-            .map(|(bt, grid)| euler::BoundaryStorage::new(bt, grid))
-            .collect();
-
-        Self {
-            fnow,
-            fnext,
-            k,
-            wb,
-            grids,
-            metrics,
-            bt,
-            eb,
-            time: 0.0,
-            operators,
-        }
-    }
-
     pub fn vortex(&mut self, t: Float, vortex_params: &euler::VortexParameters) {
         for (f, g) in self.fnow.iter_mut().zip(&self.grids) {
             f.vortex(g.x(), g.y(), t, &vortex_params);
@@ -170,7 +348,7 @@ impl System {
             }
         };
 
-        integrate::integrate::<integrate::Rk4, System, _>(
+        integrate::integrate::<integrate::Rk4, SingleThreadedSystem, _>(
             rhs,
             &self.fnow,
             &mut self.fnext,
@@ -231,191 +409,48 @@ impl System {
 
         max_dt
     }
-
-    /// Spreads the computation over n threads, in a thread per grid way.
-    /// This system can only be called once for ntime calls.
-    pub fn distribute(self, ntime: u64) -> DistributedSystem {
-        let nthreads = self.grids.len();
-        let time = 0.0;
-        // alt: crossbeam::WaitGroup
-        let b = Arc::new(Barrier::new(nthreads + 1));
-        let dt = self.max_dt();
-
-        // Build up the boundary conditions
-        let mut push_channels: Vec<Direction<Option<Sender<Array2<Float>>>>> =
-            Vec::with_capacity(nthreads);
-        let mut pull_channels: Vec<Direction<Option<Receiver<Array2<Float>>>>> =
-            vec![Direction::default(); nthreads];
-
-        // Build the set of communicators between boundaries
-        for wb in &self.bt {
-            let mut local_push = Direction::default();
-            if let euler::BoundaryCharacteristic::Grid(i)
-            | euler::BoundaryCharacteristic::Interpolate(i, _) = wb.north()
-            {
-                let (s, r) = crossbeam_channel::bounded(1);
-                pull_channels[*i]
-                    .south_mut()
-                    .replace(r)
-                    .and_then::<(), _>(|_| panic!("channel is already present"));
-                *local_push.north_mut() = Some(s);
-            }
-            if let euler::BoundaryCharacteristic::Grid(i)
-            | euler::BoundaryCharacteristic::Interpolate(i, _) = wb.south()
-            {
-                let (s, r) = crossbeam_channel::bounded(1);
-                pull_channels[*i]
-                    .north_mut()
-                    .replace(r)
-                    .and_then::<(), _>(|_| panic!("channel is already present"));
-                *local_push.south_mut() = Some(s);
-            }
-            if let euler::BoundaryCharacteristic::Grid(i)
-            | euler::BoundaryCharacteristic::Interpolate(i, _) = wb.east()
-            {
-                let (s, r) = crossbeam_channel::bounded(1);
-                pull_channels[*i]
-                    .west_mut()
-                    .replace(r)
-                    .and_then::<(), _>(|_| panic!("channel is already present"));
-                *local_push.east_mut() = Some(s);
-            }
-            if let euler::BoundaryCharacteristic::Grid(i)
-            | euler::BoundaryCharacteristic::Interpolate(i, _) = wb.west()
-            {
-                let (s, r) = crossbeam_channel::bounded(1);
-                pull_channels[*i]
-                    .east_mut()
-                    .replace(r)
-                    .and_then::<(), _>(|_| panic!("channel is already present"));
-                *local_push.west_mut() = Some(s);
-            }
-
-            push_channels.push(local_push);
-        }
-
-        let mut tids = Vec::new();
-        for ((((((((current, fut), grid), metrics), sbp), wb), bt), chan), push) in self
-            .fnow
-            .into_iter()
-            .zip(self.fnext.into_iter())
-            .zip(self.grids.into_iter())
-            .zip(self.metrics.into_iter())
-            .zip(self.operators.into_iter())
-            .zip(self.wb.into_iter())
-            .zip(self.bt)
-            .zip(pull_channels)
-            .zip(push_channels)
-        {
-            let builder = std::thread::Builder::new().name(format!("eulersolver: {}", "smth"));
-            let barrier = b.clone();
-
-            let boundary_conditions = bt.zip(chan).map(|(bt, chan)| match bt {
-                euler::BoundaryCharacteristic::This => DistributedBoundaryConditions::This,
-                euler::BoundaryCharacteristic::Grid(_) => {
-                    DistributedBoundaryConditions::Channel(chan.unwrap())
-                }
-                euler::BoundaryCharacteristic::Interpolate(_, int_op) => {
-                    DistributedBoundaryConditions::Interpolate(chan.unwrap(), int_op)
-                }
-                euler::BoundaryCharacteristic::MultiGrid(_) => unimplemented!(),
-                euler::BoundaryCharacteristic::Vortex(vp) => {
-                    DistributedBoundaryConditions::Vortex(vp)
-                }
-                euler::BoundaryCharacteristic::Eval(eval) => {
-                    DistributedBoundaryConditions::Eval(eval)
-                }
-            });
-
-            let (ny, nx) = (grid.ny(), grid.nx());
-
-            tids.push(
-                builder
-                    .spawn(move || {
-                        let mut sys = DistributedSystemPart {
-                            barrier,
-                            ntime,
-                            dt,
-                            current,
-                            fut,
-                            k: [
-                                Diff::zeros((ny, nx)),
-                                Diff::zeros((ny, nx)),
-                                Diff::zeros((ny, nx)),
-                                Diff::zeros((ny, nx)),
-                            ],
-                            boundary_conditions,
-                            grid: (grid, metrics),
-                            _output: (),
-                            push,
-                            sbp,
-                            t: time,
-
-                            wb,
-                            wb_ns: Array2::zeros((4, nx)),
-                            wb_ew: Array2::zeros((4, ny)),
-                        };
-                        sys.advance();
-                    })
-                    .unwrap(),
-            );
-        }
-        // Set up communicators
-        // Spawn a new communicator
-
-        DistributedSystem {
-            _ntime: ntime,
-            start: b,
-            sys: tids,
-        }
-    }
 }
 
-// single-threaded items: clone_from
-// sync points: scaled_add, rhs
-//
-// Could instead make every thread (of a threadpool?) carry one grid themselves,
-// and instead only wait on obtaining bc, which would make synchronisation
-// be somewhat minimal
-//
-// Difficulties/implementation notes:
-// * How to get output from each thread? Push to the multi-producer, single consumer queue
-//   with capacity of 2*N (which ensures internal synchronisation stays active), or many
-//   channels which are select!'ed. Choose instead a (name, itime, data) tuple, which is
-//   communicated. Also initialise the output file all at the start, since we know the
-//   number of time steps, and which timesteps will be used (should_output).
-// * How to balance the number of threads/jobs?
-//   Each grid will be pushed to a thread, and a thread pool created for use by all threads
-//   combined. Set affinity to limit the number of cores available (could have side effects
-//   on heat etc, although this would also be the case for multi-threaded code). A
-//   thread-per-core architecture + the output thread if we have enough cores, else let os
-//   control the scheduling for us.
-//   Use cgroups to artificially limit the number of cores.
-//   Use taskset to artificially limit the number of cores.
-// * Mechanism to wait for bc, is select available? Use a channel with 0-4 capacity, each
-//   other grid additonally pushes their direction to always use the available
-//   Use select or similar from crossbeam_channel
-// * Mechanism to push bc: try_push and then move to next bc if not available. Spawn a
-//   thread into a threadpool to handle this.
-// * Each grid is spawned onto an async threadpool with M threads allocated to itself, this should
-//   allow async waiting on the channel (or in the future from process/internet/bc processor)
-// * BC: A local thread can have a handle
 pub struct DistributedSystem {
-    /// Simple messaging system to be replaced by a more sophisticated system (e.g. run 5 steps,
+    recv: Receiver<(usize, MsgToHost)>,
-    /// collect, initialise, return something to main)
+    send: Vec<Sender<MsgFromHost>>,
-    /// This simply waits until all threads are ready, then starts the computation on all threads
+    /// All threads should be joined to mark the end of the computation
-    start: Arc<Barrier>,
-    _ntime: u64,
-    /// These should be joined to mark the end of the computation
     sys: Vec<std::thread::JoinHandle<()>>,
 }
 
 impl DistributedSystem {
-    pub fn run(self) {
+    pub fn advance(&mut self, ntime: u64) {
-        // This should start as we have n thread, but barrier blocks on n+1
+        for tid in &self.send {
-        self.start.wait();
+            tid.send(MsgFromHost::Advance(ntime));
-        self.sys.into_iter().for_each(|tid| tid.join().unwrap());
+        }
     }
+    pub fn output(&self) {
+        todo!()
+    }
+}
+
+impl Drop for DistributedSystem {
+    fn drop(&mut self) {
+        for tid in &self.send {
+            tid.send(MsgFromHost::Stop);
+        }
+        let handles = std::mem::take(&mut self.sys);
+        for tid in handles {
+            tid.join().unwrap()
+        }
+    }
+}
+
+enum MsgFromHost {
+    Advance(u64),
+    MaxDt(Float),
+    Output(u64),
+    Stop,
+}
+
+enum MsgToHost {
+    MaxDt(Float),
+    CurrentTimestep(u64),
 }
 
 // #[derive(Debug)]
@@ -441,23 +476,39 @@ struct DistributedSystemPart {
 
     current: Field,
     fut: Field,
-    k: [Diff; 4],
+
-    wb: WorkBuffers,
-    barrier: Arc<Barrier>,
-    _output: (), // hdf5::Dataset eventually,
     t: Float,
     dt: Float,
-    ntime: u64,
 
-    /// Work buffer for boundaries
+    name: String,
+    id: usize,
+    recv: Receiver<MsgFromHost>,
+    send: Sender<(usize, MsgToHost)>,
+
+    output: hdf5::Dataset,
+
+    k: [Diff; 4],
+    wb: WorkBuffers,
+    /// Work buffer for north/south boundary
     wb_ns: Array2<Float>,
+    /// Work buffer for east/west boundary
     wb_ew: Array2<Float>,
 }
 
 impl DistributedSystemPart {
-    fn advance(&mut self) {
+    fn run(&mut self) {
-        self.barrier.wait();
+        loop {
-        for _i in 0..self.ntime {
+            match self.recv.recv().unwrap() {
+                MsgFromHost::MaxDt(dt) => self.dt = dt,
+                MsgFromHost::Advance(ntime) => self.advance(ntime),
+                MsgFromHost::Output(ntime) => todo!(),
+                MsgFromHost::Stop => return,
+            }
+        }
+    }
+    fn advance(&mut self, ntime: u64) {
+        for ntime in 0..ntime {
+            self.send.send((self.id, MsgToHost::CurrentTimestep(ntime)));
             let metrics = &self.grid.1;
             let wb = &mut self.wb.0;
             let sbp = &self.sbp;
@@ -482,7 +533,6 @@ impl DistributedSystemPart {
                     s.send(y.west().to_owned()).unwrap()
                 }
 
-                use std::ops::Deref;
                 // This computation does not depend on the boundaries
                 euler::RHS_no_SAT(sbp.deref(), k, y, metrics, wb);