From 0aa8f2feb6dd3c8bc70ae60f92b86b30591120e8 Mon Sep 17 00:00:00 2001 From: Jake Faulkner Date: Mon, 5 Oct 2026 13:01:04 +1300 Subject: [PATCH 1/4] change psa solver --- IM/ims.py | 139 ++++- benches/intensity_benchmarks.rs | 59 ++- src-rust/lib.rs | 15 +- src-rust/psa.rs | 493 ++++++++---------- .../2024p220113_RLNS_HN_20/im_benchmark.csv | 14 +- .../3366146_DFHS_HN_20/im_benchmark.csv | 14 +- .../3497857_HVSC_HN_20/im_benchmark.csv | 14 +- .../3497857_PARS_HN_20/im_benchmark.csv | 14 +- .../3528839_LPCC_HN_20/im_benchmark.csv | 14 +- tests/resources/im_benchmark.csv | 14 +- tests/resources/nquist_freq/im_benchmark.csv | 14 +- tests/test_ims.py | 95 ++++ 12 files changed, 531 insertions(+), 368 deletions(-) diff --git a/IM/ims.py b/IM/ims.py index f199127d..f453d8cb 100644 --- a/IM/ims.py +++ b/IM/ims.py @@ -581,6 +581,116 @@ def ds595(waveform: Waveform, dt: float) -> xr.Dataset: return significant_duration(waveform, dt, 5, 95, IM.Ds595.value) +OSCILLATOR_SAMPLES_PER_PERIOD = 10 +"""Minimum samples per oscillator period before a record is upsampled for pSA. + +This is the conventional rule (Nigam and Jennings 1969; Boore and Goulet +2014) for resolving the peak oscillator response. +""" + +_UPSAMPLE_BUDGET = 2**24 +"""Samples of upsampling FFT (across all components) to hold in memory at once.""" + + +def _nigam_jennings_coefficients( + periods: npt.NDArray[np.float64], dt: float, damping: float = DAMPING +) -> npt.NDArray[np.float64]: + """Exact recurrence coefficients for unit-mass SDOF oscillators. + + The oscillator equation `u'' + 2 ξ ω u' + ω² u = -a_g(t)` is integrated + exactly over each step assuming `a_g` varies linearly between samples + (Nigam and Jennings 1969). This is the first-order-hold discretisation of + the state `[u, u']`, computed as the matrix exponential of the state + matrix augmented with the load and its slope, which avoids the + cancellation the textbook closed-form expressions suffer when `ω dt` is + small. + + Parameters + ---------- + periods : ndarray of float64 + Natural periods of the oscillators (s). + dt : float + Timestep of the ground acceleration (s). + damping : float, optional + Fraction of critical damping. + + Returns + ------- + ndarray of float64 + Shape `(len(periods), 8)`, one oscillator per row, laid out as + documented on `N_COEFFICIENTS` in `src-rust/psa.rs`. The state is + rescaled to `[ω² u, u']` so the recurrence yields pseudo-acceleration. + """ + w = 2 * np.pi / periods + # Augmented state [u, v, p, p'] with load p = -a_g and constant slope p'. + state = np.zeros((len(w), 4, 4)) + state[:, 0, 1] = 1.0 + state[:, 1, 0] = -(w**2) + state[:, 1, 1] = -2 * damping * w + state[:, 1, 2] = 1.0 + state[:, 2, 3] = 1.0 / dt + transition = sp.linalg.expm(state * dt) + # x[n+1] = Φ x[n] + Γ0 p[n] + Γ1 (p[n+1] - p[n]), with p = -a_g. + phi = transition[:, :2, :2] + gamma_0 = transition[:, :2, 2] + gamma_1 = transition[:, :2, 3] + coefficients = np.concatenate( + [phi.reshape(-1, 4), gamma_1 - gamma_0, -gamma_1], axis=1 + ) + # Rescale the displacement row and column to pseudo-acceleration. + coefficients[:, [1, 4, 6]] *= (w**2)[:, np.newaxis] + coefficients[:, 2] /= w**2 + return coefficients + + +def _upsampling_factors( + periods: npt.NDArray[np.float64], dt: float +) -> npt.NDArray[np.int64]: + """Power-of-two upsampling factor giving each period enough samples. + + Parameters + ---------- + periods : ndarray of float64 + Natural periods of the oscillators (s). + dt : float + Timestep of the record (s). + + Returns + ------- + ndarray of int64 + The smallest power of two `f` with `dt / f <= period / 10`. + """ + ratio = OSCILLATOR_SAMPLES_PER_PERIOD * dt / periods + exponent = np.ceil(np.log2(np.maximum(ratio, 1.0))) + return (2**exponent).astype(np.int64) + + +def _upsample_record(waveforms: np.ndarray, factor: int) -> np.ndarray: + """Band-limited (sinc) interpolation of waveforms by an integer factor. + + The record is zero padded to at least twice its length so the periodic + interpolation does not wrap the end of the record onto its start. + + Parameters + ---------- + waveforms : ndarray + Waveforms of shape `(..., nt)`. + factor : int + Upsampling factor. + + Returns + ------- + ndarray + Shape `(..., (nt - 1) * factor + 1)`, spanning the same time interval + and preserving the original samples. + """ + nt = waveforms.shape[-1] + padded = np.zeros(waveforms.shape[:-1] + (sp.fft.next_fast_len(2 * nt, real=True),)) + padded[..., :nt] = waveforms + upsampled = sp.signal.resample(padded, padded.shape[-1] * factor, axis=-1) + return upsampled[..., : (nt - 1) * factor + 1] + + def _psa_kernel( block: np.ndarray, *, @@ -603,8 +713,31 @@ def _psa_kernel( np.ndarray A chunk of solved pSA values. """ - (comp_0, comp_90, comp_ver), lead = _components(block) - psa = _core._psa(comp_0, comp_90, comp_ver, periods, dt, DAMPING) + components, lead = _components(block) + n_stations, nt = components.shape[1:] + factors = _upsampling_factors(periods, dt) + groups = { + factor: _nigam_jennings_coefficients(periods[factors == factor], dt / factor) + for factor in np.unique(factors) + } + max_factor = int(factors.max(initial=1)) + batch = ( + n_stations + if max_factor == 1 + else max(1, _UPSAMPLE_BUDGET // (len(components) * 2 * nt * max_factor)) + ) + psa = np.empty((n_stations, len(periods), len(ROTD_COMPONENTS))) + for start in range(0, n_stations, batch): + stations = slice(start, start + batch) + original = components[:, stations] + upsampled = ( + _upsample_record(original, max_factor) if max_factor > 1 else original + ) + for factor, coefficients in groups.items(): + # Keep the original samples exactly rather than their + # interpolated round trip. + chunk = original if factor == 1 else upsampled[..., :: max_factor // factor] + psa[stations, factors == factor] = _core._psa(*chunk, coefficients) return psa.reshape(lead + (len(periods), len(ROTD_COMPONENTS))) @@ -616,7 +749,7 @@ def pseudo_spectral_acceleration( """Compute pseudo-spectral acceleration (PSA) statistics. Calculates PSA for single-degree-of-freedom oscillators across various - periods using the Newmark-beta method and computes rotated (RotD) statistics. + periods. Parameters ---------- diff --git a/benches/intensity_benchmarks.rs b/benches/intensity_benchmarks.rs index 6137a24f..6299282b 100644 --- a/benches/intensity_benchmarks.rs +++ b/benches/intensity_benchmarks.rs @@ -17,8 +17,19 @@ const SAMPLE_LENGTHS: &[usize] = &[ 20_000, // 100 seconds 40_000, // 200 seconds ]; -const PSA_PERIODS: &[f64] = &[1.0]; // Should be independent of pSA period, but added here for safety in the future -const DAMPING: f64 = 0.05; +// Oscillator coefficients for T = 1 s, 5% damping at SAMPLING_RATE, from +// `IM.ims._nigam_jennings_coefficients(np.array([1.0]), 0.005)`. The cost of +// the recurrence does not depend on their values. +const PSA_COEFFICIENTS: [f64; psa::N_COEFFICIENTS] = [ + 0.9995070766804318, + 0.19704993255509468, + -0.004991333100781255, + 0.996370929600225, + -0.00032856713331003905, + -0.0024941543916031455, + -0.00016435618625817934, + -0.00249717870917811, +]; /// Benchmark CAV (Cumulative Absolute Velocity) calculations fn bench_cav(c: &mut Criterion) { @@ -131,30 +142,26 @@ fn bench_psa(c: &mut Criterion) { // PSA is expensive, so we might want to use a smaller sample size group.sample_size(10); - for &period in PSA_PERIODS { - for &stations in STATION_COUNTS { - // For PSA, use smaller sample sets to keep benchmark times reasonable - for &samples in SAMPLE_LENGTHS { - let waveforms = generate_waveforms(stations, samples); - let view = waveforms.view(); - let periods = array![period]; - let param = format!("T{:.1}s_{}stn_{}smp", period, stations, samples); - - group.throughput(Throughput::Bytes((stations * samples * 8) as u64)); - - group.bench_with_input(BenchmarkId::new("Sequential", ¶m), &view, |b, &v| { - b.iter(|| { - psa::psa( - black_box(&v), - black_box(&v), - black_box(&v), - black_box(&periods.view()), - black_box(SAMPLING_RATE), - black_box(DAMPING), - ) - }) - }); - } + let coefficients = array![PSA_COEFFICIENTS]; + for &stations in STATION_COUNTS { + // For PSA, use smaller sample sets to keep benchmark times reasonable + for &samples in SAMPLE_LENGTHS { + let waveforms = generate_waveforms(stations, samples); + let view = waveforms.view(); + let param = format!("T1.0s_{}stn_{}smp", stations, samples); + + group.throughput(Throughput::Bytes((stations * samples * 8) as u64)); + + group.bench_with_input(BenchmarkId::new("Sequential", ¶m), &view, |b, &v| { + b.iter(|| { + psa::psa( + black_box(&v), + black_box(&v), + black_box(&v), + black_box(&coefficients.view()), + ) + }) + }); } } diff --git a/src-rust/lib.rs b/src-rust/lib.rs index 02e27004..33b38f7a 100644 --- a/src-rust/lib.rs +++ b/src-rust/lib.rs @@ -13,7 +13,7 @@ use pyo3::prelude::*; /// import the module. #[pymodule] mod _core { - use numpy::{IntoPyArray, PyArray1, PyArray2, PyArray3, PyReadonlyArray1, PyReadonlyArray2}; + use numpy::{IntoPyArray, PyArray1, PyArray2, PyArray3, PyReadonlyArray2}; use pyo3::prelude::*; use crate::arias_intensity; @@ -75,8 +75,9 @@ mod _core { /// Pseudo-spectral acceleration statistics for every station and period. /// - /// The three components have shape `(ns, nt)`. Returns an `(ns, - /// n_periods, 9)` array whose last axis is laid out as + /// The three components have shape `(ns, nt)` and `coefficients` has + /// shape `(n_periods, 8)`, one oscillator per row as documented on + /// [`psa::N_COEFFICIENTS`]. Returns an `(ns, n_periods, 9)` array whose last axis is laid out as /// `ROTD_COMPONENTS`: the 000, 090, vertical and geometric mean peaks, /// then the five columns [`_rotd`] returns. #[pyfunction] @@ -85,15 +86,13 @@ mod _core { comp_0_py: PyReadonlyArray2, comp_90_py: PyReadonlyArray2, comp_ver_py: PyReadonlyArray2, - periods_py: PyReadonlyArray1, - dt: f64, - xi: f64, + coefficients_py: PyReadonlyArray2, ) -> Bound<'py, PyArray3> { let comp_0 = comp_0_py.as_array(); let comp_90 = comp_90_py.as_array(); let comp_ver = comp_ver_py.as_array(); - let periods = periods_py.as_array(); - let psa = py.detach(|| psa::psa(&comp_0, &comp_90, &comp_ver, &periods, dt, xi)); + let coefficients = coefficients_py.as_array(); + let psa = py.detach(|| psa::psa(&comp_0, &comp_90, &comp_ver, &coefficients)); psa.into_pyarray(py) } diff --git a/src-rust/psa.rs b/src-rust/psa.rs index 2261cf37..72e63d35 100644 --- a/src-rust/psa.rs +++ b/src-rust/psa.rs @@ -1,157 +1,136 @@ use ndarray::prelude::*; -use ndarray::{Array1, Ix1}; use crate::rotd::{self, Hull, N_ROTD_STATS}; -#[allow(clippy::too_many_arguments)] -fn newmark_beta_solver( - waveform: ArrayView, - dt: f64, - w: f64, - xi: f64, - gamma: f64, - beta: f64, - u0: f64, - dudt0: f64, -) -> Array1 { - let nt = waveform.dim(); - let mut u = Array1::zeros(nt); - let one_over_beta_dt_sq = 1.0 / (beta * dt * dt); - let one_over_beta_dt = 1.0 / (beta * dt); - let k = w * w; - let c = 2.0 * xi * w; - let c_gamma_over_beta_dt = (gamma * c) / (beta * dt); - let one_over_two_beta = 1.0 / (2.0 * beta); - // Constants to solve for u_n+1: - let kbar = one_over_beta_dt_sq + k + c_gamma_over_beta_dt; // u_n+1 - let a1 = c_gamma_over_beta_dt + one_over_beta_dt_sq; // u_n - let b1 = one_over_beta_dt + c * (gamma / beta - 1.0); // udot_n - let c1 = c * dt * (gamma / (2.0 * beta) - 1.0) + one_over_two_beta - 1.0; // uddot_n - - // Constants to solve for uddot_n+1 - let a2 = one_over_beta_dt_sq; // u_n+1 - u_n - let b2 = -one_over_beta_dt; // udot_n - let c2 = -c1; // uddot_n - // Constants to solve for udot_n+1 - // a'3 = 1 for uddot_n - let a3 = 1.0 - gamma; // uddot_n - let b3 = gamma; // uddot_n+1 - - u[0] = u0; - let mut udot = dudt0; - let mut uddot = -waveform[0] - (c * udot + k * u[0]); // negated because ground motion - - // ENCI335 notes solve for u''_n+1, but for numerical stability reasons we really want to solve for displacement directly and then derive velocity and acceleration from that. - // Basically the formulations in the ENCI notes solve for acceleration and then integrate to get displacement, but this involves the calculation of - // d_pti = waveform[i + 1] - waveform[i] - // which is a noisy floating-point operation. We then numerically integrate that noise twice which amplifies the noise carried into the rest of the calculations. - // Instead: implicitly solve for displacement and differentiate. The u at each time step is the smoothed response, so it is more robust to signal noise from the waveform. - // See https://collab.dvb.bayern/spaces/TUMmodsim/pages/71122788/Newmark-%CE%B2+method for the derivation when beta = 1/4, gamma = 1/2. - for i in 0..(nt - 1) { - let u_n = u[i]; - let f_next = -waveform[i + 1]; // Negated because ground motion. - let pbar = f_next + a1 * u_n + b1 * udot + c1 * uddot; - let u_next = pbar / kbar; - let uddot_next = a2 * (u_next - u_n) + b2 * udot + c2 * uddot; - let udot_next = udot + dt * (a3 * uddot + b3 * uddot_next); - u[i + 1] = u_next; - udot = udot_next; - uddot = uddot_next; - } - u -} +/// Number of coefficients describing one oscillator. +/// +/// A row of coefficients is the exact discrete-time update of a unit-mass +/// SDOF oscillator driven by a ground acceleration `ag` that varies linearly +/// between samples (Nigam and Jennings 1969, equivalently a first-order-hold +/// discretisation). The state is the pseudo-acceleration `A = ω² u` and the +/// relative velocity `v`, and the row `[a11, a12, a21, a22, b1, b2, c1, c2]` +/// steps it as +/// +/// ```text +/// A_{n+1} = a11 A_n + a12 v_n + b1 ag_n + c1 ag_{n+1} +/// v_{n+1} = a21 A_n + a22 v_n + b2 ag_n + c2 ag_{n+1} +/// ``` +/// +/// The sign of the inertial load `-ag` is folded into the `b` and `c` +/// coefficients. +pub const N_COEFFICIENTS: usize = 8; -fn choose_gamma_beta(dt: f64, w: f64) -> (f64, f64) { - let gamma = 0.5; +type Coefficients = [f64; N_COEFFICIENTS]; - let stability_constant = 0.551328895421792; - // Whilst the linear solver is theoretically stable for ratios - // dt/T up to 0.551-ish, we want to be a bit more conservative - // about when we choose the linear solver instead of the constant - // solver. During testing, we pick a conservative 80%. This means we leave - // some result accuracy on the table, but I can live with this - // because it only affects very short period pSA with large - // timesteps. - let stability_fraction = 0.8; - let effective_stability_constant = stability_fraction * stability_constant; - let pi = std::f64::consts::PI; - let beta = if dt < effective_stability_constant * (2.0 * pi) / w { - 1.0 / 6.0 - } else { - 1.0 / 4.0 +/// Peak pseudo-acceleration of each component's oscillator, starting from +/// rest. +/// +/// The 000 and 090 responses are also written to `response_0` and +/// `response_90`, which the RotD sweep needs in full. +fn respond( + components: [ArrayView1; 3], + coefficients: &Coefficients, + response_0: &mut Array1, + response_90: &mut Array1, +) -> [f64; 3] { + let [a11, a12, a21, a22, b1, b2, c1, c2] = *coefficients; + let step = |(a, v): (f64, f64), window: ArrayView1| { + ( + a11 * a + a12 * v + b1 * window[0] + c1 * window[1], + a21 * a + a22 * v + b2 * window[0] + c2 * window[1], + ) }; - - (gamma, beta) -} - -pub fn newmark_beta_method( - waveform: ArrayView, - dt: f64, - w: f64, - xi: f64, - u0: f64, - dudt0: f64, -) -> Array1 { - let (gamma, beta) = choose_gamma_beta(dt, w); - newmark_beta_solver(waveform, dt, w, xi, gamma, beta, u0, dudt0) + let [comp_0, comp_90, comp_ver] = components; + response_0[0] = 0.0; + response_90[0] = 0.0; + let mut states = [(0.0, 0.0); 3]; + let mut peaks = [0.0f64; 3]; + // The three components are independent recurrences, so stepping them + // together lets their latency-bound updates overlap. + azip!(( + window_0 in comp_0.windows(2), + window_90 in comp_90.windows(2), + window_ver in comp_ver.windows(2), + next_0 in response_0.slice_mut(s![1..]), + next_90 in response_90.slice_mut(s![1..]), + ) { + states = [ + step(states[0], window_0), + step(states[1], window_90), + step(states[2], window_ver), + ]; + *next_0 = states[0].0; + *next_90 = states[1].0; + for (peak, state) in peaks.iter_mut().zip(states) { + *peak = peak.max(state.0.abs()); + } + }); + peaks } /// Columns of a pSA row: the 000, 090, vertical and geometric mean peaks, /// then the five RotD statistics of [`rotd::rotd_stats`]. pub const N_PSA_COMPONENTS: usize = 4 + N_ROTD_STATS; -/// Peak pseudo-spectral acceleration of a displacement response. -/// -/// Multiplying by `w_squared` converts the peak relative displacement of the -/// unit-mass oscillator to a pseudo-spectral acceleration. -fn peak(response: &Array1, w_squared: f64) -> f64 { - w_squared * response.iter().fold(0.0f64, |m, &u| m.max(u.abs())) -} - /// Pseudo-spectral acceleration statistics for every station and period. +/// +/// Row `i` of `coefficients` describes the oscillator for period `i` at the +/// sampling interval of the components, as documented on [`N_COEFFICIENTS`]. /// Output shape: (stations, periods, components = 000, 090, VER, GEOM, rotd0, rotd50, rotd100, theta0, theta100). pub fn psa( comp_0: &ArrayView2, comp_90: &ArrayView2, comp_ver: &ArrayView2, - periods: &ArrayView1, - dt: f64, - xi: f64, + coefficients: &ArrayView2, ) -> Array3 { assert!( comp_0.dim() == comp_90.dim() && comp_0.dim() == comp_ver.dim(), "components must have matching shapes" ); - let ns = comp_0.nrows(); - let mut out = Array3::zeros((ns, periods.len(), N_PSA_COMPONENTS)); - let mut hull = Hull::with_capacity(comp_0.ncols()); - for (index, &period) in periods.iter().enumerate() { - let w = std::f64::consts::TAU / period; - let w_squared = w * w; - for s in 0..ns { - let mut row = out.slice_mut(s![s, index, ..]); - let row_is_finite = comp_0.row(s).iter().all(|v| v.is_finite()) - && comp_90.row(s).iter().all(|v| v.is_finite()) - && comp_ver.row(s).iter().all(|v| v.is_finite()); - if !row_is_finite { - // A non-finite input sample poisons the Newmark recursion - // from that point on, but the peak fold below would silently - // ignore the resulting NaN tail. Short-circuit instead. - row.fill(f64::NAN); + assert_eq!( + coefficients.ncols(), + N_COEFFICIENTS, + "coefficients must have {N_COEFFICIENTS} columns" + ); + let (ns, nt) = comp_0.dim(); + let mut out = Array3::zeros((ns, coefficients.nrows(), N_PSA_COMPONENTS)); + if nt == 0 { + return out; + } + // A non-finite input sample poisons the oscillator recursion from that + // point on, but the peak folds would silently ignore the resulting NaN + // tail. Short-circuit those stations instead. + let finite: Vec = (0..ns) + .map(|s| { + [comp_0, comp_90, comp_ver] + .iter() + .all(|comp| comp.row(s).iter().all(|v| v.is_finite())) + }) + .collect(); + let mut hull = Hull::with_capacity(nt); + let mut response_0 = Array1::zeros(nt); + let mut response_90 = Array1::zeros(nt); + for (index, row) in coefficients.rows().into_iter().enumerate() { + let oscillator: Coefficients = std::array::from_fn(|k| row[k]); + for (s, &is_finite) in finite.iter().enumerate() { + let mut out_row = out.slice_mut(s![s, index, ..]); + if !is_finite { + out_row.fill(f64::NAN); continue; } - let response_0 = newmark_beta_method(comp_0.row(s), dt, w, xi, 0.0, 0.0); - let response_90 = newmark_beta_method(comp_90.row(s), dt, w, xi, 0.0, 0.0); - let response_ver = newmark_beta_method(comp_ver.row(s), dt, w, xi, 0.0, 0.0); - let mut sweep = hull.peaks(response_0.view(), response_90.view()); - sweep.iter_mut().for_each(|peak| *peak *= w_squared); - let peak_0 = peak(&response_0, w_squared); - let peak_90 = peak(&response_90, w_squared); - row[0] = peak_0; - row[1] = peak_90; - row[2] = peak(&response_ver, w_squared); - row[3] = (peak_0 * peak_90).sqrt(); - row.slice_mut(s![4..]) + let [peak_0, peak_90, peak_ver] = respond( + [comp_0.row(s), comp_90.row(s), comp_ver.row(s)], + &oscillator, + &mut response_0, + &mut response_90, + ); + let sweep = hull.peaks(response_0.view(), response_90.view()); + out_row[0] = peak_0; + out_row[1] = peak_90; + out_row[2] = peak_ver; + out_row[3] = (peak_0 * peak_90).sqrt(); + out_row + .slice_mut(s![4..]) .assign(&ArrayView1::from(&rotd::rotd_stats(sweep))); } } @@ -165,141 +144,114 @@ mod tests { use approx::assert_abs_diff_eq; use std::f64::consts::PI; const XI: f64 = 0.05; - // Linear gamma and beta settings - const GAMMA: f64 = 0.5; - const BETA: f64 = 1.0 / 6.0; - const M: f64 = 1.0; - #[test] - fn test_newmark_linear_parameters() { - let dt = 0.01f64; - let w = 2.0 * PI; // T_n = 1 - let (gamma, beta) = choose_gamma_beta(dt, w); - assert_eq!((gamma, beta), (0.5, 1.0 / 6.0)); - } - - #[test] - fn test_newmark_const_parameters() { - let dt = 0.01f64; - let w = 100.0 * 2.0 * PI; // T_n = 0.01 - let (gamma, beta) = choose_gamma_beta(dt, w); - assert_eq!((gamma, beta), (0.5, 0.25)); - } - - fn differentiate(waveform: &Array, dt: f64) -> Array { - let nt = waveform.dim(); - Array1::from_shape_fn(nt, |i| { - if i == 0 { - (waveform[i + 1] - waveform[i]) / dt - } else if i == nt - 1 { - (waveform[i] - waveform[i - 1]) / dt - } else { - (waveform[i + 1] - waveform[i - 1]) / (2.0 * dt) - } - }) + /// Coefficients for angular frequency `w` and damping ratio `xi` at + /// sampling interval `dt`, from a scaled-and-squared Taylor series of the + /// matrix exponential the Python layer takes with `scipy.linalg.expm`. + fn coefficients(dt: f64, w: f64, xi: f64) -> Coefficients { + // State [u, v, p, p'] with p = -ag varying linearly over the step. + let mut m = Array2::::zeros((4, 4)); + m[[0, 1]] = dt; + m[[1, 0]] = -w * w * dt; + m[[1, 1]] = -2.0 * xi * w * dt; + m[[1, 2]] = dt; + m[[2, 3]] = 1.0; + let norm = m + .rows() + .into_iter() + .map(|row| row.iter().map(|x| x.abs()).sum::()) + .fold(0.0, f64::max); + let squarings = norm.log2().ceil().max(0.0) as i32 + 1; + let scaled = m * 0.5f64.powi(squarings); + let mut term = Array2::::eye(4); + let mut e = term.clone(); + for k in 1..=20 { + term = term.dot(&scaled) / k as f64; + e += &term; + } + for _ in 0..squarings { + e = e.dot(&e); + } + let w2 = w * w; + [ + e[[0, 0]], + e[[0, 1]] * w2, + e[[1, 0]] / w2, + e[[1, 1]], + -(e[[0, 2]] - e[[0, 3]]) * w2, + -(e[[1, 2]] - e[[1, 3]]), + -e[[0, 3]] * w2, + -e[[1, 3]], + ] } - fn ddifferentiate(waveform: &Array, dt: f64) -> Array { - let nt = waveform.dim(); - Array1::from_shape_fn(nt, |i| { - if i == 0 { - (waveform[i + 2] - 2.0 * waveform[i + 1] + waveform[i]) / (dt * dt) - } else if i == nt - 1 { - (waveform[i] - 2.0 * waveform[i - 1] + waveform[i - 2]) / (dt * dt) - } else { - (waveform[i + 1] - 2.0 * waveform[i] + waveform[i - 1]) / (dt * dt) - } - }) + /// Pseudo-acceleration response to a single component. + fn response(waveform: &Array1, coefficients: &Coefficients) -> Array1 { + let mut response = Array1::zeros(waveform.len()); + let mut unused = Array1::zeros(waveform.len()); + let view = waveform.view(); + respond([view, view, view], coefficients, &mut response, &mut unused); + response } #[test] - fn test_newmark_beta_single_zeros() { + fn test_zeros() { let waveform = Array1::::zeros(100); - let dt = 0.01; - let w = 1.0; - - let u = newmark_beta_solver(waveform.view(), dt, w, XI, GAMMA, BETA, 0.0, 0.0); - let expected = Array1::::zeros(100); - assert_eq!(u, expected); + let a = response(&waveform, &coefficients(0.01, 1.0, XI)); + assert_eq!(a, Array1::::zeros(100)); } #[test] - fn test_newmark_beta_solves_constant() { + fn test_solves_constant() { let dt = 0.001; let waveform = Array1::::ones(100_000); - let w = 2.0 * PI; - - let u = newmark_beta_solver(waveform.view(), dt, w, XI, GAMMA, BETA, 0.0, 0.0); - - // Of course we could use the exact solution here. This test is to determine - // the long-term behaviour of the solver accumulating floating point error. - let uss = -M / (M * w * w); - let uss_est = u[waveform.dim() - 1]; - let err = (uss_est - uss).abs(); - assert!( - err < 1e-4, - "NB method did not solve for steady-state solution (uss = {}): |{} - {}| = {} > 1e-4", - uss, - uss, - uss_est, - err - ); + let a = response(&waveform, &coefficients(dt, 2.0 * PI, XI)); + // The long-run steady state, u = -1 / w^2, also checks the recursion + // does not accumulate floating point error. + assert_abs_diff_eq!(a[waveform.len() - 1], -1.0, epsilon = 1e-12); } #[test] - fn test_newmark_beta_solves_equation() { - // Test a more complicated sum of frequencies - let t = Array1::::linspace(0.0, 10.0, 100_000); - let dt = t[1] - t[0]; - // W(t) = sum_i 1/i * sin(pi * i * t) - let waveform = t.map(|&t| { - (1..100) - .map(|freq| { - let freq_f = freq as f64; - 1.0 / freq_f * (freq_f * PI * t).sin() - }) - .sum() - }); - let w = 2.0 * PI; - - let u = newmark_beta_solver(waveform.view(), dt, w, XI, GAMMA, BETA, 0.0, 0.0); - - let du = differentiate(&u, dt); - let du2 = ddifferentiate(&u, dt); - let c = 2.0 * XI * w; - let k = M * w * w; - let sdof_invariant = M * du2 + c * du + k * u + M * waveform; - // Skipping the first and last value of u because the differentiation is less accurate at the boundary. - - let max_deviation = sdof_invariant - .slice(s![1..sdof_invariant.len() - 1]) - .map(|&x| x.abs()) - .fold(0.0, |x: f64, &y| x.max(y)); - assert_abs_diff_eq!(max_deviation, 0.0, epsilon = 5e-4); + fn test_is_exact_for_a_ramp() { + // A ramp is piecewise linear, so the recurrence should reproduce the + // analytical response to rounding error even at a coarse step. + let dt = 0.02; + let w = 2.0 * PI / 0.3; + let t = Array1::from_shape_fn(2000, |i| i as f64 * dt); + for xi in [0.0, XI, 0.7] { + let a = response(&t, &coefficients(dt, w, xi)); + + // u'' + 2 xi w u' + w^2 u = -t from rest. + let wd = w * (1.0 - xi * xi).sqrt(); + let c = -2.0 * xi / w.powi(3); + let d = (1.0 / (w * w) + xi * w * c) / wd; + let u = t.map(|&x| { + -(x / (w * w) - 2.0 * xi / w.powi(3)) + + (-xi * w * x).exp() * (c * (wd * x).cos() + d * (wd * x).sin()) + }); + assert_abs_diff_eq!(a, u * w * w, epsilon = 1e-9); + } } #[test] - fn test_newmark_solves_undamped_free_vibration() { + fn test_solves_critically_damped_harmonic_oscillation() { let t = Array1::::linspace(0.0, 10.0, 10000); let dt = t[1] - t[0]; - let waveform = Array1::zeros(t.dim()); - let w = 2.0 * PI; + let waveform = t.sin(); - let u = newmark_beta_solver(waveform.view(), dt, w, 0.0, GAMMA, BETA, 1.0, 0.0); - let analytical = (t * w).cos(); - assert_abs_diff_eq!(u, analytical, epsilon = 5e-4); + let a = response(&waveform, &coefficients(dt, 1.0, 1.0)); + // A sine is not piecewise linear, so expect an O(dt^2) error. + let analytical = t.map(|&x| -0.5 * (-x).exp() * (x - x.exp() * x.cos() + 1.0)); + assert_abs_diff_eq!(a, analytical, epsilon = 1e-7); } - #[test] - fn test_newmark_solves_damped_free_vibration() { - let t = Array1::::linspace(0.0, 10.0, 10000); - let dt = t[1] - t[0]; - let waveform = Array1::zeros(t.dim()); - let w = 1.0; - let xi = 1.0; - let u = newmark_beta_solver(waveform.view(), dt, w, xi, GAMMA, BETA, 1.0, 0.0); - let analytical = t.map(|&x| (-x).exp() * (x + 1.0)); - assert_abs_diff_eq!(u, analytical, epsilon = 5e-4); + fn coefficient_table(periods: &[f64], dt: f64) -> Array2 { + Array2::from( + periods + .iter() + .map(|&period| coefficients(dt, 2.0 * PI / period, XI)) + .collect::>(), + ) } #[test] @@ -309,31 +261,23 @@ mod tests { // peak ground motion path run on those responses. let t = Array1::::linspace(0.0, 2.0, 512); let dt = t[1] - t[0]; - let comp_0 = t.map(|&x| (3.0 * x).sin()).insert_axis(Axis(0)); - let comp_90 = t.map(|&x| 0.7 * (5.0 * x).cos()).insert_axis(Axis(0)); - let comp_ver = t.map(|&x| 0.2 * (7.0 * x).sin()).insert_axis(Axis(0)); - let period = 1.0; - let w_squared = (std::f64::consts::TAU / period).powi(2); + let comp_0 = t.map(|&x| (3.0 * x).sin()); + let comp_90 = t.map(|&x| 0.7 * (5.0 * x).cos()); + let comp_ver = t.map(|&x| 0.2 * (7.0 * x).sin()); let result = psa( - &comp_0.view(), - &comp_90.view(), - &comp_ver.view(), - &array![period].view(), - dt, - XI, + &comp_0.view().insert_axis(Axis(0)), + &comp_90.view().insert_axis(Axis(0)), + &comp_ver.view().insert_axis(Axis(0)), + &coefficient_table(&[1.0], dt).view(), ); - let responses: Vec> = [&comp_0, &comp_90, &comp_ver] + let oscillator = coefficients(dt, 2.0 * PI, XI); + let responses: Vec> = [&comp_0, &comp_90, &comp_ver] .iter() - .map(|comp| { - newmark_beta_method(comp.row(0), dt, w_squared.sqrt(), XI, 0.0, 0.0) - .insert_axis(Axis(0)) - }) + .map(|comp| response(comp, &oscillator)) .collect(); - let peak = |response: &Array2| { - w_squared * response.iter().fold(0.0f64, |m, &u| m.max(u.abs())) - }; + let peak = |response: &Array1| response.iter().fold(0.0f64, |m, &a| m.max(a.abs())); for (column, response) in responses.iter().enumerate() { assert_abs_diff_eq!(result[[0, 0, column]], peak(response), epsilon = 1e-12); @@ -343,11 +287,11 @@ mod tests { (peak(&responses[0]) * peak(&responses[1])).sqrt(), epsilon = 1e-12 ); - // The statistics of the scaled responses, straight off the peak - // ground motion entry point. + // The statistics of the responses, straight off the peak ground + // motion entry point. let stats = crate::rotd::rotd( - (&responses[0] * w_squared).view(), - (&responses[1] * w_squared).view(), + responses[0].view().insert_axis(Axis(0)), + responses[1].view().insert_axis(Axis(0)), ); for column in 0..N_ROTD_STATS { assert_abs_diff_eq!( @@ -360,9 +304,9 @@ mod tests { #[test] fn test_psa_is_nan_when_a_sample_is_non_finite() { - // A NaN sample poisons the Newmark recursion from that point on, but - // the peak fold used to ignore the resulting NaN tail and return the - // (finite, wrong) peak of the record up to that point instead. + // A NaN sample poisons the oscillator recursion from that point on, + // but the peak fold used to ignore the resulting NaN tail and return + // the (finite, wrong) peak of the record up to that point instead. let t = Array1::::linspace(0.0, 2.0, 512); let dt = t[1] - t[0]; let mut comp_0 = t.map(|&x| (3.0 * x).sin()); @@ -375,9 +319,7 @@ mod tests { &comp_0.view(), &comp_90.view(), &comp_ver.view(), - &array![1.0].view(), - dt, - XI, + &coefficient_table(&[1.0], dt).view(), ); for column in 0..N_PSA_COMPONENTS { @@ -388,17 +330,4 @@ mod tests { ); } } - - #[test] - fn test_newmark_solves_damped_harmonic_oscillation() { - let t = Array1::::linspace(0.0, 10.0, 10000); - let dt = t[1] - t[0]; - let waveform = t.sin(); - let w = 1.0; - let xi = 1.0; - let u = newmark_beta_solver(waveform.view(), dt, w, xi, GAMMA, BETA, 0.0, 0.0); - - let analytical = t.map(|&x| -0.5 * (-x).exp() * (x - x.exp() * x.cos() + 1.0)); - assert_abs_diff_eq!(u, analytical, epsilon = 5e-4); - } } diff --git a/tests/resources/2024p220113_RLNS_HN_20/im_benchmark.csv b/tests/resources/2024p220113_RLNS_HN_20/im_benchmark.csv index d09e478c..8effd173 100644 --- a/tests/resources/2024p220113_RLNS_HN_20/im_benchmark.csv +++ b/tests/resources/2024p220113_RLNS_HN_20/im_benchmark.csv @@ -1,8 +1,8 @@ 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diff --git a/tests/resources/3366146_DFHS_HN_20/im_benchmark.csv b/tests/resources/3366146_DFHS_HN_20/im_benchmark.csv index 0cf6ba05..abf31578 100644 --- a/tests/resources/3366146_DFHS_HN_20/im_benchmark.csv +++ b/tests/resources/3366146_DFHS_HN_20/im_benchmark.csv @@ -1,8 +1,8 @@ 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diff --git a/tests/resources/3497857_HVSC_HN_20/im_benchmark.csv b/tests/resources/3497857_HVSC_HN_20/im_benchmark.csv index 975be26e..9502d172 100644 --- a/tests/resources/3497857_HVSC_HN_20/im_benchmark.csv +++ b/tests/resources/3497857_HVSC_HN_20/im_benchmark.csv @@ -1,9 +1,9 @@ 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diff --git a/tests/resources/3497857_PARS_HN_20/im_benchmark.csv b/tests/resources/3497857_PARS_HN_20/im_benchmark.csv index e6269efb..3d11b326 100644 --- a/tests/resources/3497857_PARS_HN_20/im_benchmark.csv +++ b/tests/resources/3497857_PARS_HN_20/im_benchmark.csv @@ -1,9 +1,9 @@ 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diff --git a/tests/resources/3528839_LPCC_HN_20/im_benchmark.csv b/tests/resources/3528839_LPCC_HN_20/im_benchmark.csv index 1c718f48..7d7f0f54 100644 --- a/tests/resources/3528839_LPCC_HN_20/im_benchmark.csv +++ b/tests/resources/3528839_LPCC_HN_20/im_benchmark.csv @@ -1,9 +1,9 @@ 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diff --git a/tests/resources/im_benchmark.csv b/tests/resources/im_benchmark.csv index 68fe2753..63e17733 100644 --- a/tests/resources/im_benchmark.csv +++ b/tests/resources/im_benchmark.csv @@ -1,9 +1,9 @@ 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diff --git a/tests/resources/nquist_freq/im_benchmark.csv b/tests/resources/nquist_freq/im_benchmark.csv index b5938d48..aaafa63f 100644 --- a/tests/resources/nquist_freq/im_benchmark.csv +++ b/tests/resources/nquist_freq/im_benchmark.csv @@ -1,8 +1,8 @@ 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diff --git a/tests/test_ims.py b/tests/test_ims.py index c3a11b46..75ee6e65 100644 --- a/tests/test_ims.py +++ b/tests/test_ims.py @@ -9,6 +9,7 @@ import numpy.typing as npt import pandas as pd import pytest +import scipy as sp import xarray as xr from hypothesis import given, settings from hypothesis import strategies as st @@ -972,3 +973,97 @@ def test_rechunks_component_and_time_core_dims( expected = ims.peak_ground_acceleration(sample_waveforms) for component in expected.data_vars: assert_array_equal(expected[component].values, computed[component].values) + + +@pytest.mark.parametrize("dt", [0.0005, 0.005, 0.02]) +def test_nigam_jennings_coefficients_match_scipy_foh(dt: float) -> None: + periods = np.array([0.01, 0.05, 0.1, 0.5, 1.0, 10.0]) + coefficients = ims._nigam_jennings_coefficients(periods, dt, 0.05) + for period, row in zip(periods, coefficients): + w = 2 * np.pi / period + state = np.array([[0, 1], [-(w**2), -2 * 0.05 * w]]) + # Outputs [u, v] and load -a_g, so the feedthrough is the a_g[n+1] term. + a, b, _, d, _ = sp.signal.cont2discrete( + (state, np.array([[0.0], [-1.0]]), np.eye(2), np.zeros((2, 1))), + dt, + method="foh", + ) + b_n = b[:, 0] - a @ d[:, 0] + # Rescale the state [u, v] to [w^2 u, v]. + scale = np.array([w**2, 1.0]) + a = a * scale[:, np.newaxis] / scale + expected = np.concatenate([a.ravel(), b_n * scale, d[:, 0] * scale]) + np.testing.assert_allclose(row, expected, rtol=1e-12, atol=0) + + +@pytest.mark.parametrize( + "period,factor", [(1.0, 1), (0.05, 1), (0.049, 2), (0.025, 2), (0.02, 4), (0.01, 8)] +) +def test_upsampling_factors(period: float, factor: int) -> None: + assert ims._upsampling_factors(np.array([period]), 0.005)[0] == factor + + +def test_sinc_upsample_interpolates_band_limited_signals() -> None: + dt = 0.01 + t = np.arange(1000) * dt + # A pulse well below Nyquist, decayed to zero at both ends of the record. + envelope = np.exp(-(((t - 5) / 1.0) ** 2)) + signal = envelope * np.sin(2 * np.pi * 7.3 * t) + upsampled = ims._upsample_record(signal[np.newaxis], 8)[0] + + assert upsampled.shape == ((len(t) - 1) * 8 + 1,) + np.testing.assert_allclose(upsampled[::8], signal, atol=1e-12) + fine_t = np.arange(len(upsampled)) * dt / 8 + expected = np.exp(-(((fine_t - 5) / 1.0) ** 2)) * np.sin(2 * np.pi * 7.3 * fine_t) + np.testing.assert_allclose(upsampled, expected, atol=1e-8) + + +def test_psa_matches_an_independent_oscillator_solution() -> None: + """pSA at periods solved without upsampling must equal the peak response of + scipy's own linear-interpolation simulation of the oscillator.""" + dt = 0.005 + rng = np.random.default_rng(0) + t = np.arange(4000) * dt + ag = rng.normal(size=t.size) * np.sin(np.pi * t / t[-1]) ** 2 + waveforms = np.broadcast_to(ag, (3, 1, t.size)) + periods = np.array([0.05, 0.1, 0.3, 1.0, 3.0]) + + result = ims.pseudo_spectral_acceleration(waveforms, periods, dt) + + for period, value in zip(periods, result["000"].values[0]): + w = 2 * np.pi / period + system = sp.signal.StateSpace( + [[0, 1], [-(w**2), -2 * ims.DAMPING * w]], [[0], [-1]], [[1, 0]], [[0]] + ) + _, u, _ = sp.signal.lsim(system, ag, t, interp=True) + assert value == pytest.approx(w**2 * np.abs(u).max(), rel=1e-8) + + +def test_short_period_psa_tends_to_pga() -> None: + """A stiff oscillator moves with the ground, so pSA must approach the peak + of the continuous acceleration, including any peak between samples.""" + dt = 0.02 + + def acceleration(t: npt.NDArray[np.float64]) -> npt.NDArray[np.float64]: + envelope = np.exp(-(((t - 10) / 3) ** 2)) + return envelope * ( + np.sin(2 * np.pi * 1.1 * t) + 0.5 * np.sin(2 * np.pi * 2.3 * t) + ) + + ag = acceleration(np.arange(1000) * dt) + waveforms = np.broadcast_to(ag, (3, 1, ag.size)) + peak = np.abs(acceleration(np.arange(1000 * 64) * dt / 64)).max() + + psa = ims.pseudo_spectral_acceleration(waveforms, np.array([0.01]), dt) + assert psa["000"].item() == pytest.approx(peak, rel=1e-3) + + +def test_upsampled_psa_is_nan_only_for_the_non_finite_station( + sample_waveforms: npt.NDArray[np.float64], +) -> None: + waveforms = sample_waveforms.copy() + waveforms[ims.Component.COMP_0, 0, 10] = np.nan + psa = ims.pseudo_spectral_acceleration(waveforms, [0.02, 1.0], 0.01) + for component in ims.ROTD_COMPONENTS: + assert np.isnan(psa[component].values[0]).all() + assert np.isfinite(psa[component].values[1]).all() From 496026f4185f7423c4beae015b9449ab3f922dc1 Mon Sep 17 00:00:00 2001 From: Jake Faulkner Date: Mon, 5 Oct 2026 13:24:46 +1300 Subject: [PATCH 2/4] simplify NaN checks --- src-rust/psa.rs | 59 +++++++++++++++++++++++++++++++------------------ 1 file changed, 37 insertions(+), 22 deletions(-) diff --git a/src-rust/psa.rs b/src-rust/psa.rs index 72e63d35..9c2cd4c4 100644 --- a/src-rust/psa.rs +++ b/src-rust/psa.rs @@ -23,7 +23,7 @@ pub const N_COEFFICIENTS: usize = 8; type Coefficients = [f64; N_COEFFICIENTS]; /// Peak pseudo-acceleration of each component's oscillator, starting from -/// rest. +/// rest, or `None` if any component's response is not finite. /// /// The 000 and 090 responses are also written to `response_0` and /// `response_90`, which the RotD sweep needs in full. @@ -32,7 +32,7 @@ fn respond( coefficients: &Coefficients, response_0: &mut Array1, response_90: &mut Array1, -) -> [f64; 3] { +) -> Option<[f64; 3]> { let [a11, a12, a21, a22, b1, b2, c1, c2] = *coefficients; let step = |(a, v): (f64, f64), window: ArrayView1| { ( @@ -45,8 +45,6 @@ fn respond( response_90[0] = 0.0; let mut states = [(0.0, 0.0); 3]; let mut peaks = [0.0f64; 3]; - // The three components are independent recurrences, so stepping them - // together lets their latency-bound updates overlap. azip!(( window_0 in comp_0.windows(2), window_90 in comp_90.windows(2), @@ -65,7 +63,13 @@ fn respond( *peak = peak.max(state.0.abs()); } }); - peaks + // A non-finite sample makes the state non-finite from that step on, since + // no arithmetic turns inf or NaN finite again. The peak folds ignore NaN, + // so the final state is what reveals it. + states + .iter() + .all(|(a, v)| a.is_finite() && v.is_finite()) + .then_some(peaks) } /// Columns of a pSA row: the 000, 090, vertical and geometric mean peaks, @@ -97,33 +101,22 @@ pub fn psa( if nt == 0 { return out; } - // A non-finite input sample poisons the oscillator recursion from that - // point on, but the peak folds would silently ignore the resulting NaN - // tail. Short-circuit those stations instead. - let finite: Vec = (0..ns) - .map(|s| { - [comp_0, comp_90, comp_ver] - .iter() - .all(|comp| comp.row(s).iter().all(|v| v.is_finite())) - }) - .collect(); let mut hull = Hull::with_capacity(nt); let mut response_0 = Array1::zeros(nt); let mut response_90 = Array1::zeros(nt); for (index, row) in coefficients.rows().into_iter().enumerate() { let oscillator: Coefficients = std::array::from_fn(|k| row[k]); - for (s, &is_finite) in finite.iter().enumerate() { + for s in 0..ns { let mut out_row = out.slice_mut(s![s, index, ..]); - if !is_finite { - out_row.fill(f64::NAN); - continue; - } - let [peak_0, peak_90, peak_ver] = respond( + let Some([peak_0, peak_90, peak_ver]) = respond( [comp_0.row(s), comp_90.row(s), comp_ver.row(s)], &oscillator, &mut response_0, &mut response_90, - ); + ) else { + out_row.fill(f64::NAN); + continue; + }; let sweep = hull.peaks(response_0.view(), response_90.view()); out_row[0] = peak_0; out_row[1] = peak_90; @@ -330,4 +323,26 @@ mod tests { ); } } + + #[test] + fn test_psa_is_nan_when_the_last_sample_is_infinite() { + // The last sample only reaches the final state, which is then + // infinite rather than NaN. + let t = Array1::::linspace(0.0, 2.0, 512); + let dt = t[1] - t[0]; + let comp_0 = t.map(|&x| (3.0 * x).sin()).insert_axis(Axis(0)); + let comp_90 = t.map(|&x| 0.7 * (5.0 * x).cos()).insert_axis(Axis(0)); + let mut comp_ver = t.map(|&x| 0.2 * (7.0 * x).sin()); + comp_ver[511] = f64::INFINITY; + let comp_ver = comp_ver.insert_axis(Axis(0)); + + let result = psa( + &comp_0.view(), + &comp_90.view(), + &comp_ver.view(), + &coefficient_table(&[1.0], dt).view(), + ); + + assert!(result.iter().all(|v| v.is_nan()), "{result}"); + } } From 33fb1e90625b8c84897e87499db8e81074710b08 Mon Sep 17 00:00:00 2001 From: Jake Faulkner Date: Mon, 5 Oct 2026 13:24:52 +1300 Subject: [PATCH 3/4] add explanatory comment to the transition matrix --- IM/ims.py | 11 +++++++++-- 1 file changed, 9 insertions(+), 2 deletions(-) diff --git a/IM/ims.py b/IM/ims.py index f453d8cb..eded55e1 100644 --- a/IM/ims.py +++ b/IM/ims.py @@ -622,7 +622,7 @@ def _nigam_jennings_coefficients( rescaled to `[ω² u, u']` so the recurrence yields pseudo-acceleration. """ w = 2 * np.pi / periods - # Augmented state [u, v, p, p'] with load p = -a_g and constant slope p'. + # Augmented state [u, u', p, p'] with load p = -a_g and constant slope p'. state = np.zeros((len(w), 4, 4)) state[:, 0, 1] = 1.0 state[:, 1, 0] = -(w**2) @@ -630,7 +630,14 @@ def _nigam_jennings_coefficients( state[:, 1, 2] = 1.0 state[:, 2, 3] = 1.0 / dt transition = sp.linalg.expm(state * dt) - # x[n+1] = Φ x[n] + Γ0 p[n] + Γ1 (p[n+1] - p[n]), with p = -a_g. + # This transition matrix predicts p(n + 1) and p'(n + 1), but we only need + # the part of the transition matrix that corresponds to the u, u'. The + # matrix E = exp(state * dt) will have shape + # [Φ Γ0 Γ1; 0 0 1 1; 0 0 0 1] + # Where Φ, Γ0, Γ1 are 2 x 2 matrices. Matrix exponentiation shows [u(n + 1), + # u'(n + 1), p(n + 1), p'(n + 1)] = E [u(n), u'(n), p(n), p'(n)]. + # Noting that p'(n) = p(n + 1) - p(n), + # u(n+1) = Φ u(n) + Γ0 p(n) + Γ1 (p(n+1) - p(n)), with p = -a_g. phi = transition[:, :2, :2] gamma_0 = transition[:, :2, 2] gamma_1 = transition[:, :2, 3] From f29711fbd21745d0f0587051bdef77295d8e2909 Mon Sep 17 00:00:00 2001 From: Jake Faulkner Date: Mon, 5 Oct 2026 13:26:42 +1300 Subject: [PATCH 4/4] update tests --- tests/test_ims.py | 11 +++++++---- 1 file changed, 7 insertions(+), 4 deletions(-) diff --git a/tests/test_ims.py b/tests/test_ims.py index 75ee6e65..d3de793a 100644 --- a/tests/test_ims.py +++ b/tests/test_ims.py @@ -1032,8 +1032,11 @@ def test_psa_matches_an_independent_oscillator_solution() -> None: for period, value in zip(periods, result["000"].values[0]): w = 2 * np.pi / period - system = sp.signal.StateSpace( - [[0, 1], [-(w**2), -2 * ims.DAMPING * w]], [[0], [-1]], [[1, 0]], [[0]] + system = ( + np.array([[0.0, 1.0], [-(w**2), -2 * ims.DAMPING * w]]), + np.array([[0.0], [-1.0]]), + np.array([[1.0, 0.0]]), + np.array([[0.0]]), ) _, u, _ = sp.signal.lsim(system, ag, t, interp=True) assert value == pytest.approx(w**2 * np.abs(u).max(), rel=1e-8) @@ -1050,9 +1053,9 @@ def acceleration(t: npt.NDArray[np.float64]) -> npt.NDArray[np.float64]: np.sin(2 * np.pi * 1.1 * t) + 0.5 * np.sin(2 * np.pi * 2.3 * t) ) - ag = acceleration(np.arange(1000) * dt) + ag = acceleration(np.arange(1000, dtype=np.float64) * dt) waveforms = np.broadcast_to(ag, (3, 1, ag.size)) - peak = np.abs(acceleration(np.arange(1000 * 64) * dt / 64)).max() + peak = np.abs(acceleration(np.arange(1000 * 64, dtype=np.float64) * dt / 64)).max() psa = ims.pseudo_spectral_acceleration(waveforms, np.array([0.01]), dt) assert psa["000"].item() == pytest.approx(peak, rel=1e-3)