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We address the computational challenges of and presents results from ventricle-valve-aorta flow analysis. Including the left ventricle (LV) in the model makes the flow into the valve, and consequently the flow into the aorta, anatomically more realistic. The challenges include accurate representation of the boundary layers near moving solid surfaces even when the valve leaflets come into contact, computation with high geometric complexity, anatomically realistic representation of the LV motion, and flow stability at the inflow boundary, which has a traction condition. The challenges are mainly addressed with a Space–Time (ST) method that integrates three special ST methods around the core, ST Variational Multiscale (ST-VMS) method. The three special methods are the ST Slip Interface (ST-SI) and ST Topology Change (ST-TC) methods and ST Isogeometric Analysis (ST-IGA). The ST-discretization feature of the integrated method, ST-SI-TC-IGA, provides higher-order accuracy compared to standard discretization methods. The VMS feature addresses the computational challenges associated with the multiscale nature of the unsteady flow in the LV, valve and aorta. The moving-mesh feature of the ST framework enables high-resolution computation near the leaflets. The ST-TC enables moving-mesh computation even with the TC created by the contact between the leaflets, dealing with the contact while maintaining high-resolution representation near the leaflets. The ST-IGA provides smoother representation of the LV, valve and aorta surfaces and increased accuracy in the flow solution. The ST-SI connects the separately generated LV, valve and aorta NURBS meshes, enabling easier mesh generation, connects the mesh zones containing the leaflets, enabling a more effective mesh moving, helps the ST-TC deal with leaflet–leaflet contact location change and contact sliding, and helps the ST-TC and ST-IGA keep the element density in the narrow spaces near the contact areas at a reasonable level. The ST-SI-TC-IGA is supplemented with two other special methods in this article. A structural mechanics computation method generates the LV motion from the CT scans of the LV and anatomically realistic values for the LV volume ratio. The Constrained-Flow-Profile (CFP) Traction provides flow stability at the inflow boundary. Test computation with the CFP Traction shows its effectiveness as an inflow stabilization method, and computation with the LV-valve-aorta model shows the effectiveness of the ST-SI-TC-IGA and the two supplemental methods.

In an article published online in July 2018 it was stated that the algorithm proposed in the article is “enabling practical implementation of the space–time FEM for engineering applications.” In fact, space–time computations in practical engineering applications were already enabled in 1993. We summarize the computations that have taken place since then. These computations started with finite element discretization and are now also with isogeometric discretization. They were all in 3D space and were all carried out on parallel computers. For quarter of a century, these computations brought solution to many classes of complex problems ranging from Orion spacecraft parachutes to wind turbines, from patient-specific cerebral aneurysms to heart valves, from thermo-fluid analysis of ground vehicles and tires to turbocharger turbines and exhaust manifolds.

This work performs a numerical comparison between different discretization techniques applied to the Doyle-Fuller-Newman (DFN) Lithium-ion battery model. More specifically, the central difference approximation, Crank-Nicolson, and Chebyshev discretization methods applied to the DFN model are compared. These methods are contrasted in terms of accuracy, stability, and computational times, providing the reader with several insights regarding the selection of discretization techniques according to the type of application to be carried out, highlighting the pros and cons of the analyzed methods.

On the basis of the classical Runge-Kutta method and the complete discretization method, a Runge-Kutta-based complete discretization method (RKCDM) is proposed in the paper to predict the chatter stability of milling process, in which the regenerative effect is taken into consideration. Firstly, the dynamics model of milling process is simplified as a 2-DOF vibration system in the two orthogonal directions, which can be expressed as coefficient-varying periodic differential equations with a single time delay. Then, all parts of the delay differential equation (DDE), including delay term, time-domain term, parameter matrices, and most of all the differential terms are discretized using the classical fourth-order Runge-Kutta iteration method to replace the direct integration scheme used in the classical semi-discretization method (C-SDM) and the classical complete discretization scheme with the Euler method (C-CDSEM), which can simplify the complexity of the discretization iteration formula greatly. Lastly, the Floquet theory is adopted to predict the stability of milling process by judging the eigenvalues of the state transition matrix corresponding to certain cutting conditions. Comparing RKCDM with C-SDM and C-CDSEM, the numerical simulation results show that RKCDM has the highest convergence rate, computation accuracy, and computation efficiency. As dichotomy search rather than sequential search is used in the algorithm, the calculation time for obtaining the stability lobe diagrams (SLDs) is greatly reduced. As a result, it is practical to determine the optimal chatter-free cutting conditions for milling operation in shop floor applications.

The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length- and timescales. Often, it is computationally intractable to resolve the finest features in the solution. The only recourse is to use approximate coarse-grained representations, which aim to accurately represent long-wavelength dynamics while properly accounting for unresolved small-scale physics. Deriving such coarse-grained equations is notoriously difficult and often ad hoc. Here we introduce data-driven discretization, a method for learning optimized approximations to PDEs based on actual solutions to the known underlying equations. Our approach uses neural networks to estimate spatial derivatives, which are optimized end to end to best satisfy the equations on a low-resolution grid. The resulting numerical methods are remarkably accurate, allowing us to integrate in time a collection of nonlinear equations in 1 spatial dimension at resolutions 4× to 8× coarser than is possible with standard finite-difference methods.

We present the nonconforming virtual element method (VEM) for the numerical approximation of velocity and pressure in the steady Stokes problem. The pressure is approximated using discontinuous piecewise polynomials, while each component of the velocity is approximated using the nonconforming virtual element space. On each mesh element the local virtual space contains the space of polynomials of up to a given degree, plus suitable nonpolynomial functions. The virtual element functions are implicitly defined as the solution of local Poisson problems with polynomial Neumann boundary conditions. As typical in VEM approaches, the explicit evaluation of the nonpolynomial functions is not required. This approach makes it possible to construct nonconforming (virtual) spaces for any polynomial degree regardless of the parity, for two- and three-dimensional problems, and for meshes with very general polygonal and polyhedral elements. We show that the nonconforming VEM is inf-sup stable and establish optimal a priori error estimates for the velocity and pressure approximations. Numerical examples confirm the convergence analysis and the effectiveness of the method in providing high-order accurate approximations.

We numerically study a discretized Vicsek model (DVM) with particles orienting in q possible orientations in two dimensions. The study investigates the significance of anisotropic orientation and microscopic interaction on macroscopic behavior. The DVM is an off-lattice flocking model like the active clock model (ACM; Chatterjee et al 2022 Europhys. Lett. 138 41001) but the dynamical rules of particle alignment and movement are inspired by the prototypical Vicsek model (VM). The DVM shows qualitatively similar properties as the ACM for intermediate noise strength where a transition from macrophase to microphase separation of the coexistence region is observed as q is increased. But for small q and noise strength, the liquid phase appearing in the ACM at low temperatures is replaced in the DVM by a configuration of multiple clusters with different polarizations, which does not exhibit any long-range order. We find that the dynamical rules have a profound influence on the overarching features of the flocking phase. We further identify the metastability of the ordered liquid phase subjected to a perturbation.

The paper presented research results of a digital control system for a dynamic plant with pulse-width modulation (PWM) of control impacts. As the control PWM signal is taken the pulse duty cycle, is calculated on each current cycle of the sample from the measured values. A control algorithm is proposed based on a hybrid application of the linear-quadratic optimization procedure and the theory of observers of minimal complexity. To ensure execution that the conditions of Astatism are met, the dynamic model of the plant is supplemented with a discrete integrator. The proposed approach makes it possible to reduce hardware costs and increase the robustness of the control system due to the exclusion of operations for digital–analogue transformations of signals. The proposed algorithm for digital control of a dynamic plant with varying duty cycle values of the PWM signal shows that the PWM model turned out to be linear and practically inertia less, which makes it easy to take into account the modulator model, which significantly simplifies the solution of the problem of synthesizing a control system for a dynamic plant. The possibility of receiving a high-quality modulated control signal allows for significant suppression of signal pulsations and high control accuracy.

PDDL+ models are advanced models of hybrid systems and the resulting problems are notoriously difficult for planning engines to cope with. An additional limiting factor for the exploitation of PDDL+ approaches in real-world applications is the restricted number of domain-independent planning engines that can reason upon those models. With the aim of deepening the understanding of PDDL+ models, in this work, we study a novel mapping between a time discretisation of pddl+ and numeric planning as for PDDL2.1 (level 2). The proposed mapping not only clarifies the relationship between these two formalisms but also enables the use of a wider pool of engines, thus fostering the use of hybrid planning in real-world applications. Our experimental analysis shows the usefulness of the proposed translation and demonstrates the potential of the approach for improving the solvability of complex PDDL+ instances.

Let X N be an N -dimensional subspace of L 2 functions on a probability space ( Ω , μ ) spanned by a uniformly bounded Riesz basis Φ N . Given an integer 1 ≤ v ≤ N and an exponent 1 ≤ p ≤ 2 , we obtain universal discretization for the integral norms L p ( Ω , μ ) of functions from the collection of all subspaces of X N spanned by v elements of Φ N with the number m of required points satisfying m ≪ v ( log N ) 2 ( log v ) 2 . This last bound on m is much better than previously known bounds which are quadratic in v . Our proof uses a conditional theorem on universal sampling discretization, and an inequality of entropy numbers in terms of greedy approximation with respect to dictionaries.