Adaptive Integration Scheme Research Articles

Adaptative scheme for diffuse field response of multilayered structures

The computation of the diffuse field response of multilayered homogeneous structures leads to integrals which should be computed efficiently. This is all the more true as the frequency increases, since the integrand has localized peaks. The objective of this communication is to propose an adaptive integration scheme that aims to limit the number of resolutions of the physical problem and to maximize the reuse of previous evaluations in the course of the method. After discussing the usual integration methods, the proposed method will be presented. A first approximation of the integral is performed with two nested integration schemes with a coarse mesh of the initial integration domain. A local error indicator is derived and leads to a first partition. On each set of the latter, and depending on the local error estimated previously, an adequate numerical scheme is chosen. The error is controlled during all the process and it is iterated until convergence. The method is tested first on academic cases and then on industrial cases. The performance of the method is discussed, taking into account not only the number of evaluations of the physical function, but also the additional numerical cost generated by the adaptation process.

Adaptive conservative time integration for unsteady compressible flow

Unsteady flow computations typically rely on time integration schemes which employ the same step size everywhere. However, in cases with strong variations of wave speed and/or spatial resolution, local stability criteria and truncation errors would allow for much larger time steps in parts of the domain. To exploit this potential of improving the computational efficiency, adaptive time-stepping methods have been developed. Recently, an adaptive conservative time integration (ACTI) scheme for explicit finite volume methods was devised. Opposed to previous methods, ACTI guarantees exact conservation and periodic synchronization. Both are achieved by splitting a global time step into 2L (L≥0 is a cell dependent level) sub-steps, whereas the order of cell updates is critical. It has been demonstrated for flows in fractured porous media and for the Euler equations that ACTI can reduce the computational cost dramatically while preserving second order accuracy in space and time. In this paper, the ACTI scheme is extended to the compressible Navier-Stokes-Fourier system, and special attention is required for the spatio-temporal discretization of the convective and viscous fluxes. Numerical studies of unsteady flows involving shock boundary layer interaction demonstrate that the same accuracy is achieved with the new compressible ACTI flow solver as with classical time integration, but at much lower cost. Moreover, since the method is explicit, it has the potential for efficient computations on multiple CPUs and GPUs.

Projective integration methods in the Runge–Kutta framework and the extension to adaptivity in time

Projective Integration methods are explicit time integration schemes for stiff ODEs with large spectral gaps. In this paper, we show that all existing Projective Integration methods can be written as Runge–Kutta methods with an extended Butcher tableau including many stages. We prove consistency and order conditions of the Projective Integration methods using the Runge–Kutta framework. Spatially adaptive Projective Integration methods are included via partitioned Runge–Kutta methods. New time adaptive Projective Integration schemes are derived via embedded Runge–Kutta methods and step size variation while their accuracy, stability, convergence, and error estimators are investigated analytically and numerically.

Numerical implementation of the hypoplastic model for SPH analysis of soil structure development in extremely large deformation

The extremely large deformation of soil is a significant challenge due to the dynamic changes in soil structure, but it is poorly understood owing to inadequate effective numerical methods. This paper presents a numerical method for analysing the responses of sand-like granular soil undergoing extremely large deformation. The proposed method utilizes the Smoothed Particle Hydrodynamics (SPH) technique, which is a meshfree particle-based approach capable of circumventing the computational difficulties associated with mesh distortion. The Gudehus-Bauer (G-B) hypoplastic model, which can capture the pressure- and density- dependent mechanical behaviour of granular soil, is implemented into the in-house developed SPH code. Concerning the characteristics of large deformation, the Green-Naghdi rate is applied in this study to achieve objective stress integration. An adaptive explicit stress integration scheme, termed Modified Euler automatic Substepping with Error Control (M-E-SEC) is utilized to handle the time integration of the SPH equation system. Additionally, a stability correction method is introduced to suppress the breakdown problem that occurs at low-stress states. The effectiveness of the proposed method is validated by experimental data. Furthermore, several numerical examples are presented to elucidate the evolution of shear band structures in granular soils subject to extremely large deformation levels.

Spatially Adaptive Projective Integration Schemes For Stiff Hyperbolic Balance Laws With Spectral Gaps

Stiff hyperbolic balance laws exhibit large spectral gaps, especially if the relaxation term significantly varies in space. Using examples from rarefied gases and the general form of the underlying balance law model, we perform a detailed spectral analysis of the semi-discrete model that reveals the spectral gaps. Based on that, we show the inefficiency of standard time integration schemes expressed by a severe restriction of the CFL number. We then develop the first spatially adaptive projective integration schemes to overcome the prohibitive time step constraints of standard time integration schemes. The new schemes use different time integration methods in different parts of the computational domain, determined by the spatially varying value of the relaxation time. We use our analytical results to derive accurate stability bounds for the involved parameters and show that the severe time step constraint can be overcome. The new adaptive schemes show good accuracy in a numerical test case and can obtain a large speedup with respect to standard schemes.

An advanced control and protection integration scheme for microgrids

The increasing trend in integrating intermittent distributed energy resources (DERs) into AC microgrids presents operational challenges in stability and protection. In islanded microgrids with power electronic interfaces, protection poses a major challenge due to the reduced level of short circuit currents resulting from inverter output capabilities. However, traditional protection schemes that are utilized in distribution systems are no longer appropriate to protect the microgrid in the presence of different levels of fault currents. This paper develops an integrated control and protection framework based on state observer and fault current limiter (FCL) devices. The state observer has been developed to detect and identify the faults that occur within multiple protection zones. In addition, controlled switches, consisting of FCLs, have been utilized to limit the fault currents and provide rapid switching during the faults, thereby improving system reliability. An adaptive control and protection integration scheme proposed in this paper has been applied to islanded microgrid configuration and is demonstrated to be an effective means to protect the system and maintain the voltage and frequency within an acceptable range with the capability of power continuity during both transient and persistent faults.

Adaptive conservative time integration for transport in fractured porous media

The time step of an explicit time integration scheme for solving time-dependent hyperbolic partial differential equations in a finite volume method (FVM) framework is restricted by the Courant–Friedrichs–Lewy (CFL) criterion. Conventional time stepping integrates all grid cells with the same time step. This causes unnecessary computational costs when wave speeds and/or grid spacing vary considerably throughout the domain, and a few critical cells dictate the global time step, although most cells could be advanced with a much larger time step. Adaptive time stepping overcomes this issue by allowing different local time step sizes for each grid cell. The adaptive conservative time integration (ACTI) scheme is a recently developed adaptive time stepping method which relies on local time steps that are equal to the largest time step divided by powers of two.This work extends the ACTI scheme to tracer transport in fractured porous media. When fluid velocity within highly permeable fractures is higher than in the rock matrix and local grid refinement is applied around fractures, the CFL restriction would require prohibitively small time steps in the vicinity of fractures. For two-dimensional discrete fracture and matrix models of fracture patterns we demonstrate that ACTI reduces the computational cost by orders of magnitude compared to global time stepping while retaining solution accuracy. Empirically, we show that ACTI is stable in combination with a first-order explicit flux discretization scheme. Since combination with a standard higher-order MUSCL scheme can lead to spurious oscillations in the solution, we propose a modified MUSCL scheme relying on advection of an inclined reconstruction (MUSCL-AIR).

Temporally adaptive conservative scheme for unsteady compressible flow

An adaptive conservative time integration scheme (ACTI) for compressible flow simulations is devised. A finite volume method is used, in which every cell employs a time step size which is some power of two times smaller than a global time step. By first advancing those cells with smaller time steps, this allows to use the already computed accumulated fluxes when the adjacent cells with larger time steps catch up. Conservation and periodic synchronization are thus guaranteed by construction; both important advantages of the new scheme (especially in the presence of shock waves) compared to previously proposed adaptive time stepping methods for compressible flow. Accuracy and computational speedup are demonstrated for challenging 1D and 2D test cases. Comparisons of ACTI results with an analytical solution and with results obtained without ACTI show excellent agreement. Although maximum wave speeds and/or cell sizes vary by several orders of magnitude throughout the computational domains, the CFL numbers employed by ACTI in each cell vary not more than by approximately a factor of two. Without ACTI, on the other hand, the CFL numbers are tiny in most of the cells. Since this results in a dramatic reduction of required flux computations, high speedup gains are achieved due to ACTI; in particular, if only few cells experience a very small time step, while the time step size limitations are less severe for the majority of cells.

Numerical recipes for faster MAS-DNP simulations

Numerical simulations of Magic Angle Spinning Dynamic Nuclear Polarization (MAS-DNP) have transformed the way the DNP process is understood in rotating samples. In 2012, two methods were concomitantly developed to simulate small spin systems (< 4 spin-1/2). The development of new polarizing agents, including those containing metal centers with S > 1/2, makes it necessary to further expand the numerical tools with minimal approximations that will help rationalize the experimental observations and build approximate models. In this paper, three strategies developed in the past five years are presented: an adaptive integration scheme, a hybrid Hilbert/Liouville formalism, and a method to truncate the Liouville space basis for periodic Hamiltonian. Each of these methods enable time savings ranging from a factor of 3 to > 100. We illustrate the code performance by reporting for the first time the MAS-DNP field profiles for “AMUPol”, in which the couplings to the nitrogen nuclei are explicitly considered, as well as Cross-Effect MAS-DNP field profiles with two electrons spin 5/2 interacting with a nuclear spin 1/2.

IG-DRBEM of three-dimensional transient heat conduction problems

In this paper, the isogeometric dual reciprocity boundary element method (IG-DRBEM) is proposed to solve three-dimensional transient heat conduction problems. It is well known that the error of traditional BEM mainly comes from element dispersion, and the introduction of isogeometric ideas makes BEM become a veritable high-precision numerical method. At present, most of the problems solved by isogeometric BEM (IGBEM) are time-independent. The reason is similar to the traditional BEM, which cannot avoid solving domain integrals when solving time-dependent problems. In this paper, based on the potential fundamental solution the boundary-domain integral equation is obtained by the weighted residual method, where the classic dual reciprocity method is adopted to transform domain integrals into boundary integrals. Meanwhile, a two-level time integration scheme is used to solve the discretized differential equations. In addition, the adaptive integration scheme, the radial integral transform method and the power series expansion method are adopted to solve the boundary regular, nearly singular and singular integrals. Several classical numerical examples show that the presented method has good numerical stability and high precision by considering different factors such as the approximation function, the time step, the number of interior points and so on.

A second-order dynamic adaptive hybrid scheme for time-integration of stiff chemistry

A dynamic adaptive hybrid integration (AHI) scheme of second-order accuracy (AHI2) is proposed for time-integration of chemically reacting flows involving stiff chemistry. AHI2 is extended from a first-order AHI method (AHI1) developed in a previous study, which showed that when significant radical sources are present in the non-chemical source terms, splitting the chemical and the transport sub-systems may incur O(1) errors unless the splitting time steps are comparable to or smaller than that required for explicit integration. As such, the transport term needs to be carried during the integration of stiff chemistry to avoid the large splitting errors. In AHI, fast species and reactions that may induce stiffness are treated implicitly, while the non-stiff variables and source terms, including slow reactions and the mixing term, are treated explicitly. The separation of fast-slow chemistry is performed on-the-fly based on analytically evaluated timescales for species and reactions, such that the complexity of the implicit core in the governing equations is minimized at each time step and the time-integration can be performed with high efficiency. The newly developed AHI2 scheme combines the midpoint scheme and the trapezoidal rule to achieve second-order accuracy. The second-order scheme is tested with a toy problem, as well as auto-ignition and unsteady perfectly stirred reactors (PSR) with detailed chemistry. Results show that AHI2 can significantly improve accuracy compared with AHI1. It was further found that AHI2 can accurately predict extinction of unsteady PSRs while the Strang splitting scheme fails to control the error, showing the necessity not to split the chemistry and transport source terms for prediction of extinction or forced-ignition problems involving significant radical sources. Further analysis of numerical efficiency shows that for auto-ignition AHI2 reduces computational cost primarily through the reduction in the number of variables to be solved implicitly, and the time-saving increases with the mechanism size, reaching approximately 70% for the 111-species USC-Mech II compared with a fully implicit scheme. For unsteady PSR involving homogeneous mixing, AHI2 achieved speedup factors of 20 to 30 compared with the Strang splitting scheme. Furthermore, sparse matrix techniques are integrated into AHI2 (AHI2-S) to achieve high computational efficiency. It is shown that the computational cost of AHI2-S is overall linearly proportional to the mechanism size and is comparable to that of evaluating reaction rates using CHEMKIN-II subroutines. It is further shown that AHI2-S achieves a speed-up factor of around two compared with the efficient fully implicit sparse solver LSODES with analytic Jacobian.

Non-conforming interface coupling and symmetric iterative solution in isogeometric FE–BE​ analysis

In this paper, a three-dimensional (3D) isogeometric finite element (FE) and boundary element (BE) coupling approach based on non-uniform rational B-splines (NURBS) is developed to simulate the deformed body. Based on the geometric exactness and higher smoothness of isogeometric analysis (IGA), NURBS discretization is applied to the approximation of geometric representation, displacement field and traction field simultaneously. An improved power series expansion method (M-PSEM) is used to evaluate the singular integral and is suitable for high order (≥3) NURBS basis functions, which is the limitation of the traditional method. Adaptive integration scheme based on quadtree subdivision technique is adopted to enable the non-singular integration computation more flexible and efficient at optimal computational cost. The relationship between the tractions and nodal external forces on the interface is established by using the virtual work principle. On the other hand, displacement coupling constraints among the non-conforming (or incomplete coincident) surface meshes are constructed by a virtual knot insertion technique. The main advantages of this method are simplicity and robustness, as it is problem-independent and only depends on the NURBS meshes on both sides of the interface. Introducing an enhanced symmetric matrix related to the BE subdomain, the symmetric iterative coupling method (SICM) is extended to solve the final system coupling equation. The final coupling matrix obtained by this iterative method is symmetric, and its dimension is the same as that of the stiffness matrix of IGA finite element method (IGAFEM). Numerical examples are investigated to assess the accuracy and efficiency of the presented method.

Mathematical analysis of the spatial coupling of an explicit temporal adaptive integration scheme with an implicit time integration scheme

AbstractThe Reynolds‐averaged Navier–Stokes equations and the large eddy simulation equations can be coupled using a transition function to switch from a set of equations applied in some areas of a domain to the other set in the other part of the domain. Following this idea, different time integration schemes can be coupled. In this context, we developed a hybrid time integration scheme that spatially couples the explicit scheme of Heun and Crank–Nicolson's implicit scheme using a dedicated transition function. This scheme is linearly stable and second‐order accurate. In this article, an extension of this hybrid scheme is introduced to deal with a temporal adaptive procedure. The idea is to treat the time integration procedure with unstructured grids as it is performed with Cartesian grids and local mesh refinement. Depending on its characteristic size, each mesh cell is assigned to a rank. And for two cells from two consecutive ranks, the ratio of the associated time steps for time marching the solutions is 2. As a consequence, the cells with the lowest rank iterate more than the other ones to reach the same physical time. In a finite‐volume context, a key ingredient is to keep the conservation property for the interfaces that separate two cells of different ranks. After introducing the different schemes, the article recalls briefly the coupling procedure, and details the extension to the temporal adaptive procedure. The new time integrator is validated with the propagation of 1D wave packet, the Sod's tube, and the advection of 2D vortex.

Efficient higher-order cycle jump integration of a continuum fatigue damage model

Simulating high-cycle fatigue with continuum models offers the possibility to model stress-redistributions, consider 3D-stress states and simplifies extensions to multi-physics problems. The computational cost of conventional cycle-by-cycle time integrations is reduced by reformulating the fatigue problem as an ordinary differential equation for the material state and solving it with high-order adaptive time integration schemes. The computational cost of calculating the change of the material state in one cycle is further reduced by a high-order fatigue-specific time integration. The approach is exemplarily demonstrated for a fatigue extension of the implicit gradient-enhanced damage model in 3D and compared to experimental Wöhler lines.

Stepwise analysis of pantographic beams subjected to impulsive loads

Materials based on pantographic unit cells have very interesting mechanical peculiarities. For these reasons they are largely studied from a theoretical, experimental, and numerical point of view. Numerical simulations furnish an important contribution for the the design and optimization of such materials and, more generally, for metamaterials. Here, we consider the influence of inertial forces, removing the hypothesis of quasistatic loading. By using an intrinsically discrete model, inspired by Hencky’s ideas, already tested in a series of published works, here we add the contribution of inertial forces and, in the framework of stepwise schemes, we re-experience an adaptive integration scheme capable of reconstructing the best structural response corresponding to a prefixed time step. Several numerical simulations, although preparatory, inspire some remarks on materials based on pantographic cells and outline the way for future challenges.

Enhanced numerical integration scheme based on image-compression techniques: application to fictitious domain methods

In the present work, we propose a new approach, the so-called compressed adaptive integration scheme (C-AIS), for the computation of the stiffness and mass matrices in fictitious domain methods requiring the integration of discontinuous functions. The novel approach extends the conventional quadtree-decomposition-based adaptive integration scheme (AIS) by an additional step, in which established image-compression techniques are exploited to decrease the number of integration sub-cells. The benefits of the C-AIS are manifold: First, the compression of the sub-cells inevitably leads to significant savings in terms of computational time required by the numerical integration. Second, the compression procedure, which is executed directly after the quadtree-decomposition algorithm, can be easily included in existing codes. Third, if applied to polynomial integrands, the C-AIS yields exactly the same accuracy as the conventional AIS. Finally, the fourth advantage is seen in the fact that the C-AIS can readily be combined with other approaches seeking a reduction of the number of integration points such as the Boolean-FCM. The efficiency of the C-AIS approach is presented in the context of the FCM based on Cartesian meshes applied to problems of linear elastostatics and modal analysis, while it is also suitable for the quadrature in other fictitious domain approaches, e.g., CutFEM and cgFEM.

Density-Matrix Based Extended Lagrangian Born-Oppenheimer Molecular Dynamics.

Extended Lagrangian Born-Oppenheimer molecular dynamics [ Phys. Rev. Lett. 2008, 100, 123004] is presented for Hartree-Fock theory, where the extended electronic degrees of freedom are represented by a density matrix, including fractional occupation numbers at elevated electronic temperatures. In contrast to regular direct Born-Oppenheimer molecular dynamics simulations, no iterative self-consistent field optimization is required prior to the force evaluations. To sample regions of the potential energy landscape where the gap is small or vanishing, which leads to particular convergence problems in regular direct Born-Oppenheimer molecular dynamics simulations, an adaptive integration scheme for the extended electronic degrees of freedom is presented. The integration scheme is based on a tunable, low-rank approximation of a fourth-order kernel, [Formula: see text], that determines the metric tensor, [Formula: see text], used in the extended harmonic oscillator of the Lagrangian that generates the dynamics of the electronic degrees of freedom. The formulation and algorithms provide a general guide to implement extended Lagrangian Born-Oppenheimer molecular dynamics for quantum chemistry, density functional theory, and semiempirical methods using a density matrix formalism.

An adaptive Gaussian quadrature for the Voigt function

We evaluate an adaptive Gaussian quadrature integration scheme suitable for the numerical evaluation of generalized redistribution in frequency functions. The latter are indispensable ingredients for “full non-LTE” radiation transfer computations, assuming potential deviations of the velocity distribution of massive particles from the usual Maxwell–Boltzmann distribution. A first validation is made with computations of the usual Voigt profile.

Time adaptive conservative finite volume method

Finite volume methods for time dependent problems typically employ time integration schemes with constant time step sizes. The latter, in order to ensure stability and time accurate solutions, have to be chosen small enough, such that the Courant-Friedrichs-Lewy (CFL) number stays below a critical value everywhere. In many cases, however, this time step size limitation leads to huge numbers of very small time steps, which renders simulations very expensive. Motivated by this drawback of conventional time stepping schemes, various sub-time stepping algorithms have been proposed. However, most of them are inherently asynchronous, require small local CFL numbers or are not strictly conservative. In this paper a new adaptive time integration scheme for finite volume methods, which is conservative, of high spatial and temporal order, robust and easy to implement is presented. It relies on local sub-time steps which are fractions of a global time step by powers of two, i.e., the grid cells proceed asynchronously in the order of their termination time, but since they all synchronize at the end of each global time step, it is possible to guarantee continuity of the mean fluxes and thus strict conservation at the global time step resolution. Numerical experiments with 1D and 2D test cases demonstrate that the adaptive conservative time integration (ACTI) scheme can achieve extreme speed-up factors over conventional time integration, while still maintaining high spatial and temporal accuracy.

An adaptive integration scheme for heat conduction in additive manufacturing

Abstract Solidification dynamics are important for determining final microstructure in additively manufactured parts. Recently, researchers have adopted semi-analytical approaches for predicting heat conduction effects at length and time scales not accessible to complex multi-physics numerical models. The present work focuses on improving a semi-analytical heat conduction model for additive manufacturing by designing an adaptive integration technique. The proposed scheme considers material properties, process conditions, and the inherent physical behavior of the transient heat conduction around both stationary and moving heat sources. Both the adaptive integration scheme and a technique for calculating only the points within the melt pools are described in detail. The full algorithm is then implemented and compared against a simple Riemann sum integration scheme for a variety of cases that highlight process and material variations relevant to additive manufacturing. The new scheme is shown to have significant improvements in computational efficiency, solution accuracy, and usability.