Lagrangian Formulation Research Articles

Hydro-structural response of complex elevated pile cap substructures to strom surge impact by smoothed particle hydrodynamics: Part 2: Model results

Extreme waves, i.e., tsunamis and storm surges, can generate tremendous hydraulic bores capable of inflicting considerable damage to near-shore bridge infrastructure, often leading to structural failure. The conventional numerical approach to studying hydrodynamic impact is based on Eulerian grid-based, e.g., finite volume or diffused element algorithms. These perform well with perfectly defined simulation domains. However, in case of severe surface deformation, these grids must be continually reconstructed, which is time-intensive and introduces cumulative system errors. As Smoothed Particle Hydrodynamic (SPH) integrates the hydrodynamic equations of motion on each particle in Lagrangian formalism, it can handle complex wave non-linearity without mesh constraints. In this study opensource DualSPHysics model is applied to study the hydrodynamic response and group effect of elevated pile cap substructures to impulsive surge impact. Two conventional grids (2 × 2, 2 × 3 array-layout) and a hexagonal pile arrangement under three submersion levels are considered in studying the pile group effect on hydrodynamic forcing mechanism, flow characteristics, and factors governing pressure variability. The uplift thrust on the pile cap was found to be dependent on the surge fluid mass, incident velocity, and vertical distance between the surge breaking point and cap bottom. At impulsive impact, gridded pile arrangement offered greater resistance to incident surge flow. In reference to free flow condition, hexagonal layout can reduce the average flow velocity by 8.22–19.76%, while for a gridded system, it is between 14.00% and 23.52% depending on the submersion level. However, in a quasi-steady state, the flow resistance of the hexagonal system was greater. The study also compared the numerical results to accepted offshore design standards, which were found to underestimate the pile's group effect, specifically for complex hexagonal layouts. Later, it proposes rationalized force diagrams to be applied in design of complex elevated structures. The paper's finding is expected to contribute to formulating new guidelines for resilient hydraulic design of coastal and deep-water elevated structures.

Efficient analytical gradients of property-based diabatic states: Geometry optimizations for localized holes.

We present efficient analytical gradients of property-based diabatic states and couplings using a Lagrangian formalism. Unlike previous formulations, the method achieves a computational scaling that is independent of the number of adiabatic states used to construct the diabats. The approach is generalizable to other property-based diabatization schemes and electronic structure methods as long as analytical energy gradients are available and integral derivatives with the property operator can be formed. We also introduce a scheme to phase and reorder diabats to ensure their continuity between molecular configurations. We demonstrate this for the specific case of Boys diabatic states obtained from state-averaged complete active space self-consistent field electronic structure calculations with GPU acceleration in the TeraChem package. The method is used to test the Condon approximation for the hole transfer in an explicitly solvated model DNA oligomer.

N3LO spin-orbit interaction via the EFT of spinning gravitating objects

We present the derivation of the third subleading order (N3LO) spin-orbit interaction at the state of the art of post-Newtonian (PN) gravity via the EFT of spinning objects. The present sector contains the largest and most elaborate collection of Feynman graphs ever tackled to date in sectors with spin, and in all PN sectors up to third subleading order. Our computations are carried out via advanced multi-loop methods. Their most demanding aspect is the imperative transition to a generic dimension across the whole derivation, due to the emergence of dimensional-regularization poles across all loop orders as of the N3LO sectors. At this high order of sectors with spin, it is also critical to extend the formal procedure for the reduction of higher-order time derivatives of spin variables beyond linear order for the first time. This gives rise to a new unique contribution at the present sector. The full interaction potential in Lagrangian form and the general Hamiltonian are provided here for the first time. The consequent gravitational-wave (GW) gauge-invariant observables are also derived, including relations among the binding energy, angular momentum, and emitted frequency. Complete agreement is found between our results, and the binding energy of GW sources, and also with the extrapolated scattering angle in the scattering problem, derived via traditional GR. In contrast with the latter derivation, our framework is free-standing and generic, and has provided theory and results, which have been critical to establish the state of the art, and to push the precision frontier for the measurement of GWs.

Preface to the topical issue on applied and nonlinear dynamics: Part II

The current special issue of the GAMM Mitteilungen, which is the second of a two-part series, contains several contributions on the topic of Applied and Nonlinear Dynamics. We are very happy that several teams of authors have accepted our invitation to report on recent developments, research highlights and emerging application areas in Applied and Nonlinear Dynamics. This second part of the topical issue on Applied and Nonlinear Dynamics includes five interesting papers. These are devoted to numerical and experimental methods in applied and nonlinear dynamics as well as advanced applications of multibody systems and optimal control methods to dynamical systems. Contribution 3 deals with stationary solutions in applied dynamics. Thereby a unified framework for the numerical calculation and stability assessment of periodic and quasi-periodic solutions based on invariant manifolds is presented. Paper 2 gives an overview of dynamic human body models in vehicle safety, a unique application of multibody dynamics. In paper 1, a family of total Lagrangian Petrov-Galerkin Cosserat rod finite element formulations is presented. Paper 5 discusses continuation methods for lab experiments of nonlinear vibrations. Finally paper 4 deals with the optimal operation of dielectric elastomer wave energy converters under harmonic and stochastic excitation.

Research on the fusion of continuous fiber reinforced thermoplastic filaments for fused filament fabrication

The printed continuous fiber reinforced thermoplastic composites have drawn extensive attention due to their excellent mechanical properties. However, interlayer and intralayer voids, which are caused by the insufficient fusion of composite filaments, diminish the mechanical behaviors of the parts. In this work, the influence mechanism of continuous fiber on the fusion process is revealed. A numerical framework based on Lagrangian formulation for simulating the fusion process of pre-impregnated filaments is proposed and verified by experiments. Then, the influence of fiber volume fraction, fiber shape, and fiber eccentricity on the fusion process is studied by the numerical model. The concepts of normalized distance and effective volume fraction are proposed to quantify the influence of the fibers. This study is helpful to further understand the process of filament fusion and void formation in material extrusion additive manufacturing.

The MLS based numerical manifold method for bending analysis of thin plates on elastic foundations

Compared with the finite element method, H2-regularity in the Galerkin based approximation to the Kirchhoff thin plate model can be easily realized using either the moving least squares (MLS) or the generalized moving least squares (GMLS), which take the Lagrange form and the Hermite form, respectively. Coupling (G)MLS with the numerical manifold method (NMM) can greatly improve numerical properties of NMM in the treatment of plates of complicated shape, thereby denoted by MLS-NMM and GMLS-NMM. In the (G)MLS-NMM, the mathematical cover is composed of simply connected and partially overlapped mathematical patches that are the influence domains of (G)MLS nodes. GMLS-NMM appears to better fit to the Kirchhoff plate because it is equipped with rotation angle degrees of freedom. Through numerical tests and theoretical analysis in solving problems of thin plates on elastic foundations, however, this study shows that MLS-NMM is much more advantageous over GMLS-NMM from the aspects of both accuracy and memory usage.

Notes on the L∞-approach to local gauge field theories

It is well known that a Q-manifold gives rise to an L∞-algebra structure on the tangent space at a fixed point of the homological vector field. From the field theory perspective this implies that the expansion of a classical Batalin-Vilkovisky (BV) formulation around a vacuum solution can be equivalently cast into the form of an L∞-algebra. In so doing, the BV symplectic structure determines a compatible cyclic structure on the L∞-algebra. Moreover, L∞ quasi-isomorphisms correspond to so-called equivalent reductions (also known as the elimination of generalized auxiliary fields) of the respective BV systems. Although at the formal level the relation is straightforward, its implementation in field theory requires some care because the underlying spaces become infinite-dimensional. In the case of local gauge theories, the relevant spaces can be approximated by nearly finite-dimensional ones through employing the jet-bundle technique. In this work we study the L∞-counterpart of the jet-bundle BV approach and generalise it to a more flexible setup of so-called gauge PDEs. It turns out that in the latter case the underlying L∞-structure is analogous to that of Chern-Simons theory. In particular, the underlying linear space is a module over the space-time exterior algebra and the higher L∞-maps are multilinear. Furthermore, a counterpart of the cyclic structure turns out to be degenerate and possibly nonlinear, and corresponds to a compatible presymplectic structure the gauge PDE, which is known to encode the BV symplectic structure and hence the full-scale Lagrangian BV formulation. Moreover, given a degenerate cyclic structure one can consistently relax the L∞-axioms in such a way that the formalism still describes non-topological models but involves only finitely-generated modules, as we illustrate in the example of Yang-Mills theory.

Constraining [formula omitted] gravity with gravitational baryogenesis

Gravitational baryogenesis is one of the mechanisms which help us to explore more about our early universe, especially baryon-anti-baryon asymmetry. As we know, modified theories of gravity are very successful in describing the present accelerated scenario of the universe. Therefore, in this letter, we aim to constrain the generalized torsion-based modified theory of gravity, namely, f(T,T) gravity with gravitational baryogenesis, where T, T are the torsion scalar, trace of the energy-momentum tensor, respectively. For this, we examine how the various Lagrangian forms of f(T,T) affect the baryogenesis. We also impose the constraints on the extra degrees of freedom induced by modified theory with the observational values of the baryon-to-entropy ratio. In addition, we further explore how more generalized gravitational baryogenesis can attribute in a physically viable and consistent way to the cosmologies of the modified theory of gravity.

Humanitarian Demining Serial-Tracked Robot: Design and Dynamic Modeling

The paper proposes an original mechanical structure of a serial-tracked robot, subject of national invention patent number RO132301, B1/2021, destinated for humanitarian demining operations: anti-personnel mine detection by using a detection device mounted on the bottom’s tracked platform, demining and clearing the land of exploded mines using a TRTTR robot structure. The dynamic model of the robot structure is determined and numerically validated. A novel approach based on the Lagrange formalism and mechanical design equations has been used in the calculus and selection of robot driving motors. The obtained results for robot translation modules are presented and analyzed.

A discontinuous Galerkin formulation for nonlinear analysis of multilayered shells refined theories

A novel pure penalty discontinuous Galerkin method is proposed for the geometrically nonlinear analysis of multilayered composite plates and shells, modelled via high-order refined theories. The approach allows to build different two-dimensional equivalent single layer structural models, which are obtained by expressing the covariant components of the displacement field through-the-thickness via Taylor’s polynomial expansion of different order. The problem governing equations are deduced starting from the geometrically nonlinear principle of virtual displacements in a total Lagrangian formulation. They are addressed with a pure penalty discontinuous Galerkin method using Legendre polynomials trial functions. The resulting nonlinear algebraic system is solved by a Newton–Raphson arc-length linearization scheme. Numerical tests involving plates and shells are proposed to validate the method, by comparison with literature benchmark problems and finite element solutions, and to assess its features. The obtained results demonstrate the accuracy of the method as well as the effectiveness of high-order elements.

Quantum cosmology of pure connection general relativity

We study homogeneous cosmological models in formulations of general relativity with cosmological constant based on a (complexified) connection rather than a spacetime metric, in particular in a first order theory obtained by integrating out the self-dual two-forms in the chiral Plebański formulation. Classical dynamics for the Bianchi IX model are studied in the Lagrangian and Hamiltonian formalism, where we emphasise the reality conditions needed to obtain real Lorentzian solutions. The solutions to these reality conditions fall into different branches, which in turn lead to different real Hamiltonian theories, only one of which is the usual Lorentzian Bianchi IX model. We also show the simpler case of the flat Bianchi I model, for which both the reality conditions and dynamical equations simplify considerably. We discuss the relation of a real Euclidean version of the same theory to this complex theory. Finally, we study the quantum theory of homogeneous and isotropic models, for which the pure connection action for general relativity reduces to a pure boundary term and the path integral is evaluated immediately, reproducing known results in quantum cosmology. An intriguing aspect of these theories is that the signature of the effective spacetime metric, and hence the interpretation of the cosmological constant, are intrinsically ambiguous.

A CNN-supported Lagrangian ISPH model for free surface flow

As a popular method for modeling violent free surface flow, the incompressible Smoothed Particle Hydrodynamics (ISPH) based on the Lagrangian formulation has attracted a great attention worldwide. The Lagrangian ISPH solves the unsteady Navier-Stokes and continuity equations using the projection method, in which the pressure is obtained by solving the pressure Poisson's equation (PPE) that is the most time-consuming part in the ISPH procedure. In this paper, the Convolutional Neural Network (CNN) is combined with ISPH and used to predict the fluid pressure instead of solving the PPE directly. Although limited attempts of using CNN for solving the PPE in Eulerian formulation (referred to as the Eulerian CNN framework) in mesh-based methods are found in the public domain, the present model is the first ISPH model supported by CNN in a Lagrangian formulation. The proposed model overcome several challenges associated with combining CNN with ISPH, including selecting the input parameters, formulating the objective functions, producing the training dataset and dealing with boundary conditions. Two classic free surface problems, i.e. the dam breaking and the wave propagation, are simulated to evaluate the performance of the present model. Quantitative assessments of the numerical error in terms of both the free surface profile and the pressure field are carried out. The assessments show that the new model does not only give results with satisfactory accuracy, but also requires much less computation time for estimating pressure if the number of particles is large, e.g., 100 thousands particles that is usually required in the practical ISPH simulation for free surface flow.

Multiplier waveform inversion: A reduced-space full-waveform inversion by the method of multipliers

Full-waveform inversion (FWI) is a nonlinear optimization problem that addresses the estimation of subsurface model parameters by matching the predicted to the observed seismograms. We formulate FWI as a constrained optimization problem in which the regularization term is minimized subject to the nonlinear data matching constraints. Unlike standard FWI, which solves this regularized problem with a penalty method, we use an augmented Lagrangian formulation in the framework of the method of multipliers (MM). This leads to a two-step recursive algorithm, which is called multiplier waveform inversion (MWI). First, the primal step solves an unconstrained minimization problem, which is similar to the standard FWI but with a modified data term. These modified data are obtained by adding the Lagrange multipliers to the data residuals of the classical FWI. Then, the dual step updates the Lagrange multipliers with basic gradient ascent steps. In this framework, they reduce to the running sum of the data residuals of the previous iterations. The performance of the overall algorithm is improved by considering that the MM does not require the exact solution of the primal step at each iteration. In fact, convergence is attained when the primal subproblem is solved with one (without an inner loop) iteration of a gradient-based method at each iteration of the MWI. The new algorithm can be easily implemented in existing FWI codes without computational overhead by adding the Lagrange multipliers to the data residuals at each iteration. We find with numerical examples that MWI has a faster convergence speed and much improved stability compared with FWI and can converge to an accurate solution of the inverse problem in the absence of low-frequency data, even with a constant step size.

Practical implications of the erroneous treatment of exposure time in the Eulerian hemolysis power law model.

Eulerian and Lagrangian power-law formulations are both widely used for computational fluid dynamics (CFD) to predict flow-induced hemolysis in blood-contacting medical devices. Both are based on the same empirical power-law correlation between hemolysis and the shear stress and exposure time. In the Lagrangian approach, blood damage is predicted by tracking both the stress and exposure time along a finite number of pathlines in the domain. In the Eulerian approach, a scalar transport equation is solved for a time-linearized damage index within the entire domain. Previous analytical work has demonstrated that there is a fundamental problem with the treatment of exposure time in the Eulerian model formulation such that the only condition under which the model correctly represents the true exposure time is in a flow field with no streamwise velocity variation. However, the practical implications of this limitation have yet to be thoroughly investigated. In this study, we demonstrate the inaccuracy of Eulerian hemolysis power-law model predictions due to the erroneous treatment of exposure time by systematically considering four benchmark test cases with increasing degrees of flow acceleration: Poiseuille flow through a straight tube, inclined Couette flow, and flow through a converging tube with two different convergence ratios. Compared with Lagrangian predictions, we show that, as flow acceleration becomes more pronounced, the resultant inaccuracy in the Eulerian predictions increases. For the inclined Couette flow case, there is a small degree of flow acceleration that yields a discrepancy in the range of 10% between Lagrangian and Eulerian predictions. For flows with a larger degree of acceleration, as occurs in the converging tube flow cases, the discrepancy is considerably larger (up to 257%). The inaccuracy of hemolysis predictions due to the erroneous treatment of exposure time in the Eulerian power-law model can be significant when there is streamwise velocity variation in the flow field. These results may partially explain the extremely large variability in CFD hemolysis predictions reported in the literature between Lagrangian and Eulerian models.

Model references frequency-domain design based fractional order PD controller for quadrotor control: attitude, altitude, and position studies

This article proposes a control strategy based on a Fractional Order Proportional Derivative (FOPD) controller to enhance the performance and robustness of quadrotors operating in trajectory tracking mode. The tuning method proposed for the FOPD controller is based on an ideal Bode transfer function in a closed loop as a fractional reference model. The latter represents the quadrotor desired closed-loop response, and its specification is the key design decision. In this study, we first appropriately set model reference parameters to realistic desired time response characteristics (i.e. overshoot and time constant). Then, the parameters of the FOPD controller are computed using the direct synthesis design approach. The dynamic model of the quadrotor is developed using Lagrange formulation. The model is used to test the performance of the proposed FOPD-based control strategy for the attitude, position, and altitude of the flying robot. The results show that the proposed controller achieved accurately set points tracking with transient responses (i.e. overshoot and time constant) that satisfactorily meet those assigned for their respective reference models. These results validate the effectiveness of our proposed control strategy. Moreover, our comparative studies of the proposed FOPD controller with the conventional PD controller show that the former satisfies better-desired design requirements.

Lagrangian formalism and the intrinsic geometry of PDEs

A notion of internal Lagrangians of a system of differential equations is introduced. A spectral sequence related to internal Lagrangians is obtained. A connection between internal Lagrangians and presymplectic structures is investigated. An interpretation of the term E23,n−2 of Vinogradov's C-spectral sequence is given for irreducible Lagrangian gauge theories.

Semi-analytical implicit direct time integration method for 1-D gas dynamic problem

Sharp wave treatment for 1-D gas dynamic problem is still a chellenge for modern numerical methods. They often require too many space and time steps, produce spurious oscillation of solution, exhibit a strong numerical dissipation or divergence of results. This paper is further extension of authors’ idea of employment the analytical solution for space coordinate, where time step is a parameter which used in the space solution. Its peculiarity consists in development of additional procedure of linearization of dependence between the pressure and density. It is performed in premise that actual pressure for each space element is close to the basic pressure, attained at previous moment of time. The efficiency of method is tested on the very popular task of Sod, where two different ideal gases in a tube are separated by diaphragm, which is suddenly broken. The problem considered in Lagrangian coordinates formulation. The results obtained show the very good efficiency of method, which requires the essentially lesser time and space steps, leads to no spurious oscillation and give consistent and predictable results with respect to meshing. The accuracy of method is mostly controlled by time step, which should be larger than clearly stated theoretical lower limit. Other advantage of method is that it can calculate the process to any desired moment of time, and space meshing can be variable in time and space and can be easily adapted during the process of calculation.

Recursive inverse dynamics sensitivity analysis of open-tree-type multibody systems

We present a first-order recursive approach to sensitivity analysis based on the application of the direct differentiation method to the inverse Lagrangian dynamics of rigid multibody systems. Our method is simple and efficient and is characterized by the following features. Firstly, it describes the kinematics of multibody systems using branch connectivity graphs and joint-branch connectivity matrices. For most mechanical systems with an open-tree kinematic structure, this method turns out to be more efficient compared to other kinematic descriptions employing joint or link connectivity graphs. Secondly, a recursive sensitivity analysis is presented for a dynamic system with an open-tree kinematic structure and inverse dynamic equations described in terms of the Lagrangian formalism. Thirdly, known approaches to recursive inverse dynamic and sensitivity analyses are modified to include dynamic systems with external forces and torques acting simultaneously at all joints. Finally, the proposed method for sensitivity analysis is easy to implement and computationally efficient. It can be utilized to evaluate the derivatives of the dynamic equations of multibody systems in gradient-based optimization algorithms. It also allows less experienced users to perform sensitivity analyses using the power of high-level programming languages such as MATLAB. To illustrate the method, simulation results for a human body model are discussed. The shortcomings of the method and possible directions for future work are outlined.

Accelerated nonlinear finite element method for analysis of isotropic hyperelastic materials nonlinear deformations

In this work, an approximate Jacobian matrix is proposed based on the total Lagrangian formulation of Finite Element Method for isotropic hyperelastic materials. The approximate Jacobian matrix can take the place of the exact Jacobian matrix in the Newton–Raphson method to avoid frequent construction and factorization of the Jacobian matrix. A new Quasi–Newton method employing the approximate Jacobian matrix is developed to significantly improve the convergence rate. The proposed method was tested on three examples with the combinations of three hyperelastic material models and three element types. The results show that the proposed method is more efficient than ABAQUS and CALCULIX (an open-source FEM software package) on all of the tests without loss of accuracy. It is up to 100 times faster than the traditional Quasi–Newton method, and at least 2.5 times faster than ABAQUS.

Nonlinear vibrations of multiscale composite beams on a nonlinear softening foundation

This paper describes a new approach to evaluate the instability of beams on a softening elastic foundation. The basis of this approach is solving a nonlinear frequency problem. The beam is made of multiscale composite materials based on carbon nanotubes (CNTs) and long fibers. Homogenization methods are utilized in a sequential process to obtain the effective material properties of these composites. Furthermore, the thermal environment's effect on the beam's response is examined. Reddy's third-order shear deformation theory approximates the induced displacements in the composite beam. Also, the kinetic and strain energies of the beam are calculated based on the von Kármán nonlinear strains and linear thermoelasticity theories. A recently reported Lagrangian nodal weak formulation (LNWF) method is implemented to obtain the linear and nonlinear stiffness and mass matrices governing the beam's vibrations. The modified coefficients for nonlinear stiffness matrices are obtained using the residual Galerkin method across time. Finally, the nonlinear natural frequencies are calculated by employing the iterative displacement control strategy. In the results section, in addition to examining various parameters on nonlinear free vibrations, the effect of these parameters on the displacement required for the loss of stability of the beam placed on the softening foundation is also examined.