Reduction In Degrees Of Freedom Research Articles

Reduction in Degrees of Freedom for Large-Scale Nonlinear Systems in Computer-Assisted Proofs

In many physical systems, it is important to know the exact trajectory of a solution. Relevant applications include celestial mechanics, fluid mechanics, robotics, etc. For cases where analytical methods cannot be applied, one can use computer-assisted proofs or rigorous computations. One can obtain a guaranteed bound for the solution trajectory in the phase space. The application of rigorous computations poses few problems for low-dimensional systems of ordinary differential equations (ODEs) but is a challenging problem for large-scale systems, for example, systems of ODEs obtained from the discretization of the PDEs. A large-scale system size for rigorous computations can be as small as about a hundred ODE equations because computational complexity for rigorous algorithms is much larger than that for simple computations. We are interested in the application of rigorous computations to the problem of proving the existence of a periodic orbit in the Kolmogorov problem for the Navier–Stokes equations. One of the key issues, among others, is the computation complexity, which increases rapidly with the growth of the problem dimension. In previous papers, we showed that 79 degrees of freedom are needed in order to achieve convergence of the rigorous algorithm only for the system of ordinary differential equations. Here, we wish to demonstrate the application of the proper orthogonal decomposition (POD) in order to approximate the attracting set of the system and reduce the dimension of the active degrees of freedom.

A simplified method for calculating the accumulation of irreversible ro-tations of wind turbine shallow foundations

Wind turbines represent a convenient source of renewable energy and their diffusion is a fundamental step for reducing carbon dioxide emissions, main responsible for global warming. Turbine towers are characterized by large heights, therefore the wind load induces considerable bending actions at the base of the structure. Although, especially offshore, deep foundations are usually adopted, shallow foundations are still an option in case of onshore installations [1]. These circular shallow foundations are massive concrete structures (Figure 1) that counteract the bending moment with their own weight. The optimization of the design and the extension of the service life are fundamental from both economic and environmental perspectives. These will increase the competitivity of energy production from renewable wind sources over fossil ones, thus boosting the transition toward sustainable energy.
 From a geotechnical perspective, the isolated foundations of wind turbines are a peculiar case since the accumulation of vertical settlements and horizontal displacements are not a main concern. On the contrary, the accumulation of excessive rotations due to the wind cyclic loads may reduce the efficiency of the turbine and, in particularly critical cases, define the end of service of the structure.
 The accumulation of irreversible rotation during the exercise cyclic loads may be induced by both the development of irreversible strain in the soil and the formation of cracks in the concrete foundation. This second aspect can however be neglected, since, in the current practice, very over-conservative approaches are used to design the reinforcements. Recent experimental small scale tests (Figure 1) proved that, even when stirrups are completely removed, exercise loads do not induce the formation of cracks [2]. Therefore, the accumulation of cyclic irreversible rotation is only due to the soil.
 Recently, numerous constitutive relationships accounting for the cyclic behaviour of the soil were proposed [3,4]. These in principle allow to perform finite element analyses capable of reproducing the soil-structure interaction. However, during the foundation design life a very large number of load cycles (up to 108) is expected, leading to unacceptable computational times.
 A way to circumvent this problem is the employment of models conceived in the framework of the macroelement approach [5]. This is based on the assumption of reproducing the response of a complex system by using a very small number of degrees of freedom and by introducing an upscaled constitutive law relating generalized stresses (forces and moments) and strains (displacements and rotations). The abrupt reduction in degrees of freedom significantly reduces the computational times. However, the definition of the upscaled constitutive laws and the calibration of its parameters are particularly critical since they will be affected not only by the soil behaviour, but also by the foundation geometry and mechanical properties.
 In this work, the authors intend to propose a new one-dimensional macroelement to reproduce the moment-rotation response of shallow foundations for wind turbines. This generalized constitutive law, inspired to the one proposed in [6] for reproducing the lateral response of piles, is conceived in the framework of the bounding surface plasticity theory. The main ingredients of this model are a bounding surface and a mixed isotropic-kinematic generalized strain hardening controlling the ratcheting.
 The model is capable of qualitatively reproduce (Fig. 2) the experimental results of [2] and seems to be a very promising approach for design purposes. In addition, the model can also simply be introduced in structural finite element codes allowing to solve soil-structure interaction problems also accounting the turbine tower. This can be used to adequately reproduce the whole system response and to also optimize the tower design to dynamic loads.

PODTherm-GP: A Physics-Based Data-Driven Approach for Effective Architecture-Level Thermal Simulation of Multi-Core CPUs

A thermal simulation methodology derived from the proper orthogonal decomposition (POD) and the Galerkin projection (GP), hereafter referred to as PODTherm-GP, is evaluated in terms of its efficiency and accuracy in a multi-core CPU. The GP projects the heat transfer equation onto a mathematical space whose basis functions are generated from thermal data enabled by the POD learning algorithm. The thermal solution data are collected from FEniCS using the finite element method (FEM) accounting for appropriate parametric variations. The GP incorporates physical principles of heat transfer in the methodology to reach high accuracy and efficiency. The dynamic power map for the CPU in FEM thermal simulation is generated from gem5 and McPACT, together with the SPLASH-2 benchmarks as the simulation workload. It is shown that PODTherm-GP offers an accurate thermal prediction of the CPU with a resolution as fine as the FEM. It is also demonstrated that PODTherm-GP is capable of predicting the dynamic thermal profile of the chip with a good accuracy beyond the training conditions. Additionally, the approach offers a reduction in degrees of freedom by more than 5 orders of magnitude and a speedup of 4 orders, compared to the FEM.

A continuous field adaptive mesh refinement algorithm for isogeometric topology optimization using PHT-Splines

This study proposes a continuous density-field based isogeometric topology optimization (ITO) using Polynomial splines over Hierarchical T-meshes (PHT-Splines). Taking the benefit of isogeometric methods, the design density values for the mesh are defined using spline basis functions as a continuous density function (CDF) through the control points. The material density at the control points will then appear as a facade over the domain, which is further employed to generate a very smooth topology and also to propose a marking criterion for an adaptive mesh refinement (AMR) strategy during the optimization process. During this adaptive meshing, the isogeometric domain mesh is refined during each adaptive step, properly tracking the density variation over the elements and sharpening the boundaries of the optimized topologies. By utilizing the rationale of the Geometry Independent Field approximaTion (GIFT) framework, any complex multi-patch geometry constructed in an industry-standard CAD package using Non-Uniform Rational B-Splines (NURBS) may be imported and discretized using PHT-Splines, including higher dimensions. This helps to maintain the geometrical accuracy through an initial coarse mesh and computational accuracy of the optimization iterations through the local refinement feature of the PHT-Splines. An adaptive first-neighbourhood smoothening algorithm based on the Shepard function is also proposed to obtain manufacturable topologies and to circumvent post-processing stages. The proposed method was observed to provide remarkably smooth topologies with an initial coarse mesh, thereby obtaining a computationally efficient domain to be solved. Compared to existing element density based methods, the proposed method demonstrates a substantial minimization in Degree’s-of-Freedom (DoF) requirement to arrive at clear and realistic solutions. Also, when compared to globally refined solutions, a 50%–90% reduction in DoF is achieved using the adaptive refinement strategy.

The Effect of the Degree of Freedom and Weight of the Hand Exoskeleton on Joint Mobility Function

This study aims to investigate the effects of the degree of freedom (DOF) and weight of the hand exoskeleton (HE) on hand joint mobility function (ease of movement, movement range) in fine hand use activities. A three-digit passive HE prototype was built to fit each of the 12 participants. Two DOF setups (three DOF, two DOF), two digits’ weight levels (70 g, 140 g), and barehand conditions were tested. A productivity task (performed with Standardized-Nine Hole Peg Test) and motion tasks, both performing the tip pinch and tripod pinch, were conducted to measure the task completion time and the range of motion (ROM) of the digit joints, respectively, using a motion capture system. The perceived ease rating was also measured. The results showed that DOF reduction and weight addition caused a significant task completion time increase and rating drop (p < 0.05). Meanwhile, the DOF reduction increased the ROM reduction of the proximal interphalangeal joints; however, the weight addition caused a correction of the ROM reduction of several joints (p < 0.05) at the tripod pinch. In conclusion, wearing an HE reduces hand joint mobility, especially in lower DOF. However, a certain weight addition may improve joint mobility in terms of the fingers’ movement range.

Protection Against False Data Injection Attacks Considering Degrees of Freedom in Attack Vectors

In a false data injection attack to compromise the results of state estimation, the attacker intends to alter the meter readings communicated to the control center to his/her desired values to realize an objective. However, to avoid detection, the manipulated readings should not violate physical laws. Therefore, even if an attacker is able to attack a set of meters, he/she may not be able to inject an arbitrary magnitude of attack into all meters of this set. In this paper, we study the degrees of freedom in the attack vectors available to the attacker; higher degrees of freedom pose a higher risk to the utility. We propose an algorithm to determine the degrees of freedom in attack vectors and a method to find the set of meters to be secured in order to minimize the degrees of freedom in all attack vectors. Our results show that a significant reduction in degrees of freedom in all attack vectors is possible by securing a small set of meters.

Vorticity-based polynomial adaptation for moving and deforming domains

This paper introduces a novel non-dimensional vorticity-based polynomial adaptation indicator for moving and deforming domains using a high-order unstructured spatial discretization. We verify the utility of this approach when applied to the Arbitrary Lagrangian–Eulerian (ALE) form of the compressible Navier–Stokes equations for a range of applications on moving and deforming domains. Specifically, we verify the ALE implementation by performing simulations of an Euler Vortex (EV), and then, illustrate the accuracy and efficiency of the adaptation routine by performing simulations of flow over an oscillating circular cylinder with two different flow settings, dynamic stall of a 2D NACA 0012 airfoil undergoing heaving and pitching motions, shallow dynamic stall of a 3D SD 7003 airfoil undergoing heaving and pitching motions, and flow over a Vertical Axis Wind Turbine (VAWT) composed of two NACA 0012 airfoils. Results demonstrate that the non-dimensional vorticity indicator can track regions of interest, such as vortices and boundary layers, and yields a significant reduction in degrees of freedom when paired with polynomial adaptation.

Pushing nanomaterials up to the kilogram scale – An accelerated approach for synthesizing antimicrobial ZnO with high shear reactors, machine learning and high-throughput analysis

Novel materials are the backbone of major technological advances. However, the development and wide-scale introduction of new materials, such as nanomaterials, is limited by three main factors—the expense of experiments, inefficiency of synthesis methods and complexity of scale-up. Reaching the kilogram scale is a hurdle that takes years of effort for many nanomaterials. We introduce an improved methodology for materials development, combining state-of-the-art techniques—multi-objective machine learning optimization, high yield microreactors and high throughput analysis. We demonstrate this approach through the optimization of ZnO nanoparticle synthesis, simultaneously targeting high yield and high antibacterial activity. In fewer than 100 experiments, we developed a 1 kg day−1 continuous synthesis for ZnO (with a space-time-yield of 62.4 kg day−1 m−3), having an antibacterial activity comparable to hydrothermally synthesized nano-ZnO and cetrimonium bromide. Following this, we provide insights into the mechanistic factors underlying the performance-yield tradeoffs of synthesis and highlight the need for benchmarking machine learning models with traditional chemical engineering methods. Methods for increasing model accuracy at steep pareto fronts, in this case at yields close to 1 kg per day, should also be improved. To project the next steps for process scale-up and the potential advantages of this methodology, we conduct a scalability analysis in comparison to conventional batch production methods, in which there is a significant reduction in degrees of freedom. The proposed method has the potential to significantly reduce experimental costs, increase process efficiency and enhance material performance, which culminate to form a new pathway for materials discovery.

Temporal splitting algorithms for non-stationary multiscale problems

In this paper, we study temporal splitting algorithms for multiscale problems. The exact fine-grid spatial problems typically require some reduction in degrees of freedom. Multiscale algorithms are designed to represent the fine-scale details on a coarse grid and, thus, reduce the problems' size. When solving time-dependent problems, one can take advantage of the multiscale decomposition of the solution and perform temporal splitting by solving smaller-dimensional problems, which is studied in the paper. In the proposed approach, we consider the temporal splitting based on various low dimensional spatial approximations. Because a multiscale spatial splitting gives a “good” decomposition of the solution space, one can achieve an efficient implicit-explicit temporal discretization. We present a recently developed theoretical result in [15] and adopt in this paper for multiscale problems. Numerical results are presented to demonstrate the efficiency of the proposed splitting algorithm.

Intra-Segment Coordination Variability in Road Cyclists during Pedaling at Different Intensities

Background: The purpose of this study is to examine the lower extremity intra-segment coordination and variability of road cyclists during pedaling at different intensities. Methods: Eleven semi-professional road cyclists perform four trials at workloads of 50%, 75%, 100%, and 125% of their maximum power output in a randomized order. Thigh, shank, and foot range of motions (ROM) and the coordination and variability of these segments are compared across different pedaling intensities. Results: Foot ROM in the sagittal plane and shank ROM in the transverse plane are significantly different between different pedaling intensities. Moreover, specific coupling patterns and variabilities are observed across the pedaling cycle; however, they were not significantly different across different pedaling intensities in four pedaling phases. Conclusion: The results highlight the role of knee extensors and plantar flexors during the first and second half of the propulsion phase of pedaling, respectively. Thigh abduction dominancy with faster movement compared to the shank indicates a more valgus stress during the propulsive phase of pedaling, which can increase the risk of overuse injuries in the knee. Moreover, the smaller variability during the transition between the propulsive and recovery phases indicates a reduction in degrees of freedom and may increase the risk of overuse injuries.

Using coarse-grained models to examine structure-property relationships of diblock-arm star polymers

Molecular dynamics simulations were used to verify that diblock-arm star polymers (SPs) can be accurately described using coarse-grained models. Each arm of the SPs examined consist of an inner polyethylene block and an outer polyethylene oxide block. A sixteen-arm SP with arm chemistry -CH224-CH2OCH26-H was mapped to a MARTINI coarse-grained representation and compared to results from atomistic simulations of an identical chemistry and architecture. Results for radius of gyration, radial distribution functions, and occupied volumes all show agreement with the atomistic model and reproduce general trends with respect to temperature. Additional measurements of interfacial area and anisotropy reveal that our model does not demonstrate crystallization of the polyethylene blocks at low temperatures as observed in the atomistic model which can be ascribed to the reduction in degrees of freedom that follow from coarse-graining. Calculated values for relaxation times agree with the general trends from the atomistic simulations. Nineteen additional coarse-grained star polymers were examined in order to investigate structure-property relationships that arise by varying arm number and arm length. The ability to model these chemical species at this resolution will allow for simulations of polymeric nanocarriers in more complex environments including at increased concentrations and near surfaces. This work demonstrates an important first step in work that will enable improved design of these polymeric nanoparticles for drug delivery applications.

Alternative multiscale material and structures modeling by the finite-element method

In this study, we present an alternative finite element to multiscale analysis. In this strategy, strain energy comes only from semi-discrete or lattice elements immersed in a continuum without stiffness, enabling mechanical analysis from molecular scales to macroscopic scales. Some characteristics of the proposed element are: (1) geometrically non-linear exact description that allows the presence of large displacements and large strain, (2) general mapping that allows curved and distorted elements generation with automatic immersions, (3) total compatibility with standard finite elements, and (4) huge degrees of freedom reduction with a small loss of continuum mobility. Throughout the text, the proposed strategy is presented in detail and applied in the determination of suitable meshes for any scale of analysis, which is an important information for future applications. To be direct, a Lennard–Jones-like (LJL) potential is chosen to build different crystalline-like structures that, when immersed in finite elements without stiffness, results in the desired continuous behavior. In this sense, some space of the paper is used to determine the energy constant of the LJL potential for these different "crystalline" structures at any scale. Taking advantage of the total compatibility of the proposed element with continuum elements, the multiscale strategy is straightforward applied. Selected examples are used to demonstrate the good behavior of the proposed element and its applicability. Future developments to enhance applications are commented at the conclusion section.

Efficient topology optimization based on DOF reduction and convergence acceleration methods

This paper proposes a highly efficient topology optimization using two accelerated methods, which can reduce the degrees of freedom (DOFs) of the finite element equations and accelerate the iteration convergence of the topology optimization. For the DOF reduction, a method based on the empty elements and the displacement change during the topology iterations is presented to remove the DOFs from the finite element equations. For the convergence acceleration, a gray-scale suppression method is proposed to accelerate the polarization of design variables which accelerates the iteration convergence of the topology optimization. Three numerical examples including 2D and 3D cases are tested, and the results show that the proposed method can significantly improve the efficiency of the topology optimization and obtain the optimization results with the same accuracy. The computational time is only about 7% - 29% compared to the conventional topology optimization method.

Argyris type quasi-interpolation of optimal approximation order

The paper investigates a quasi-interpolation framework for the construction of operators that provide approximations for bivariate functions from the space of Argyris macro-element splines on a general triangulation. An operator of such type is determined by local approximation operators that produce quintic polynomials and are naturally associated with vertices and triangles of the triangulation. The quasi-interpolant resembles the properties of the standard Argyris interpolant, especially in the determination of data corresponding to the vertices of the triangulation. The main difference is in the definition of the cross-boundary derivatives, where a novel approach with degree raising is used to eliminate the need for additional interpolation of derivatives at the edges. Despite reduction in degrees of freedom, a suitable choice of local approximation operators ensures that the quasi-interpolation operator has quintic precision and is of optimal approximation order. To demonstrate the usability of the presented approach, three approximation methods are derived based on local Hermite and Lagrange interpolation and least square fitting.

Ultra-weak symmetry of stress for augmented mixed finite element formulations in continuum mechanics

In this paper we propose a novel way to prescribe weakly the symmetry of stress tensors in weak formulations amenable to the construction of mixed finite element schemes. The approach is first motivated in the context of solid mechanics (using, for illustrative purposes, the linear problem of linear elasticity), and then we apply this technique to reduce the computational cost of augmented fully-mixed methods for thermal convection problems in fluid mechanics, in the case where several additional variables are defined. We show that the new approach allows to maintain the same structure of the mathematical analysis as in the original formulations. Therefore we only need to focus on ellipticity of certain bilinear forms, as this property provides feasible ranges for the stabilization parameters that complete the description of augmented methods. In addition, we present some numerical examples to show that these methods perform better than their counterparts that include vorticity, and emphasize that the reduction in degrees of freedom (and therefore, in computational cost) does not affect the quality of numerical solutions.

Putting Scaffolding Into Action: Preschool Teachers\u2019 Actions Using Interactive Whiteboard

This study aimed to explore preschool teachers’ actions in order to support children’s learning processes in a context where an interactive whiteboard (IWB) is used. Five preschool teachers and 22 children aged 4–6 were video observed in 2017 and early spring 2018 over a period of 5 months. The findings of the study revealed 21 scaffolding actions which preschool teachers used including: Concretizing, Questioning, Instructing, Providing space, Affirming, Providing feedback, Inviting, Watching, Laughing together, Approaching, Standing/sitting beside, Simplifying, Filling in the blanks, Confirming, Participating, Challenging perception, Challenging thought, Explaining facts, Displaying, Explaining solutions, and Referring back. By characterizing teachers’ actions in relation to different scaffolding functions, the relationship between action and scaffolding function was particularly clarified. Six of the functions, including recruitment, direction maintenance, marking critical features, reduction in degrees of freedom, frustration control and demonstration were aligned with Wood et al.’s (Child Psychol Psychiatry 17:88–100, 1976) theoretical framework. By identifying two additional functions, i.e., mutual enjoyment and participation in the activity, more importantly the study contributed to the development of Wood et al.’s (Child Psychol Psychiatry 17:88–100, 1976) theoretical framework. It can be said that the findings of the study expanded and deepened our understanding regarding scaffolding processes and the ways they can be implemented in teaching practices.

Effects of treadmill running velocity on lower extremity coordination variability in healthy runners

With a growing interest in coordination variability and its role in endurance running, it is important to identify the effect of running velocity. The purpose of the current study was to investigate the effect of treadmill running velocity on the coordination and variability of coordination of lower extremity couplings of healthy runners during stance. Fourteen apparently healthy runners ran on a split-belt force instrumented treadmill at five different velocities. Continuous relative phase (CRP) was used to quantify coordination and variability (vCRP) between lower extremity couplings of the right limb (thigh-shank, thigh-foot, shank-foot) during three phases of stance (loading, mid stance, and propulsion). Multiple one-way repeated measure ANOVAs were conducted to identify differences among velocity conditions at each phase and discrete events (initial foot contact, peak knee flexion during stance, and toe-off). Thigh internal/external rotation (IR/ER)-Shank abduction/adduction (AB/AD) coupling was different during the propulsive phase (p = 0.02). Thigh flexion/extension-Shank flexion/extension showed the greatest differences in vCRP across velocity conditions with differences occurring during loading phase, mid stance, propulsive phase, and peak flexion (p < 0.05). Additionally, significant differences were seen in Thigh FL/EX-Shank FL/EX (toe-off, p = 0.01) and Thigh FL/EX-Foot inversion/eversion (IN/EV) (toe-off, p = 0.032). Interestingly, the decreases in vCRP values were accompanied by changes in center of mass vertical motion during stance, but not knee flexion angles. Increases in running velocity led to a more constrained running pattern through a reduction in degrees of freedom.

Dynamic Reductions for the Nonlinear Soil-Foundation-Structure System Interactions

The article presents new dynamic superelements for DOF (degrees of freedom) reduction. Proposed spectral superelements (SES) are based on the method of normal coordinates. The developed superelements with basis nodes (SEB) for the inhomogeneous systems are also shown. Proposed superelements ensure to obtain the diagonal matrix of masses which allow to use the efficiency of explicit method. The results of nonlinear analysis using proposed reduction method for the group of high-rise building are presented. The calculations were performed by explicit method with the help of Automated System of Scientific Research (ASSR) “VESNA-DYN”.

An extension of Carrera unified formulation in polar coordinate for analysis of circular sandwich plate with FGM core using GDQ method

The Carrera Unified Formulation (CUF) is a technique that, in addition to eliminate the imperfections of other theories, allows us to achieve a large class of models, such as equivalent single layer, layer wise, and mixed formulation in a unified manner. For this reason, this formulation can be very effective in different analyses, and according to the desired analysis, an appropriate model could be chosen. In this research, for the first time, the Carrera Unified Formulation is extended in the polar coordinates for analyzing the sandwich circular plate with the functionally graded material core. In order to apply variations in the properties of the functionally graded material, the variable kinematic method is used in the frame work of CUF, which ultimately leads to a reduction in degrees of freedom and an increase in the accuracy of the results. The functionally graded material is modeled as a mixture of ceramics and metal, whose properties change according to a power distribution in the direction of thickness. In this research, the generalized differential quadrature (GDQ) method is used to solve the governing equations. The obtained results are compared with the existing three-dimensional results as well as the generalized Zig-Zag theory, which indicates the high accuracy of the CUF formulation for circular plates in the polar coordinates. In addition, new results are provided for different geometries, thickness ratios, boundary conditions as well as deflection, radial displacements, annular and radial stresses, and transverse stresses along the thickness.

Application of a high-order CFD harmonic balance method to nonlinear aeroelasticity

An Aeroelastic-Harmonic Balance (A-HB) formulation of the Euler flow equations using a high-order spatial discretization scheme coupled with structural dynamic equations is proposed. The main objective of this new approach is to dramatically reduce the computational cost required to predict unsteady, periodic problems such as limit cycle oscillations (LCO). To this end, a new solver based on the Monotonicity Preserving limiter together with the AUSM+-up flux function is developed for the harmonic balance equations. The use of high-order CFD schemes allows the reduction of the number of degrees of freedom required to achieve a given desired accuracy, with respect to lower order schemes. In this paper, the reduction in degrees of freedom of the fluid system is exploited in the context of a CFD based Harmonic-Balance framework using a frequency updating procedure to determine the limit cycle conditions. The standard A-HB methodology has shown over one order of magnitude speed-up over time-marching methods; by employing the proposed high-order scheme in conjunction with coarser grids, the LCO computational time is halved without compromising accuracy.