Method for solving differential equations with coefficients in the form of the Heaviside step function

2021 ◽  
Vol 13 (3) ◽  
pp. 41-47
Author(s):  
K. O. Guk ◽  
O. A. Myltcina
Keyword(s):  
Differential Equations ◽  
Step Function ◽  
Heaviside Step Function

A Meshless Local Petrov-Garlerkin Method for Solving the Biharmonic Equation

2014 ◽  
Vol 931-932 ◽  
pp. 1488-1494
Author(s):  
Supanut Kaewumpai ◽  
Suwon Tangmanee ◽  
Anirut Luadsong
Keyword(s):  
Relative Error ◽  
Biharmonic Equation ◽  
Interpolation Method ◽  
Step Function ◽  
Special Treatment ◽  
Maximum Relative Error ◽  
Heaviside Step Function ◽  
Treatment Techniques ◽  
Point Interpolation Method ◽  
Point Interpolation

A meshless local Petrov-Galerkin method (MLPG) using Heaviside step function as a test function for solving the biharmonic equation with subjected to boundary of the second kind is presented in this paper. Nodal shape function is constructed by the radial point interpolation method (RPIM) which holds the Kroneckers delta property. Two-field variables local weak forms are used in order to decompose the biharmonic equation into a couple of Poisson equations as well as impose straightforward boundary of the second kind, and no special treatment techniques are required. Selected engineering numerical examples using conventional nodal arrangement as well as polynomial basis choices are considered to demonstrate the applicability, the easiness, and the accuracy of the proposed method. This robust method gives quite accurate numerical results, implementing by maximum relative error and root mean square relative error.

The Mesoscopic Investigation on Moduli Prediction in Visco-Elastic Particle Reinforced Composites

2008 ◽  
Vol 385-387 ◽  
pp. 329-332
Author(s):  
Xue Zhong Ding ◽  
Li Qiang Tang
Keyword(s):  
Volume Fraction ◽  
Step Function ◽  
Elastic Behavior ◽  
Loading Path ◽  
Elastic Model ◽  
Reinforced Composites ◽  
Heaviside Step Function ◽  
Equivalent Inclusion ◽  
Inclusion Theory ◽  
Elastic Particle

The visco-elastic mechanism of particles reinforced composites has been investigated through revised Eshelby equivalent inclusion theory. A visco-elastic model is applied. Furthermore, by introducing Heaviside step function and Laplace transform, the creep constitutional equation related to strain rate effect is achieved. Finally, by equivalent inclusion theory, introducing secant modulus, the material moduli with time and volume fraction concerning Glass/ED6 particles reinforced materials have been given. The results show that the visco-elastic property of composite material is mainly determined by the visco-elastic behavior of the matrix, which meet experiment results well. It can be concluded from the results that there exits close relationship between the inclusion shape, volume fraction and loading path.

scholarly journals An Asymptotic Approach for the Elastodynamic Problem of a Plate under Impact Loading

2010 ◽  
Vol 2010 ◽  
pp. 1-10
Author(s):  
Penelope Michalopoulou ◽  
George A. Papadopoulos
Keyword(s):  
Laplace Transform ◽  
Impact Loading ◽  
Step Function ◽  
Infinite Strip ◽  
Asymptotic Approach ◽  
Heaviside Step Function ◽  
Inertia Effects ◽  
Elastodynamic Problem ◽  
Line Force ◽  
The One

An approach is presented for analyzing the transient elastodynamic problem of a plate under an impact loading. The plate is considered to be in the form of a long strip under plane strain conditions. The loading is taken as a concentrated line force applied normal to the plate surface. It is assumed that this line force is suddenly applied and maintained thereafter (i.e., it is a Heaviside step function of time). Inertia effects are taken into consideration and the problem is treated exactly within the framework of elastodynamic theory. The approach is based on multiple Laplace transforms and on certain asymptotic arguments. In particular, the one-sided Laplace transform is applied to suppress time dependence and the two-sided Laplace transform to suppress the dependence upon a spatial variable (along the extent of the infinite strip). Exact inversions are then followed by invoking the asymptotic Tauber theorem and the Cagniard-deHoop technique. Various extensions of this basic analysis are also discussed.

scholarly journals Characterization and Mitigation of Model Bias in Parametric Mapping of Dopamine Response to Behavioral Challenge

2021 ◽  
Author(s):  
Michael A Levine ◽  
Joseph B Mandeville ◽  
Finnegan Calabro ◽  
David Izquierdo-Garcia ◽  
Julie C Price ◽  
...  
Keyword(s):  
Nucleus Accumbens ◽  
Step Function ◽  
Binding Potential ◽  
Reference Tissue ◽  
Heaviside Step Function ◽  
Average Bias ◽  
Adult Participants ◽  
Single Target ◽  
First Pass ◽  
Initial Uptake

Compartmental modeling of 11C-raclopride (RAC) is commonly used to measure dopamine response to intra-scan behavioral tasks. Bias in estimates of binding potential (BPND) and its dynamic changes (ΔBPND) can arise when the selected compartmental model deviates from the underlying biology. In this work, we characterize the bias associated with assuming a single target compartment and propose a model for reducing this bias by selectively discounting the contribution of the initial uptake period. Methods: 69 healthy young adult participants were scanned using RAC PET/MR while simultaneously performing a rewarded behavioral task. BPND and ΔBPND were estimated using an extension of the Multilinear Reference Tissue Model (MRTM2) with the task challenge encoded as a Heaviside step function. Bias was estimated using simulations designed to match the acquired data and was reduced by introducing a new model (DE-MRTM2) that reduces the biasing influence of the initial uptake period in the modeled estimation of BPND for both simulations and participant data. Results: Bias in ΔBPND was observed to vary both spatially with BPND and with the assumed value of k4. At the most likely value of k4 (0.13 min-1), the average bias and the maximum voxel bias magnitude in the nucleus accumbens were estimated to be 1.2% and 3.9% respectively. Simulations estimated that debiasing the contribution of the first 27 minutes of acquired data reduced average bias and maximum voxel bias in the nucleus accumbens ΔBPND to -0.3% and 2.4% respectively. In the acquired participant data, DE-MRTM2 produced modest changes in the experimental estimates of striatal ΔBPND, while extrastriatal bias patterns were greatly reduced. DE-MRTM2 also considerably reduced the dependence of ΔBPND upon the first-pass selection of k2'. Conclusion: Selectively discounting the contribution of the initial uptake period can help mitigate BPND- and k4-dependent bias in single compartment models of ΔBPND, while also reducing the dependence of ΔBPND on the first-pass estimation of k2'.

scholarly journals Applying the Heaviside step function to simulate the changes of temperature in automotive batteries

2021 ◽  
Vol 2061 (1) ◽  
pp. 012001
Author(s):  
N S Zakharov ◽  
N O Sapozhenkov ◽  
R V Tyan ◽  
V P Nazarov
Keyword(s):  
Thermal State ◽  
Electrical Equipment ◽  
Step Function ◽  
Heaviside Step Function ◽  
Large Cities ◽  
Battery Charge ◽  
Operating Modes ◽  
Control Units ◽  
Parking Lot ◽  
Nominal Capacity

Abstract Discharged batteries do not provide the specified voltage in the car’s power supply system during parking, which can cause malfunctions of electrical equipment and an increase in the quiescent current in the on-board network, due to incorrect operation of electronic control units responsible for the operation of self-diagnosis systems, anti-theft alarm, multi-media, maintaining a thermal state, etc. Therefore, to ensure a reliable start of the ICE and the proper operation of the electrical equipment of a car at low temperatures, it is required to maintain the battery in a charged state. Vehicle generator is selected taking into account the nominal capacity of the battery, power and operating modes of electrical consumers, which excludes the battery operation with a low level of charge. However, when operating cars in large cities in winter, the battery charge level decreases. Deterioration of the battery charging characteristics, increased power consumption of additional equipment and low speed of movement of cars in the city with frequent stops at intersections are the reasons for the decrease in the efficiency of the battery charge. In such conditions, the battery can be discharged not only by starting the ICE and turning on consumers in the parking lot, but also when the ICE is idling and at low crankshaft speeds while driving on city routes and during rush hours. Considering that the operational characteristics of the battery change significantly with decreasing temperature, studies aimed at establishing and predicting the battery temperature during operation are relevant.

A Universal Adjoint-Weighted Algorithm for Geometric Sensitivity Analysis of K-Eigenvalue Based on Continuous-Energy Monte Carlo Method

2018 ◽  
Author(s):  
Hao Li ◽  
Ganglin Yu ◽  
Shanfang Huang ◽  
Kan Wang
Keyword(s):  
Monte Carlo ◽  
Cross Section ◽  
Geometric Parameter ◽  
Neutron Transport ◽  
Delta Function ◽  
Step Function ◽  
Universal Method ◽  
Analytic Geometry ◽  
Heaviside Step Function ◽  
Dirac Delta

There exists a typical problem in Monte Carlo neutron transport: the effective multiplication factor sensitivity to geometric parameter. In several methods attempting to solve it, Monte Carlo adjoint-weighted theory has been proven to be quite effective. The major obstacle of adjoint-weighted theory is calculating derivative of cross section with respect to geometric parameter. In order to fix this problem, Heaviside step function and Dirac delta function are introduced to describe cross section and its derivative. This technique is crucial, and it establishes the foundation of further research. Based on above work, adjoint-weighted method is developed to solve geometric sensitivity. However, this method is limited to surfaces which are uniformly expanded or contracted with respect to its origin, such as vertical movement of plane or expansion of sphere. Rotation and translation are not allowed, while these two transformation types are more common and more important in engineering projects. In this paper, a more universal method, Cell Constraint Condition Perturbation (CCCP) method, is developed and validated. Different from traditional method, CCCP method for the first time explicitly articulates that the perturbed quantity is the parameter of spatial analytic geometry equations that used to describe surface. Thus, the CCCP can treat arbitrary one-parameter geometric perturbation of arbitrary surface as long as this surface can be described by spatial analytic geometry equation. Furthermore, CCCP can treat the perturbation of the whole cell, such as translation, rotation, expansion and constriction. Several examples are calculated to confirm the validity of CCCP method.

A Logical Function Method for Dynamic and Design Sensitivity Analysis of Mechanical Systems With Intermittent Motion

1982 ◽  
Vol 104 (1) ◽  
pp. 90-100 ◽  
Author(s):  
P. E. Ehle ◽  
E. J. Haug
Keyword(s):  
Sensitivity Analysis ◽  
Design Sensitivity ◽  
Step Function ◽  
Design Sensitivity Analysis ◽  
Logical Function ◽  
Smooth Functions ◽  
Heaviside Step Function ◽  
Approximate Problem ◽  
Logical Functions ◽  
Intermittent Motion

Dynamic and design sensitivity analysis of mechanisms and machines with intermittent motion are accomplished through introduction of “logical functions” to approximate discontinuities and special features of system motion. The Heaviside step function and the delta and unit doublet distributions are introduced to represent discontinuities and to determine the values of certain functions of isolated times. These functions and distributions are approximated by smooth functions, and validity of the approximation is argued both mathematically and physically. A numerical method is then presented for analysis of the approximate problem. An elementary and a complex, realistic example are presented to illustrate applications of the method.

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