The Finite Element Method

Pepper, Darrell W.; Heinrich, J. C.

doi:10.1201/9781315395104

Cited by 60 publications

(23 citation statements)

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“…, n, assuming that the integrals are existing and finite. In the subsequent sections, these functions are used as test functions in a similar sense to the case of the FEM [10].…”

Section: Local Green Functions Of Ode For Arbitrary Partitioningmentioning

confidence: 99%

“…, n -1. On the first interval I 0 , only the test function ψ 0 is considered, while on the last interval I n , we define only the test function ϕ n similar to the FEM [10]. Initially, it appears as though the local Green functions are not all independent because…”

Section: Local Green Functions Of Ode For Arbitrary Partitioningmentioning

confidence: 99%

“…Furthermore, as indicated by (2.8) and demonstrated in the following subsections, the test functions are handled in a similar manner as in the FEM [10].…”

Section: Local Green Functions Of Ode For Arbitrary Partitioningmentioning

confidence: 99%

“…Well-known numerical methods, such as the finite difference method (FDM) [7], finite volume method [8], and boundary element method [9], can be used to resolve equations in form (1.1) by approximating the analytical solution of the ODE. Among the classic numerical methods for second-order differential equations, at present, the finite element method (FEM) is the most widely used [10]. This method has also been used to develop approaches involving a higher order of convergence, particularly with the application of higher-order elements [10].…”

Section: Introductionmentioning

confidence: 99%

“…Among the classic numerical methods for second-order differential equations, at present, the finite element method (FEM) is the most widely used [10]. This method has also been used to develop approaches involving a higher order of convergence, particularly with the application of higher-order elements [10].…”

Section: Introductionmentioning

confidence: 99%

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Exact schemes for second-order linear differential equations in self-adjoint cases

et al. 2020

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When working with mathematical models, to keep the model errors as small as possible, a special system of linear equations is constructed whose solution vector yields accurate discretized values for the exact solution of the second-order linear inhomogeneous ordinary differential equation (ODE). This case involves a 1D spatial variable x with an arbitrary coefficient function $\kappa (x)$ κ ( x ) and an arbitrary source function $f(x)$ f ( x ) at each grid point under Dirichlet or/and Neumann boundary conditions. This novel exact scheme is developed considering the recurrence relations between the variables. Consequently, this scheme is similar to those obtained using the finite difference, finite element, or finite volume methods; however, the proposed scheme provides the exact solution without any error. In particular, the adequate test functions that provide accurate values for the solution of the ODE at arbitrarily located grid points are determined, thereby eliminating the errors originating from discretization and numerical approximation.

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“…, n, assuming that the integrals are existing and finite. In the subsequent sections, these functions are used as test functions in a similar sense to the case of the FEM [10].…”

Section: Local Green Functions Of Ode For Arbitrary Partitioningmentioning

confidence: 99%

Section: Local Green Functions Of Ode For Arbitrary Partitioningmentioning

confidence: 99%

“…Furthermore, as indicated by (2.8) and demonstrated in the following subsections, the test functions are handled in a similar manner as in the FEM [10].…”

Section: Local Green Functions Of Ode For Arbitrary Partitioningmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exact schemes for second-order linear differential equations in self-adjoint cases

et al. 2020

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Visualizing Quasi‐Static Electric Fields with Flexible and Printed Organic Transistors

Shoji

Wada

Uto

et al. 2021

Adv Materials Technologies

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quasi-static E-fields have rarely been utilized in visualization or communication tools. Sharks and stingrays are some of the unique creatures that can feel quasistatic E-fields when searching for their prey. [4] Detecting the imperceptible quasistatic E-fields has the potential to open a new method of visualization. [5] For example, monitoring static electricity can avoid electrostatic discharge, which causes the breakdown of electronic devices [6] and fire accidents. [7] Proximity sensing based on the E-field is useful in robotics and human-machine interfaces. [8] A combination of E-field sensors and electrets, which have a quasi-permanent charge or dipole polarization, can be used to develop a new human-machine interface, such as touchscreen devices. The difficulty in visualizing quasi-static E-fields is mainly due to low-energy photons and the lack of integrable devices with high sensitivity to detect them. Furthermore, while there are lenses capable of perceiving different wavelengths of electromagnetic radiation, no such technology exists to see quasi-static E-fields. Conventional methods for measuring quasi-static E-fields or charges include gold-leaf electroscopes, [9] Faraday cages, and surface potential sensors. [10] A typical surface potential sensor uses metal mechanical choppers to modulate the electric flux and precisely measure the E-field. However, these systems are not suitable for application in 2D imaging because of their size (>1 cm 3 ), weight (>10 g), and cost (>$100) for each pixel.In recent studies, it has been shown that E-field sensing using organic semiconductors are realizable. Wang et al. used an organic semiconductor, a single rubrene crystal with two gold electrodes that was not a transistor, to detect the proximity of an atomic force microscope tip, a human finger, and a plastic ruler, each at a distance of several millimeters. [11] Liu et al. used a thin-film organic semiconductor polymer; poly[2,5-(2-octyl dodecyl)-3,6-diketopyrrolopyrrole-alt-5,5-(2,5-di(thien-2-yl)thieno[3,2-b]thiophene)], as a field-effect transistor with a silver wire connected to the gate electrode, and demonstrated proximity sensing up to a distance of several centimeters. [12] Furthermore, Lv et al. used a dinaphtho[2, thiophene organic semiconductor film with two gold electrodes (not a transistor) deposited in a vacuum to sense the proximity of charged objects. [13] The same study also demonstrated a sensor array on an ultra-flexible substrate. These sensors utilize the so-called field effect, [14] where charge carriers are induced in the organic semiconductor layers by the E-field. Although these studies have successfully detected the proximity of various charged objects, the sensitivity of the sensors is not sufficiently Unlike electromagnetic radiation such as radio waves, infrared-visibleultraviolet, and X-rays, static or quasi-static electric fields are rarely utilized for visualization in people's lives and industries. The difficulty in measuring quasi-static electric fields is mainly due to lo...

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Interfacial wettability and mass transfer characterizations for gas–liquid–solid triple‐phase catalysis

et al. 2022

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Heterogeneous catalysis is inseparable from interfacial mass transfer and chemical reaction processes determined by the structure and microenvironment. Different from hightemperature thermochemical processes, photo-and electrocatalysis operated at mild conditions often involve both gas and liquid phases, making it important but challenging to characterize the reaction process typically occurring at the gas-liquid-solid interface. Herein, we review the scope, feasibility, and limitation of ten types of currently available technologies used to characterize interfacial wettability and mass transfer properties of various triple-phase catalytic reactions. The review summarizes techniques from macroscopic contact angle measurement to microscopic environment electron microscopy for investigating the wettability-controlled structure of triple-phase interfaces. Experimental and computational methods in revealing the interfacial mass transfer process have also been systematically discussed, followed by a perspective on the opportunities and challenges of advanced characterization methods to help understand the fundamental reaction mechanism of triple-phase catalysis.

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The Finite Element Method

Cited by 60 publications

References 0 publications

Exact schemes for second-order linear differential equations in self-adjoint cases

Exact schemes for second-order linear differential equations in self-adjoint cases

Visualizing Quasi‐Static Electric Fields with Flexible and Printed Organic Transistors

Interfacial wettability and mass transfer characterizations for gas–liquid–solid triple‐phase catalysis

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