Application of the Biot model to ultrasound in bone: Direct problem

The article considers direct and inverse problems of a system of nonlinear differential equations. Such problems are often found in various fields of science, especially in medicine, chemistry and economics. One of the main methods for solving nonlinear differential equations is the numerical method. The initial direct problem is solved by the Rune-Kutta method with second accuracy and graphs of the numerical solution are shown. The inverse problem of finding the coefficients of a system of nonlinear differential equations with additional information on solving the direct problem is posed. The numerical solution of this inverse problem is reduced to minimizing the objective functional. One of the methods that is applicable to nonsmooth and noisy functionals, unconditional optimization of the functional of several variables, which does not use the gradient of the functional, is the Nelder-Mead method. The article presents the NellerMead algorithm. And also a numerical solution of the inverse problem is shown.

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A Nonlinear Inverse Design Problem for a Pipe Type Heat Exchanger Equipped with Internal Z-Shape Lateral Fins and Ribs

Energies ◽

10.3390/en13236424 ◽

2020 ◽

Vol 13 (23) ◽

pp. 6424

Author(s):

Cheng-Hung Huang ◽

Chih-Yang Kuo

Keyword(s):

Heat Exchanger ◽

Thermal Performance ◽

Design Problem ◽

Direct Problem ◽

Numerical Solutions ◽

Three Dimensional ◽

Design Variables ◽

Non Linear ◽

Inverse Design Problem ◽

Η Value

A non-linear three-dimensional inverse shape design problem was investigated for a pipe type heat exchanger to estimate the design variables of continuous lateral ribs on internal Z-shape lateral fins for maximum thermal performance factor η. The design variables were considered as the positions, heights, and number of ribs while the physical properties of air were considered as a polynomial function of temperature; this makes the problem non-linear. The direct problem was solved using software package CFD-ACE+, and the Levenberg–Marquardt method (LMM) was utilized as the optimization tool because it has been proven to be a powerful algorithm for solving inverse problems. Z-shape lateral fins were found to be the best thermal performance among Z-shape, S-shape, and V-shape lateral fins. The objective of this study was to include continuous lateral ribs to Z-shape lateral fins to further improve η. Firstly, the numerical solutions of direct problem were solved using both polynomial and constant air properties and then compared with the corrected solutions to verify the necessity for using polynomial air properties. Then, four design cases, A, B, C and D, based on various design variables were conducted numerically, and the resultant η values were computed and compared. The results revealed that considering continuous lateral ribs on the surface of Z-shape lateral fins can indeed improve η value at the design working condition Re = 5000. η values of designs A, B and C were approximately 13% higher than that for Z-shape lateral fins, however, when the rib numbers were increased, i.e., design D, the value of η became only 11.5 % higher. This implies that more ribs will not guarantee higher η value.

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Direct Problem-Based Learning (DPBL): A Framework for Integrating Direct Instruction and Problem-Based Learning Approach

International Education Studies ◽

10.5539/ies.v11n1p119 ◽

2017 ◽

Vol 11 (1) ◽

pp. 119 ◽

Cited By ~ 3

Author(s):

Sri Winarno ◽

Kalaiarasi Sonai Muthu ◽

Lew Sook Ling

Keyword(s):

Direct Instruction ◽

Direct Problem ◽

Creative Thinking ◽

Problem Based Learning ◽

Knowledge Score ◽

Thinking Skills ◽

Face To Face ◽

Quasi Experimental ◽

Teamwork Skills ◽

Instruction Approach

Direct instruction approach has been widely used in higher education. Many studies revealed that direct instruction improved students’ knowledge. The characteristics of direct instruction include the subject delivered through face-to-face interaction with the lecturers and materials that sequenced deliberately and taught explicitly. However, direct instruction resulted in low creative thinking and teamwork skills among students. Therefore, problem-based learning activities were adapted to reform and create an innovation of a direct instruction approach in developing the new situation.Objective: This study aimed at exploring lecturers’ and students’ perspectives towards Direct Problem-Based Learning (DPBL) activities as a new approach for activities in the classroom.Design: A quasi-experimental design was used.Participants: Third-year students (N = 276) who signed up for Computer Networks subject from Dian Nuswantoro University, Indonesia and five lecturers were involved.Findings and Results: Learning outcomes were significantly positively (Sig. p=.00). Creative thinking skills score increased 8.4%, Teamwork skills score increased 11.5%, and knowledge score increased 25.9% of DPBL approach. The majority of students have difficulties in the direct instruction approach 4.71(.472). Whereas, 1.99(.655) students have low difficulty in DPBL approach. Expert participants agreed that DPBL approach can enhance creative thinking and teamwork skills 4.70(.50).

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The Optimization for the Shape Profile of the Slider Surface Under Ultra-Thin Film Lubrication Conditions by the Rarefied-Flow Model

Journal of Mechanical Design ◽

10.1115/1.3213528 ◽

2009 ◽

Vol 131 (10) ◽

Cited By ~ 3

Author(s):

Chin-Hsiang Cheng ◽

Mei-Hsia Chang

Keyword(s):

Thin Film ◽

Conjugate Gradient ◽

Gradient Method ◽

Direct Problem ◽

Problem Solver ◽

Thin Film Lubrication ◽

Rarefied Flow ◽

Slider Surface ◽

Inverse Knudsen Number ◽

Film Lubrication

The optimization of the surface shape for a slider to meet the specified load demands under an ultra-thin film lubrication condition has been performed in this study. The optimization process is developed based on the conjugate gradient method in conjunction with a direct problem solver, which is built based on the rarefied-flow theory. The direct problem solver is able to predict the pressure distributions of the rarefied gas flows in the slip-flow, transition-flow, and molecular-flow regimes with a wide range of characteristic inverse Knudsen number. First, the validity of the direct problem solver has been verified by a comparison with the existing information for some particular cases, and then the developed direct problem solver is incorporated with the conjugate gradient method for optimizing the shape profile of the slider surface. The performance of the present optimization approach has also been evaluated. Results show that the shape profile of the slider surface can be efficiently optimized by using the present approach. Thus, a number of cases under various combinations of influential parameters, involving the characteristic inverse Knudsen number and the bearing numbers in the x- and y-directions, are investigated.

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Direct Problem: The Euler Equation Approach

SpringerBriefs in Mathematics - Discrete–Time Stochastic Control and Dynamic Potential Games ◽

10.1007/978-3-319-01059-5_2 ◽

2013 ◽

pp. 11-34

Author(s):

David González-Sánchez ◽

Onésimo Hernández-Lerma

Keyword(s):

Euler Equation ◽

Direct Problem ◽

Equation Approach

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On investigation of the inverse problem for a parabolic integro-differential equation with a variable coefficient of thermal conductivity

Vestnik Udmurtskogo Universiteta Matematika Mekhanika Komp yuternye Nauki ◽

10.35634/vm200403 ◽

2020 ◽

Vol 30 (4) ◽

pp. 572-584

Author(s):

D.K. Durdiev ◽

J.Z. Nuriddinov

Keyword(s):

Thermal Conductivity ◽

Inverse Problem ◽

Variable Coefficient ◽

Direct Problem ◽

Local Existence ◽

Problem Solution ◽

Volterra Type ◽

Integral Term ◽

Local Existence And Uniqueness ◽

Coefficient Of Thermal Conductivity

The inverse problem of determining a multidimensional kernel of an integral term depending on a time variable $t$ and $ (n-1)$-dimensional spatial variable $x'=\left(x_1,\ldots, x_ {n-1}\right)$ in the $n$-dimensional heat equation with a variable coefficient of thermal conductivity is investigated. The direct problem is the Cauchy problem for this equation. The integral term has the time convolution form of kernel and direct problem solution. As additional information for solving the inverse problem, the solution of the direct problem on the hyperplane $x_n = 0$ is given. At the beginning, the properties of the solution to the direct problem are studied. For this, the problem is reduced to solving an integral equation of the second kind of Volterra-type and the method of successive approximations is applied to it. Further the stated inverse problem is reduced to two auxiliary problems, in the second one of them an unknown kernel is included in an additional condition outside integral. Then the auxiliary problems are replaced by an equivalent closed system of Volterra-type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the Hölder class of functions, we prove the main result of the article, which is a local existence and uniqueness theorem of the inverse problem solution.

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Another point of view on proportional navigation

Mathematical Problems in Engineering ◽

10.1155/s1024123x98000799 ◽

1998 ◽

Vol 4 (3) ◽

pp. 201-231

Author(s):

E. Duflos ◽

P. Penel ◽

P. Vanbeeghe ◽

P. Borne

Keyword(s):

Inverse Problem ◽

Direct Problem ◽

Point Of View ◽

Proportional Navigation ◽

Guidance Laws ◽

The Way

Proportional navigation is one of the most popular and one of the most used of the guidance laws. But the way it is studied is always the same: the acceleration needed to reach a known target is derived or analyzed. This way of studying guidance laws is called “the direct problem” by the authors. On the contrary, the problem considered here is to find, from the knowledge of a part of the trajectory of a maneuvering object, the target of this object. The authors call this way of studying guidance laws “the inverse problem”.

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