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Application of Numerical Computation of Partial Differential Equations in Interactive Design of Virtual Reality Media

Publié en ligne: 15 Jul 2022
Volume & Edition: AHEAD OF PRINT
Pages: -
Reçu: 18 Apr 2022
Accepté: 09 Jun 2022
Détails du magazine
License
Format
Magazine
eISSN
2444-8656
Première parution
01 Jan 2016
Périodicité
2 fois par an
Langues
Anglais
Introduction

Nowadays, virtual reality (WIF) has been rapidly progressed and studied in depth in recent years. It is a three-dimensional digital model of the real world simulated through advanced scientific and technological means to process, display and achieve human-computer interaction[1]. At present, virtual reality technology has been applied to medical simulation, which has strong vitality with the function of super large-scale integrated circuit graphics simulation and high precision, which makes it one of the popular applications in the field of computer; at the same time, the continuous development and improvement of 3D printing technology and related software, which also promotes the progress of virtual reality theory research and engineering calculation method research to a certain extent[2,3]. Virtual reality technology in the field of computer simulation has also been rapidly developed. At present, virtual reality theory has been widely used in medical simulation, and the system is based on 3D modeling as the core, supported by relevant computational models to achieve human-computer interaction[4].

Virtual reality technology (LabVIE) is a new advanced science and technology developed, which breaks the traditional computer can only achieve the measurement and study of real-world objects through simulation programs, without the need of physical models to observe their internal structures[5]. In recent years, as the knowledge of computer, network and communication and other related disciplines have been expanded and integrated, people can use the virtual instrument system to observe the changes of the surrounding environment in real time and make timely response to obtain the required information[6].

Virtual reality technology is the product of the combination of computer hardware and software, which mainly utilizes the human-computer interface to achieve the interrelationship between human and machine through software. Virtual reality technology has two ways to show the digital media contains parameters calculation, image processing and other applications[7]. The most widely used and most common to common means in the actual work process is based on the network platform for simulation simulation analysis and related algorithm research; and for some complex environments need to solve various types of problems will use this method, the advantage is to achieve fast and low cost, but there are also some disadvantages. For example, changes in the simulation process can affect the results, which leads to slow algorithm calculation speed[8].

Partial differential equation and its numerical solution algorithm

Partial differential equation is a very important concept in computing, which is also called numerical solution method. It is mainly used in digital image processing and simulation, and through the study of virtual reality technology, we can understand the role played by computers in practical applications[9]. The common partial differential equation models are shown below.

Thermal diffusion equation: This equation is shown in Equation (1). ftκΔf=0 {{\partial f} \over {\partial t}} - \kappa \Delta f = 0

Laplace's equation: Laplace's equation, also known as the harmonic equation or potential equation in physics, is mainly used in mathematical modeling in electromagnetism and fluid science, and also used as a descriptive equation for image smoothing in the field of images. This equation is shown in Equation (2). 2f=0 {\nabla^2}f = 0

Finite difference method

The basic idea of the finite difference method is to use the difference to calculate the partial derivative, specifically, it is to approximate the partial derivative of a function with respect to a variable as the ratio of the difference between the values of two functions at finite distances from adjacent points to the distance between them[10]. For example, in digital image processing forward (Forword) difference is often used to approximate the partial derivative of a function with respect to time, as shown in Equation (3). ut|inuin+1uinΔt=Dt(+)u|in \left. {{{\partial u} \over {\partial t}}} \right|_i^n \approx {{u_i^{n + 1} - u_i^n} \over {\Delta t}} = \left. {D_t^{\left(+ \right)}u} \right|_i^n

According to the previous description, the difference approximation of the first-order partial derivative can be expressed as Equation (4): ut|inuin+1uinΔx:=Dx(+)u|inut|inuinui1nΔx:=Dx()u|inut|inui+1nui1n2Δx:=Dx(0)u|in \matrix{{\left. {{{\partial u} \over {\partial t}}} \right|_i^n \approx {{u_i^{n + 1} - u_i^n} \over {\Delta x}}: = \left. {D_x^{\left(+ \right)}u} \right|_i^n} \hfill \cr {\left. {{{\partial u} \over {\partial t}}} \right|_i^n \approx {{u_i^n - u_{i - 1}^n} \over {\Delta x}}: = \left. {D_x^{\left(- \right)}u} \right|_i^n} \hfill \cr {\left. {{{\partial u} \over {\partial t}}} \right|_i^n \approx {{u_{i + 1}^n - u_{i - 1}^n} \over {2\Delta x}}: = \left. {D_x^{\left(0 \right)}u} \right|_i^n} \hfill \cr}

The representations in the above formula are called forward difference, backward difference, and center difference in turn. If u(x) is differentiable, use Taylor expansion to get Equation (5): u(x+Δx)=u(x)+uxΔx+122ux2(Δx)2+ux=u(x+Δx)u(x)Δx122ux2Δx=Dx(+)u+O(Δx) \matrix{{u\left({x + \Delta x} \right) = u\left(x \right) \cdot + {{\partial u} \over {\partial x}}\Delta x + {1 \over 2}{{{\partial^2}u} \over {\partial {x^2}}}{{\left({\Delta x} \right)}^2} + \cdots} \hfill \cr {\Rightarrow {{\partial u} \over {\partial x}} = {{u\left({x + \Delta x} \right) - u\left(x \right)} \over {\Delta x}} - {1 \over 2}{{{\partial^2}u} \over {\partial {x^2}}}\Delta x - \cdots} \hfill \cr {= D_x^{\left(+ \right)}u + O\left({\Delta x} \right)} \hfill \cr}

It can be obtained from the above formula that the accuracy of the forward difference and the backward difference are both first-order[11]. Similarly, Equation (6) is obtained: u(x+Δx)=u(x)+uxΔx+122ux2(Δx)2+u(xΔx)=u(x)uxΔx+122ux2(Δx)2+ux=u(x+Δx)u(xΔx)2ΔxO((Δx)2) \matrix{{u\left({x + \Delta x} \right) = u\left(x \right) + {{\partial u} \over {\partial x}}\Delta x + {1 \over 2}{{{\partial^2}u} \over {\partial {x^2}}}{{\left({\Delta x} \right)}^2} + \cdots} \hfill \cr {u\left({x - \Delta x} \right) = u\left(x \right) - {{\partial u} \over {\partial x}}\Delta x + {1 \over 2}{{{\partial^2}u} \over {\partial {x^2}}}{{\left({\Delta x} \right)}^2} + \cdots} \hfill \cr {\Rightarrow {{\partial u} \over {\partial x}} = {{u\left({x + \Delta x} \right) - u\left({x - \Delta x} \right)} \over {2\Delta x}}O\left({{{\left({\Delta x} \right)}^2}} \right) - \cdots} \hfill \cr}

It can be concluded that the accuracy of the central difference is second-order.

In addition, using the finite difference method, it is also possible to approximate the numerical estimation of the second order partial derivative in the partial differential equation. We first use the central difference to find the first-order partial derivatives at the two adjacent half-points to obtain Equation (7): [ux]i+1/2nui+1nuinΔx,[ux]i1/2nuinui1nΔx \matrix{{\left[{{{\partial u} \over {\partial x}}} \right]_{i + 1/2}^n \approx {{u_{i + 1}^n - u_i^n} \over {\Delta x}},} \hfill \cr {\left[{{{\partial u} \over {\partial x}}} \right]_{i - 1/2}^n \approx {{u_i^n - u_{i - 1}^n} \over {\Delta x}}} \hfill \cr}

Next, the result obtained by doing the central difference between the two first-order differences is the approximation of the second-order partial derivatives, as shown in Equation (8): [2ux2]in([ux]i+12n[ux]i12n)/Δx=ui+1n2uin+ui1n(Δx)2=Dxx(0)u|in \matrix{{\left[{{{{\partial^2}u} \over {\partial {x^2}}}} \right]_i^n \approx \left({\left[{{{\partial u} \over {\partial x}}} \right]_{i + {1 \over 2}}^n - \left[{{{\partial u} \over {\partial x}}} \right]_{i - {1 \over 2}}^n} \right)/\Delta x} \hfill \cr {= {{u_{i + 1}^n - 2u_i^n + u_{i - 1}^n} \over {{{\left({\Delta x} \right)}^2}}} = \left. {D_{xx}^{\left(0 \right)}u} \right|_i^n} \hfill \cr}

Similarly, the half-point central difference method can also be used to approximate the second-order partial derivatives in two different directions along the x-axis and the y-axis, as shown in Equation (9): [2uxy]i,j=([ux]i,j+1/2(ux)i,j1/2)/Δy(ui+1,j+1/2ui1,j+1/22Δxui+1,j1/2ui1,j1/22Δx)/Δy \matrix{{{{\left[{{{{\partial^2}u} \over {\partial x\partial y}}} \right]}_{i,j}} = \left({{{\left[{{{\partial u} \over {\partial x}}} \right]}_{i,j + 1/2}} - {{\left({{{\partial u} \over {\partial x}}} \right)}_{i,j - 1/2}}} \right)/\Delta y} \hfill \cr {\approx \left({{{{u_{i + 1,\,j + 1/2}}{u_{i - 1,\,j + 1/2}}} \over {2\Delta x}} - {{{u_{i + 1,\,j - 1/2}}{u_{i - 1,\,j - 1/2}}} \over {2\Delta x}}} \right)/\Delta y} \hfill \cr}

Principle of variational partitioning

A generalized function can be viewed simply as a function with a function as the independent variable. In a one-dimensional space, a generalized function can be represented by the following expression, as shown in Equation (10). E(u)=x0x1F(x,u,ux)dx E\left(u \right) = \int_{{x_0}}^{{x_1}} {F\left({x,u,{u_x}} \right)dx}

In order to find the first-order variant E′ of the generalized function E(u), one can add a small increment v(x) to the assumed optimal solution u(x) to obtain u(x)+v(x). If both v(x) and v′(x) satisfy the condition of being sufficiently small, then using the Taylor expansion we can obtain Equation (11). F(x,u+v,u+v)=F(x,u,u)+Fuv+Fuv+ F\left({x,u + v,{u^{'}} + {v^{'}}} \right) = F\left({x,u,{u^{'}}} \right) + {{\partial F} \over {\partial u}}v + {{\partial F} \over {\partial u}}{v^{'}} + \cdots

Combining equation (10) and integrating both sides of the above equation yields Equation (12): E(u+v)E(u)+x0x1(vFu+vFu)dx E\left({u + v} \right) \cong E\left(u \right) + \int_{{x_0}}^{{x_1}} {\left({v{{\partial F} \over {\partial u}} + {v^{'}}{{\partial F} \over {\partial {u^{'}}}}} \right)dx}

From the endpoint boundary conditions we have: u(x0)+(x0)=a,u(x1)+v(x1)=b, so v(x0)=0,v(x1)=0. So by the divisional integration method we get Equation (13). x0x1vFudx=x0x1Fudv=vFu|x0x1x0x1vddx(Fu)dx=x0x1vddx(Fu)dx \matrix{{\int_{{x_0}}^{{x_1}} {{v^{'}}{{\partial F} \over {\partial {u^{'}}}}dx = \int_{{x_0}}^{{x_1}} {{{\partial F} \over {\partial u'}}} dv}} \hfill \cr {\left. {= v{{\partial F} \over {\partial {u^{'}}}}} \right|_{{x_0}}^{{x_1}} - \int_{{x_0}}^{{x_1}} {v{d \over {dx}}\left({{{\partial F} \over {\partial {u^{'}}}}} \right)dx}} \hfill \cr {= - \int_{{x_0}}^{{x_1}} {v{d \over {dx}}\left({{{\partial F} \over {\partial {u^{'}}}}} \right)dx}} \hfill \cr}

Substituting the above equation into equation (14) can be obtained to equation (14). E(u+v)=E(u)+x0x1[vFuvddx(Fu)]dx E\left({u + v} \right) = E\left(u \right) + - \int_{{x_0}}^{{x_1}} {\left[{v{{\partial F} \over {\partial u}} - v{d \over {dx}}\left({{{\partial F} \over {\partial {u^{'}}}}} \right)} \right]dx}

In the above equation, when the generalized function E(u) is taken to the extreme value, if the perturbation vx) is sufficiently small, the value of the generalized function E is kept constant, at which point Equation (15) is obtained. Fuddx(Fu)=0 {{\partial F} \over {\partial u}} - {d \over {dx}}\left({{{\partial F} \over {\partial {u^{'}}}}} \right) = 0

The above expression can be called the Euler-Lagrange equation that must be satisfied when the generalized function in the one-dimensional case takes its extreme value[12].

Similarly, in the two-dimensional case, the generalized function E(u) can be expressed as shown in Equation (16). E(u)=ΩF(x,y,u,ux,uy)dxdy E\left(u \right) = \int\!\!\!\int\limits_\Omega {F\left({x,y,u,{u_x},{u_y}} \right)dxdy}

Using a derivation method similar to that in the one-dimensional case, the Euler-Lagrange equation corresponding to the extreme value of the functional in the two-dimensional case can also be obtained, that is, Equation (17) is obtained: Fuddx(Fu)ddx(Fuy)=0 {{\partial {\rm{F}}} \over {\partial u}} - {d \over {dx}}\left({{{\partial {\rm{F}}} \over {\partial {u^{'}}}}} \right) - {d \over {dx}}\left({{{\partial {\rm{F}}} \over {\partial {u_y}}}} \right) = 0

Gradient descent flow

For the partial differential equation we need to solve, it is assumed that its solution varies with time, so the time variable t can be introduced, and the perturbation term v(•) in Equation (16) can be regarded as from time t to t + At The amount of change that occurs in the function u(•, t) in the process of, then the Equation (18) is obtained: v=utΔt v = {{\partial u} \over {\partial t}}\Delta t

So Equation (18) can be rewritten as Equation (19): E(,t+Δt)=E(,t)+Δtx0x1ut[Fuddx(Fu)]dx \matrix{{E\left({\bullet,t + \Delta t} \right)} \hfill \cr {= E\left({\bullet,t} \right) + \Delta t\int_{{x_0}}^{{x_1}} {{{\partial u} \over {\partial t}}\left[{{{\partial F} \over {\partial u}} - {d \over {dx}}\left({{{\partial F} \over {\partial {u^{'}}}}} \right)} \right]dx}} \hfill \cr}

For the above formula, the first derivative can be made as Equation (20): ut=[Fuddx(Fu)]=ddx(Fu)Fu {{\partial u} \over {\partial t}} = - \left[{{{\partial F} \over {\partial u}} - {d \over {dx}}\left({{{\partial F} \over {\partial {u^{'}}}}} \right)} \right] = {d \over {dx}}\left({{{\partial F} \over {\partial {u^{'}}}}} \right) - {{\partial F} \over {\partial u}}

Substitute it into Equation (19) to get Equation (21): ΔE=E(,t+Δt)E(,t)=Δt[Fuddx(Fu)]2dx0 \matrix{{\Delta E = E\left({\bullet,t + \Delta t} \right) - E\left({\bullet,t} \right)} \hfill \cr {= - \Delta t\int {{{\left[{{{\partial F} \over {\partial u}} - {d \over {dx}}\left({{{\partial F} \over {\partial {u^{'}}}}} \right)} \right]}^2}dx \le 0}} \hfill \cr}

Based on the above formula, the solution of the function u(o, t) when the time variable reaches a steady state after the introduction of the time variable is also the solution of the static partial differential equation u(•), and the Equation (22) can be obtained at this time: ut=0Fuddx(Fu)=0 {{\partial u} \over {\partial t}} = 0 \Rightarrow {{\partial F} \over {\partial u}} - {d \over {dx}}\left({{{\partial F} \over {\partial {u^{'}}}}} \right) = 0

Similarly, applying the previous derivation process to the two-dimensional variational problem, the gradient descent flow for solving the two-dimensional variational problem can be obtained as shown in Equation (23): ut=ddx(Fux)+ddx(Fuy)Fu {{\partial u} \over {\partial t}} = {d \over {dx}}\left({{{\partial F} \over {\partial {u_x}}}} \right) + {d \over {dx}}\left({{{\partial F} \over {\partial {u_y}}}} \right) - {{\partial F} \over {\partial u}}

The application of network virtual reality technology in interaction design
Development tools

Partial differential equations as a huge branch of calculation, its numerical results are difficult to process. The WPF module structure is shown in Figure 1. The software provides a powerful background function and interface support system for the WPF module, which can realize each major part of the virtual reality media interaction design: interface display, parameter setting and state management, etc.; the program runs automatically and completes the simulation experiment data collection and analysis; the simulation image can be analyzed to visualize the load situation and change trend under the 3D practical training environment[13].

Figure 1

WPF module structure

NISQL is a software that provides data for virtual reality systems. It is used to obtain information about solid models in the virtual world and to interact them with programs. This software is mainly designed for users. The plug-in includes a variety of functional modules such as keys, keyboard and mouse, as well as related parameter setting methods and algorithm implementation; and through the USB interface connection so that users can select the required options to complete the corresponding operations according to their needs; NIAccess database is also a file format used to save data stored in MYSQL, so that it is convenient to protect the system when it is maintained in the future.

Application software of the network virtual reality system

The application software of the virtual reality system is computer hardware and software, which has a complete network interaction platform, including servers, clients and terminal devices. The application can be set to display the corresponding parameter values on different functional interfaces, so as to achieve the required data storage and file management and other functions; it can also perform data backup processing and store them in the database, so that it is convenient to find the lost or faulty information in the next use; this can also provide video conference system, voice synthesis player, which is convenient for people to watch virtual reality programs and game recording effects[14]. The composition structure of a common helmet display-based virtual implementation system is shown in Figure 2.

Figure 2

Helmet display-based virtual realization system

The server is one of the most core and important parts of the network interaction platform, and virtual reality technology is a hot spot in the current computer network research. It has strong real-time in desktop interaction, which can be combined with real life and can realize the user's problem solving and environment simulation. The server module used in this paper is shown in Figure 3.

Figure 3

Server module

Application Interface

Computer simulation software development should take into account some of the issues involved in the design of the interactive interface of virtual reality media. The first is the process of computer simulation. In this process, the model needs to be modified accordingly. Then the simulation results are compared with the actual situation to determine whether the error exists; finally, the difference between the parameter values shown on the simulation drawings and the programmed values is analyzed, and if there is a difference, it is necessary to find out the cause of the difference and make adjustments in time, so that the accuracy of the virtual reality media interactive interface can be improved and the accuracy of the relevant parameters can be improved.

The design of the virtual realization system based on partial differential equations

Virtual reality interactive system is a multimedia interaction based on partial differential equations. In computer technology, one can use suitable digital signal processing software so that the virtual environment can be displayed in real time and synchronized with the real world. The design of virtual reality media interactive system is a very critical and indispensable part of the whole computer software engineering, whether its function is perfect and reasonable, which is directly related to the overall software development quality. The principle of the virtual multimedia interactive system is shown in Figure 4.

Figure 4

The principle of virtual multimedia interactive system

System software structure diagram: Combined with the schematic diagram of multimedia interactive control system, considering the execution efficiency of software modules and the performance balance of each terminal of the system, the system software structure diagram can be represented in Figure 5:

Figure 5

Piranha processor architecture

Web virtual reality media interaction design examples
Virtual reality scenes

The interaction design in virtual reality systems is mainly proposed for a large number of problems in the real world, which can make the simulated environment and the actual situation match each other to a certain extent, and at the same time, it can solve various unexpected things that appear or may be encountered in the real life. In this thesis, we designed a simulation experiment based on partial differential equation numerical computation technique and combined with software for virtual reality scenarios. Firstly, it displays all the information related to the virtual system interaction interface containing partial differential equations on the web screen by using the SLOG plug-in; secondly, it uses the python2015 development environment to write programs to display the information related to the virtual reality system interaction containing partial differential equations on the web interface. Finally, the compiled program is simulated numerically by NIRE simulation software and its causes are analyzed. Finally, it is concluded that the program can accurately reflect the real-world partial problem.

Virtual reality media interaction method

Virtual reality technology is a computer-based interactive design system that simulates a virtual world through one or more real environments and then displays it on the screen. When people enter this space they will have many ideas: firstly, they need to connect their sensory organs with digital devices; secondly, they need to express the human visual process with animation (e.g. video); and finally, they use computer technology to realize interactive design and other functions to express virtual reality, which is virtual reality technology. Virtual reality is a new computer system that allows people to use the software without using any equipment, and it is not affected by environmental conditions as well as other human factors thus producing errors or even stopping the working state; and by combining it with traditional design methods to achieve interactive computing, parallel operation and other features can be a good solution to this problem.

Web-based Virtual Reality Media Interaction Design Validation

In this example, our main research object is the virtual reality media interaction design. The simulation system includes two parts: the simulation platform and the virtual world. The simulation environment is a fixed value; while the real life is a three-dimensional space built from several real objects, there is a certain degree of difference between the model and the real object: the model is composed of three different materials, thickness and shape of the same and independent of each other. In this process, we can intuitively feel that each object in the real world is presented exactly the same but with differences. By calculating the 3D model, we can visualize the real image of each object in the virtual reality media and choose the appropriate parameters according to the actual situation.

Conclusion

The implementation of virtual reality technology is a novelty in the field of computing, which brings many conveniences to human beings. In this paper, we study the application of partial differential equation numerical computation in digital media interactive design. Firstly, the concept of virtual simulation and its theoretical basis are introduced; secondly, the results are analyzed and summarized for the simulation software, and the problems and solutions are pointed out; finally, the author proposes his own solutions based on the conclusions obtained. Partial differential equations can be used to simulate the real-world program operation and improve it so that it can achieve the expected effect and can be applied in the actual engineering projects. The future trend of computer development is to make digitalization, intelligence and image processing a new field; the emergence of virtual reality technology makes it have a broad prospect in engineering applications.

Figure 1

WPF module structure
WPF module structure

Figure 2

Helmet display-based virtual realization system
Helmet display-based virtual realization system

Figure 3

Server module
Server module

Figure 4

The principle of virtual multimedia interactive system
The principle of virtual multimedia interactive system

Figure 5

Piranha processor architecture
Piranha processor architecture

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