Quantifying diffusion for an ultrasonic wire bonding process by applying the theory of material forces

Ultrasonic wire bonding is a method applied in electronic packaging to fabricate interconnections between two devices at ambient temperature. In order to investigate the material diffusion during this process, the occurring thermal and mechanical mechanisms at and around the interface of the formed bond were studied by means of coupled thermo-mechanical FE simulations. Within the framework of material forces the local jump of the Eshelby tensor was compared with the thickness of the formed intermetallic phase for various bonding parameters. This allows us to predict an effective diffusion constant which takes temperature and mechanical driving forces into account. After this relation has been established a subsequent objective of our investigations is to optimize the growth of the Au8Al3 intermetallic phase in terms of bonding parameters. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A thermomechanical analysis of interfaces between the mating parts in ultrasonic wire bonding

Ultrasonic welding (USW) is an alternative solution for the bonding process especially in automotive industry. Ultrasonic welding of metals is a joining technique as a combination of applying pressure and frictional vibrations within the range of ultrasonic frequencies. In automotive industry, ultrasonic welding is often used for wired connections. As an alternative for crimping technology of multi-strand aluminum cables in wire bonding, ultrasonic welding is used. This work presents a thermomechanical analysis of the interface between two mating parts in USW. For this reason, the temperature distribution at bonding locations inside a wire bundle due to frictional vibrations and pressure is investigated using the finite element method (FEM). The obvious difference in microsections from different welding samples, which originates from different local temperature rises, was the motivation for this study to further investigate the thermomechanical aspects of the USW by use of finite element simulations. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite element analysis of effects of ultrasonic welding parameters on the temperature distribution in wire bonding

Wire bonding is an essential process in the automotive industry. Multi-strand flexible aluminium cables are used for connection of different electronic components and electrical centres in cars. As an alternative for crimping technology in wire bonding, ultrasonic welding (USW) is applied, which is a rapid manufacturing process used to create solid joints between mating materials at low energy consumption compared to the known welding processes, such as oxy-fuel welding and arc welding. An ultrasonic welding machine consists of different parts, such as pneumatic cylinder, piezoelectric converter, booster, welding sonotrode and anvil. Despite of the simplicity of the USW process, choosing the right machine and process parameters, like pressure of the pneumatic cylinder, welding time as well as vibration amplitude of the piezo-converter, is a tricky and complicated task for obtaining an adequate bond. Experimental investigations done in this area are extremely time-consuming and require a lot of effort. Therefore, some new approaches must be developed to understand the process in more detail. The present study focuses on the influence of the ultrasonic welding parameters, such as sonotrode pressure and vibration amplitude, on the temperature distribution at interfaces of two mating pieces in wire bonding [1,2]. Investigations are done by means of FEM simulations as well as by experiments. The results are then extended to thermo-mechanical analysis of multi-strand models. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive estimation of an ergodic diffusion process based on sampled data

We consider adaptive maximum likelihood type estimation of both drift and diffusion coefficient parameters for an ergodic diffusion process based on discrete observations. Two kinds of adaptive maximum likelihood type estimators are proposed and asymptotic properties of the adaptive estimators, including convergence of moments, are obtained.     

An extended thermodynamic description of mass diffusion in elastic solids

We present here a non-conventional model for a themoelastic body, based on the Extended Irreversible Thermodynamics, of physical processes in which mass diffusion occurs. Then we linearize the field equations around a constant state in order to obtain the dispersion relation pointing out the interaction between diffusive field and thermal field.     

An extended thermodynamic description of mass diffusion in elastic solids

We present here a non-conventional model for a themoelastic body, based on the Extended Irreversible Thermodynamics, of physical processes in which mass diffusion occurs. Then we linearize the field equations around a constant state in order to obtain the dispersion relation pointing out the interaction between diffusive field and thermal field.     

Non parametric estimation of the diffusion coefficient of a diffusion process

This paper deals with the nonparametric estimation of the diffusion coefficient of a diffusion process in the case when it depends only on time. the estimator is based on the regularization of the quadratic mean through multiscale analysis. We then define estimators of the second derivative of the diffusion coefficient, in order to obtain optimal bandwith and to perform a test of linearity. The results we obtain are quite similar to those of the estimation of the density function of an i.i.d. sequence     

Analysis of the time fractional nonlinear diffusion equation from diffusion process

Under investigation in this paper is a time fractional nonlinear diffusion equation which can be utilized to express various diffusion processes. The symmetry of this considered equation has been obtained via fractional Lie group approach with the sense of Riemann-Liouville (R-L) fractional derivative. Based on the symmetry, this equation can be changed into an ordinary differential equation of fractional order. Moreover, some new invariant solutions of this considered equation are found. Lastly, utilising the Noether theorem and the general form of Noether type theorem, the conservation laws are yielded to the time fractional nonlinear diffusion equation, respectively. Our discovery that there are no conservation laws under the general form of Noether type theorem case. This result tells us the symmetry of this equation is not variational symmetry of the considered functional. These rich results can give us more information to interpret this equation.     

On the uniqueness and reconstruction for an inverse problem of the fractional diffusion process

Consider an inverse problem for the time-fractional diffusion equation in one dimensional spatial space. The aim is to determine the initial status and heat flux on the boundary simultaneously from heat measurement data given on the other boundary. Using the Laplace transform and the unique extension technique, the uniqueness for this inverse problem is proven. Then we construct a regularizing scheme for the reconstruction of boundary flux for known initial status. The convergence rate of the regularizing solution is established under some a priori information about the exact solution. Moreover, the initial distribution can also be recovered approximately from our regularizing scheme. Finally we present some numerical examples, which show the validity of the proposed reconstruction scheme.     

The first boundary exit point of a diffusion process with small diffusion

A diffusion process with a drift b(x) and small diffusion sooner or later reaches the boundary of a region containing a stable position of equilibrium of the dynamical system x(t)=b(x_t). The first exit point belongs to a small part of the boundary (see [1, 2]). An estimate of the order of the size of this part is given. Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 411–420, September, 1977.     

经济环境下带扩散扰动古典风险过程的破产概率

对于经济环境下带扩散扰动古典风险过程的重要性质进行了讨论,给出了有限时间的破产概率的上界估计.     

An explicit description of Kallenberg's lattice type point process

The asymptotic behaviour of solutions of nonlinear VOLTERRA integral equations is studied in a real BANACH spaces. The nonlinear operator is assumed to satisfy some accretivity-type conditions.     

ENTROPY PRODUCTION RATE OF THE MINIMAL DIFFUSION PROCESS

The entropy production rate of stationary minimal diffusion processes with smooth coefficients is calculated. As a byproduct, the continuity of paths of the minimal diffusion processes is discussed, and that the point at infinity is absorbing is proved.     

Singular stochastic control of a singular diffusion process

This paper studies the monotone follower problem for a one-dimensional singular diffusion process. The dynamic programming principle is established. It is shown that the value function is continuous and satisfies the Hamilton-Jacobi-Bellman equation in the viscosity sense     

Emptiness formation probability of XX-chain in diffusion process

We study the distribution of emptiness formation probability of XX-model in the diffusion process. There exits a Gaussian decay as well as an exponential decay. The Gaussian decay is caused by the existence of zero point in the Fermi distribution function. The correlation length for each point of scaling factor varies up to the initial condition, monotonically or non-monotonically.     

The boundary value problem and the Jensen inequality for an entropy functional of a Markov diffusion process

The paper introduces the recent results related to an entropy functional on trajectories of a controlled diffusion process, expressed through an additive functional of the diffusion process, with a Lagrangian, determined by the parameters of a controlled stochastic equation. These results include a minimum condition for the entropy functional and the functional's Jensen inequality, which both are useful for the solution of important mathematical and applied problems.     

Competitive diffusion process of repurchased products in knowledgeable manufacturing

This paper presents a diffusion model to explain the competitive diffusion of the repurchased products in knowledgeable manufacturing. The acute market competition accelerates the products’ improvement, which requires that the manufacturing enterprises be highly capable of rapid reaction by means of knowledgeable manufacturing. To forecast the diffusion behavior effectively enables the realization of knowledgeable manufacturing system (KMS) which targets T (time), Q (quality), C (cost), S (service), and E (environment). Various diffusion models have emerged since Bass model was firstly proposed in 1969. A nonlinear model of the repurchased competitive products is proposed on the basis of the product diffusion analysis. By taking the frequently purchased products as example, the stability of the model is examined in light of the qualitative theory of differential equations and proved by the approximate linearization method. As the qualitative analysis reveals, between the two frequently purchased products competing in the same market, one undoubtedly occupies a fixed market share while the other may finally be eliminated from the market. A special case of the problem is that both products are one-time-purchased. With the corresponding model given, the qualitative analysis shows that either of the products occupies a market share, the size of which is determined by the product’s competitive strength and the new product’s time-to-market. A system dynamics model is then established and simulated by vensim. The result is consistent with that of the qualitative analysis.