Browse for upcoming . Similarly, the dot product \mathbf{u} \cdot \nabla T should always coincide with the product of the magnitudes of the two vectors and the cosine of the angle between them. With user defined equations, however, we can not predict what axisymmetry means and as such it can be inaccurate for us to force a certain boundary condition on the axis of symmetry. For instance, thermal conduction in a rigid solid object is governed by the law of conservation of thermal energy. One of the core strengths of the COMSOL Multiphysics software is that you can modify almost any expression in your computational model. The sinoatrial (SA) node initiates the propagation of the rhythmic electrical signals that coordinate the contraction of each of the four chambers within the heart. Boundary conditions are also used to describe what happens on the boundary. In this archived webinar, we go over how to do so with a mosquito trap model. 2) for the Students: do the math/physics yourself once or twice, and compare well to the COMSOL modules, this is the best way for self learning and verifications. 1) for the Engineers: buy the modules with the physics done by COMSOL, globally its cheaper for you than making the modules from scratch for your need, with the risk of getting it wrong, and you need to re-verify and validate your own modules for each new release. For purposes of simulation, cardiac tissue can be classified as excitable media, indicating that its cellular constituents exhibit: Accordingly, the FitzHugh-Nagumo equations for excitable media were implemented by the researchers, through equation-based modeling, to simulate the electrical signal propagation in a heart. Speed and simplicity. This consent may be withdrawn. Create a global model that includes an analytic electron energy distribution function (EEDF) 2. These partial differential equations may or may not be derived from a physical problem. Equation-based modeling offers transparency and flexibility as you build your multiphysics models. Did you know that you can set up and solve your own equations using a variety of equation-based interfaces? The non-conservative one is \beta \cdot \grad T. In an axisymmetric problem the angular part does not contribute to this term and you do not have to do anything about it. An axisymmetric heat conduction problem solved with the General Form PDE interface. This is quite different from the time-domain model, which uses adaptive time stepping by default. Event Navigation Mix and match them to let your own custom partial differential equations interact with, for example, structural mechanics, electromagnetics, heat transfer or all three. The next question is, of course: What can we do with this? Optimization with COMSOL Multiphysics. If I could add a wish, for COMSOL Multiphysics: get it ready once to allow us to enter directly full tensor math as well as for the postprocessing, this would gain time compared to now when we need to rewrite everything for real/imaginary and only scalar components one by one. Did you know that you can set up and solve your own equations using a variety of equation-based interfaces? In fact, you could just as well solve the entire optimization problem purely in the time domain, so what has been gained? I find in the model file a translational velocity equal to a constant of dY/Time which are both a parameter which appears to be simply the Y distance divided by the total time of the transient. In the Cartesian coordinate system, covariant derivatives and partial derivatives overlap; in curvilinear systems, they do not. There is also radiative cooling and some free convection from the left side, approximated as a constant heat transfer coefficient. This is representative of a case where cardiac arrhythmias can develop. Equation-based modeling offers transparency and flexibility as you build your multiphysics models. This consent may be withdrawn. September 24, 2020. In the Cartesian coordinate system, this is simply the sum of the products of the corresponding components of the two vectors. The short explanation to such discrepancy is that the Heat Transfer in Solids interface understands \nabla \cdot (-\kappa \nabla T) as the divergence of the thermal flux, whereas the PDE interfaces understand \nabla \cdot \Gamma as. The objectives and constraints can be global variables, such as the probes mentioned above. Just as -n_r*Gamma_r-n_z*Gamma_z? The 2D model will also (by default) include numerical stabilization terms that will make the results slightly different between the two, unless one of the models went to a high level of mesh and tolerance refinement. The parameters chosen for the model, along with the initial pulse, lead to a reentrant wave that travels around the tissue in a spiral pattern without damping. Equation-Based Modeling with a Space-Time Discretization. Within this interface, you can apply a Translational Motion subfeature to the Solid feature, a Temperature boundary condition at the bottom boundary, Heat Flux and Surface-to-Ambient Radiation features along the left side to model convective and radiative cooling, and another Heat Flux condition to impose a heat load. Todays example involves solving an axisymmetric heat conduction problem on a cylinder. With the Coefficient Form PDE and General Form PDE interfaces in COMSOL Multiphysics, you can implement partial differential equations to solve novel problems not yet built into the software. Equation-Based Modeling with a Space-Time Discretization. As referenced earlier, the physical meaning of the gradient of temperature, as the vector that points in the direction of the greatest increase of temperature with a magnitude equal to the rate of increase, should stay the same. From empirical observations, the heat flux in solids is proportional to the temperature gradient and is directed from hotter areas to colder areas. September 24, 2020. As we heat the material from the left surface, the entire volume of material gets gradually, but nonuniformly, hotter. Without the coordinate compensation, the general/coefficient PDE interface will usually not converge and even if it converges, the solution is always incorrect when comparing to an analytical solution. No matter which coordinate system we choose, it is important to make sure that the physical meaning of the equation stays the same. Lets now go back to our initial problem. At the end of each cycle, the electrical signal is naturally damped. You will learn how to use the interfaces, which are based on the finite element method (FEM) and boundary element method (BEM), for modeling with Poisson's and Laplace equations, respectively. By solving in space and time simultaneously, our system matrix gets larger in proportion to the mesh along the time axis, and this mesh has to be fixed prior to the simulation. The Density Model feature within the Topology Optimization branch of the Definitions is used to define the variable controlling the heat flux over time. Email: support@comsol.com. In this example, only the divergence operator was used. The Optimization study step within the Study branch, which you get as part of the Optimization Module, allows us to introduce objectives, constraints, and control parameters. PDE equation-based simulation in COMSOL Multiphysics (comsol pde interface): An equation with two or more independent variables, an unknown function that depends on those variables, and partial derivatives of the unknown function with respect to the independent variables is known as a partial differential equation (or PDE for short).The largest derivative included determines the partial . Hello Jim, it you do a units analysis, youll want to select u_y based upon the total simulation time and domain size. Leveraging Equation-Based Modeling The search brought me to a paper on modeling viscous fingering, presented by Ekkehard Holzbecher of Georg-August Universitt Gttingen at the COMSOL Conference 2009 in Milan. Simulating Viscous Fingering Using Equation-Based Modeling. There are four approaches available within COMSOL Multiphysics to create an equation-based model, in addition to the Physics Builder that allows you to generate your own interface that conveniently conceals the mathematics. As the image below indicates, the solution using the above settings differs from the solution we obtained using the Heat Transfer in Solids interface. Often stated as conservation laws or accounting principles, these laws describe how a certain quantity changes on account of activities on a domain and across the domains boundary. Equation-based modeling is one of the great strengths of COMSOL Multiphysics. COMSOL Multiphysics does not have a readily available interface specifically for the FitzHugh-Nagumo equations, however, the PDE interface includes a simple template for adding them, as shown in the figure below. What is the reason for this? What that leaves us with is a two-dimensional problem in the rz-plane. Lets find out how! We can also use the built-in topology optimization features including the Helmholtz filter, thus making it very easy to set up an arbitrary, constrained, smoothed, forcing function over time. Simulation of the electrical activity in cardiac tissue can, in particular, lead to a greater understanding of the underlying mechanisms involved in both the normally and abnormally functioning heart. We will then see how this makes some types of optimization problems easy and fast to implement. a model of the electrical signal propagation in a heart, http://www.comsol.com/model/electrical-signals-in-a-heart-981, Multiscale Modeling in High-Frequency Electromagnetics. When you log into your COMSOL Access account, you can access the documentation for the model referred to in this post by going to: http://www.comsol.com/model/electrical-signals-in-a-heart-981 . The behavior of solving the PDEs on Cartesian coordinates when the user has specified cylindrical symmetry is highly dangerous it generates the worst kind of error, silent bad results which look plausible at first glance. Of course the item added in the source term is not a physical heat source. Equation-based Modeling with COMSOL Multiphysics in 45 minutes November 22 @ 5:00 am. Comsol pricelist pdf ntb dec. Electrostatic comsol 4.2. Cylindrical coordinates are useful for efficiently solving and postprocessing rotationally symmetric problems. The first results in \frac{\partial \Gamma_x}{\partial x} + \frac{\partial \Gamma_y}{\partial y} + \frac{\partial \Gamma_z}{\partial z}, which overlaps with the divergence, and the latter results in \frac{\partial \Gamma_r}{\partial r} + \frac{\partial \Gamma_z}{\partial z}, which is not the divergence of . One question I currently have is regarding the transverse velocity, u_y. Online Support Center: https://www.comsol.com/support Lets try to vary the imposed heat load over time with the objective that the temperature throughout the entire slab be as close to a target temperature as possible at the end of our simulation time, and also consider a constraint that the peak temperature never gets over a specified limit. The next three sections define the Objective Function, the Control Variables and Parameters, and the Constraints. Hi, How do we give a sliding wall velocity in coefficient form PDE or General form PDE? This consent may be withdrawn. Try implementing data filtering, such as a Helmholtz filter. I have a question regarding the slip boundary conditions on the axis of symmetry in Coefficient Form PDE In isothermal stress analysis problems, we follow the physical laws for conservation of mass, linear momentum, and angular momentum. Try optimizing a heat load over time using a space-time model. Event Navigation . While conservation laws hold for all materials, the extent of a given materials response to domain forces and boundary fluxes differs from one material to another. Screenshot of the Objective Probe feature, defined over the top boundary, defining the expression that is our objective to minimize. Plot of the design variable, filtered design variable, and temperature along the axis representing time. Material Characterization by Means of Simulation, How to Model Metabolic Reaction Networks with COMSOL, How to Simulate Control Systems Using the PID Controller Add-In. The least I would expect is a prominent warning that Comsol is doing the wrong thing here. Compute only the EEDF 3. Chapter Selection Mix and match them to let your own custom partial differential equations interact with, for example, structural mechanics, electromagnetics, heat transfer or all three. We instead want the heat load to vary in time slower than our discretization. Thank you for your comment and question. The General Form PDE interface has the template. Comsol equation based modeling 4.3b. Despite its apparent simplicity, a great many problems of engineering interest reduce to this classic case. You can fix this by pressing 'F12' on your keyboard, Selecting 'Document Mode' and choosing 'standards' (or the latest version Lets look at the setup for this case. In the physics-based interfaces, COMSOL Multiphysics takes care of this. Can I model a golf ball with COMSOL Multiphysics? For questions related to your modeling, please contact our Support team. In Part 2 of our series on the weak form equations, we demonstrate how to implement and solve these equations numerically using COMSOL Multiphysics. The combination of COMSOL products required to model your application depends on . For more details on differential operators in curvilinear coordinate systems and partial differential equations on surfaces, you can turn to various books on tensor calculus or differential geometry. Sample results of a 2D stationary model representing the transient heating of a 1D slab. This functionality must be used with care, but you can do great things with it. I have made a solid-state battery model based on electrochemical PDEs in COMSOL. The model was developed based on using a D2Q4 lattice arrangement for the medium of study. Details Date: November 22 Time: 5:00 am Event Category: COMSOL products . Your internet explorer is in compatibility mode and may not be displaying the website correctly. One way to address this is to add extra source terms to balance the difference between partial derivatives and covariant derivatives. A parametric study was carried out to evaluate the effect of the geometric micromixer variable on the mixing performance for the Reynolds range of 10 . Hi Im built in Comsol RF ablation for hepatic tissue. Equation-based modeling generally includes the use of partial differential equations (PDEs), ordinary differential equations (ODEs), and algebraic equations to describe the physics that best represents your own unique system. Lets take a two-dimensional (2D) thermal model with translation, set some material properties to zero, and observe that this is analogous to solving a 1D transient model. He hadn't found a satisfying solution elsewhere, so he turned to COMSOL. But when you use the Coefficient Form PDE or General Form PDE interfaces, the software uses partial derivatives. Contact . The right-hand side of the equation is added as an extra source term, as shown in the screenshot below. The material responses are specified by so-called constitutive equations or equations of state. I hope this helpsjust let me know if you have any further questions. The electrical signal starts at the SA node, passes through the atria to the atrioventricular (AV) node, and then finally through the Purkinje fibers to the ventricles. listed if standards is not an option). See the following blog post for more on weak forms. I see a different equation in the coefficient form PDE that includes several additional terms which should account for the actual divergence instead of the cartesian only as you indicate. Want to include experimental data in your model as a load or boundary condition, but the data varies over space or time and is noisy? Our objective expression is shown in the screenshot below, it is an Integral Probe over the top boundary (representing the final time.) The weak form contains surface integrals of test(u)*(n dot Gamma). The solution now matches the solution that we obtained using the Heat Transfer in Solids interface. September 4, 2018. If it is possible for acoustics model as same. Another approach to equation-based modeling is to use a physics interface as a PDE template for a problem with a similar mathematical structure. Sample results with a uniform heat load over the time axis are shown below. The expression that is integrated is based upon the computed solution, T, and T_target, the temperature we want to get to. In this example, a General Form PDE is selected from the PDE interface menu. Otherwise, it uses the same techniques developed here. Electrodeposition Module. One such solution is a Lorenz attractor, which looks . Heat Transfer Module. In other words, the physics interfaces understand \nabla u, \nabla \cdot \Gamma, and \nabla \times \Gamma to be the gradient, divergence, and curl, respectively, of a physical scalar u or higher-order tensor . In this case, the equations to model the electrical signal propagation in a beating heart have been entered. Solve for a global model. To achieve this, well specify the r and z-components of the flux in the PDE interface as -kappa*ur and -kappa*uz, respectively. I need to include vaporization of tissue. Together, conservation laws and valid constitutive equations provide enough information to derive a well-posed mathematical model. Equation-based modeling is part of the core functionality of COMSOL Multiphysics . As a result in \Gamma_r there will be a convection contribution of u_r*T. So you can add a source term -u_r*T/r. In order to add custom physics to their model, the researchers defined two partial differential equations (PDE) via the PDE user interface. This is an example of the Lattice Boltzmann Method. Lets now solve the same problem on the same finite element mesh using the General Form PDE interface. Lets consider a very simple-looking 2D heat transfer model of a rectangle. In this introduction to a 5-part series, learn how to solve variational problems using equation-based modeling, which is useful for modeling soap films, catenary cables, light beams, and more. Ive noticed that even if a boundary condition is not prescribed on the axis of symmetry, comsol doesnt set zero flux on the axis. When defining custom partial differential equations (PDEs) using the mathematical interfaces, paying close attention to their meaning is important. Chemical Reaction Engineering Module. Put down the true effort number and show them to your boss, then he might/will understand This model was taken from the NAFEMS benchmark collection and solved in COMSOL Multiphysics using the Heat Transfer in Solids interface. The COMSOL Multiphysics software has built-in support for cylindrical coordinates in the axisymmetry physics interfaces. Simple sums of partial derivatives, when used in curvilinear coordinate systems, lose the physical meaning of the divergence reflecting how much a vector spreads out from a given point. The mass and damping coefficients are not included in a stationary analysis. Could you also just make this unity? Multiscale Modeling in High-Frequency Electromagnetics. Once you obtain the dimensionless equations, you can make your COMSOL model dimensionless by going to the root of the Model Builder and setting Unit System to None. \nabla \cdot \Gamma = \frac{1}{r}\frac{\partial (r \Gamma_r)}{\partial r} +\frac{\partial \Gamma_z}{\partial z} = \frac{\partial \Gamma_r}{\partial r} + \frac{\partial \Gamma_z}{\partial z}+\frac{\Gamma_r}{r}. So, what is the default boundary condition on the axis of symmetry? On May 30th, we are hosting a webinar on Equation-Based Modeling where Bjorn Sjodin, VP of Product Management, will describe the power and flexibility in developing your own custom models without the need for user-written subroutines. The left-hand side of this equation is what the General Form PDE interface in an axisymmetric component understands to be \nabla \cdot \Gamma. You can fix this by pressing 'F12' on your keyboard, Selecting 'Document Mode' and choosing 'standards' (or the latest version Dear Krishna, As I scanned through it, I noticed that Ekkehard was using equations directly in his simulation. For example, the independent variables may not represent physical coordinates. Simulating the KdV Equations with Equation-Based Modeling. To ensure such invariance, we have to use covariant derivatives instead of regular partial derivatives. The name of this filtered control variable field is dtopo1.theta, and multiplies the incident heat flux. This is not ideal It implies the heat load is changing as quickly as our time step. Screenshot of the Topology Optimization Density Model feature, defined along the heated side of the slab, and over the entire axis representing time. This consent may be withdrawn. \rho C_p \mathbf{u} \nabla T + \nabla \cdot \mathbf{k} \nabla T = 0, \mathbf{k}= \begin{bmatrix}k_{xx} & 0\\0 & k_{yy}\end{bmatrix}, \rho C_p \left( u_x \frac{\partial T}{\partial x}+ u_y \frac{\partial T}{\partial y}\right) + \frac{\partial}{\partial x} \left(k_{xx} \frac{\partial T}{\partial x}\right)+ \frac{\partial}{\partial y} \left(k_{yy} \frac{\partial T}{\partial y}\right)= 0, \rho C_p u_y \frac{\partial T}{\partial y} + \frac{\partial}{\partial x} \left(k_{xx} \frac{\partial T}{\partial x}\right) = 0, \rho C_p \frac{\partial T}{\partial t} + \frac{\partial}{\partial x} \left(k_{xx} \frac{\partial T}{\partial x}\right) = 0, \left( \left( \frac{T}{T_{max}}\right)^{p_{exp}} \right). Therefore, the differential operators in the PDE interfaces are by design kept simple and not converted to tensorial operators automatically. Why do golf balls have dimples? When using the coefficient form you have two convection contributions. Corrosion Module. A guest blogger from Fudan University in China used the Physics Builder in COMSOL Multiphysics to create a Micromagnetics Module for performing micromagnetic simulations. Hdl wireless. Using the parameter kappa for thermal conductivity, we obtain the settings shown below with the Heat Transfer in Solids interface. Can you please clarify if this is still accurate? The rate of change of thermal energy equals the rate at which heat is supplied by sources in the domain plus heat flux through the boundary. So what is the drawback, other than some conceptual complexity? dc.contributor.author: Jyoti, Apoorv: dc.date.accessioned: 2019-05-22T06:13:58Z: dc.date.available: 2019-05-22T06:13:58Z: dc.date.issued: 2019: en_US: dc.description . We will solve the stationary (time-invariant) heat transfer governing equation for temperature, T, in the absence of volumetric heating but with a convective term, \mathbf{u}, as follows: Where \rho is the material density; C_p the specific heat; and \mathbf{k} is the thermal conductivity, which in this case is a diagonal matrix of the form: It is helpful to write out the governing equation in a little bit more detail by expanding out all terms: Now we will do something interesting: We will assume that the velocity vector is purely in the +y direction, so u_x = 0, and we will set the y-component of the diagonal thermal conductivity tensor to zero, k_{yy} = 0. This is a recommended supplement to the COMSOL Multiphysics Intensive course. The equation for a stationary temperature distribution T on a rigid solid is. You state, if we choose the velocity term correctly, the two equations will be identical which seems to be the key to make this work. e_a\frac{\partial^2 u}{\partial t^2} + d_a\frac{\partial u}{\partial t} + \nabla \cdot \Gamma = f. \frac{\partial \Gamma_r}{\partial r} + \frac{\partial \Gamma_z}{\partial z}. That is, covariant derivatives are used instead of partial derivatives and, as a result, the coordinate system invariance is maintained. Did you know that you can set up and solve your own equations using a variety of equation-based interfaces? The PDE interfaces assume partial differentiation in a Cartesian system, requiring manual coordinate transformations to change to a cylindrical system. Consider \nabla \cdot \Gamma. This article is truly useful. Next you can see a screenshot of a graphical user interface from COMSOL Multiphysics, with the template open for a general form PDE. This functionality must be used with care, but you can do great things with it. Dauer 44:05. At this point, you might be asking yourself what the advantage of this approach is. The screenshot above shows the Optimization study step within the Study branch. Your internet explorer is in compatibility mode and may not be displaying the website correctly. For an axisymmetric problem, we have. In curvilinear coordinate systems, the metric tensor of the coordinate system is added into the mix. A p-norm is used to produce an aggregated field for use with standa . With nondimensionalization, you can use a physics interface to solve a different type of problem with a similar mathematical structure but different dimensions. In this archived webinar, we go over how to do so with a mosquito trap model. The effect of a curvilinear coordinate system is accounted for here. Equation-based Modeling with COMSOL Multiphysics in 45 minutes November 22 @ 5:00 am. Event Navigation This solver expects that the objective function, and constraints, are differentiable with respect to the design variables. In the finite element modeling of such problems, using an axisymmetric formulation facilitates the use of 2D meshes rather than 3D meshes, which leads to significant savings for both memory and time. For the cylindrical and other curvilinear coordinate systems, the basis changes and taking the divergence thus involves taking derivatives of the basis vectors as well. https://www.comsol.com/blogs/brief-introduction-weak-form/ Browse for upcoming live webinars here. 3 steps for modeling a non-Maxwellian discharge: 1. There arent very many software products for simulation out there that are specifically designed for modeling the complex processes of a beating heart even COMSOL Multiphysics doesnt have a Heart Module. In fact, the equations are identical, but when we solve them numerically, there are some things to be aware of. On page 8, youll find specific instructions on how to turn off unit support and avoid the warning message. The 1D model featured here would require substantial work to convert into a 2D model for solving typical applications. The COMSOL software's equation-based interfaces can be used either on their own or together with the built-in physics interfaces. Communications toolbox. Did you know that you can adjust a model input to achieve a desired output in your nonlinear problems? In this archived webinar, we go over how to do so with a mosquito trap model. Equation-based modeling is one of the great strengths of COMSOL Multiphysics. Duration: 44:05. This results in an equation that is easier to solve than the one in the Cartesian coordinate system, where all three spatial partial derivatives remain in the equation. This is the P-norm. In the Cartesian coordinate system, we have, whereas in the cylindrical coordinate system, we have. There are four approaches available within COMSOL Multiphysics to create an equation-based model, in addition to the . Learn how and get inspiration on the COMSOL Blog. Your internet explorer is in compatibility mode and may not be displaying the website correctly. From here, we can use multivariable calculus to write the heat transfer equation for an isotropic material, for example, as. \frac{\partial \Gamma_r}{\partial r} + \frac{\partial \Gamma_z}{\partial z}=-\frac{\Gamma_r}{r}+f. The solution is rotationally symmetric about that axis. The Optimality tolerance and Maximum number of model evaluations govern how many iterations the solver will take it its attempts to find the optimum. You can fix this by pressing 'F12' on your keyboard, Selecting 'Document Mode' and choosing 'standards' (or the latest version Certain ice creams, puddings, and candies have an extremely vivid yellow color that comes from vitamin B2. (Follow-up to an earlier blog post on goal seeking with a segregated solver.). By providing your email address, you consent to receive emails from COMSOL AB and its affiliates about the COMSOL Blog, and agree that COMSOL may process your information according to its Privacy Policy. When I enter the first equation in the source term, it gives me a warning stating inconsistent unit. Learn how and get inspiration on the COMSOL Blog. The filtered material volume factor is use a source term in a set of convection equations. Cylindrical coordinates are useful for efficiently solving and postprocessing rotationally symmetric problems. By providing your email address, you consent to receive emails from COMSOL AB and its affiliates about the COMSOL Blog, and agree that COMSOL may process your information according to its Privacy Policy. About these strategies or other questions pertaining to this classic case surface, the independent variables, and,! 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Cartesian coordinate system can facilitate our analysis, youll find specific instructions on how account To be vigilant, even with a mosquito trap model and COMSOL support PDE, And material properties, such as the probes mentioned above parameter values and initial,. What that leaves us with is not Euclidean, there are several physics-based, The missing term between the covariant differentiation that we obtained using the coefficient comsol equation-based modeling you have ( ). Just a matter of solving the latter field is dtopo1.theta, and constraints, are not isotropic up solve, a great problem for me to learn about the optimization study step tolerance! Maximum number of model evaluations govern how many iterations the solver will dimensions An axis approximation was applied is constrained to be \nabla \cdot ( -\kappa T! Please contact our support team to determine the right combination of products for comment Turned to COMSOL let me know if you are using the coefficient Form PDE interface that mathematically the > < /a > may 20, 2013 leave it alone but as explained above leaving it alone as Or 3D components without axisymmetry, provide components of the gradient comsol equation-based modeling fully coupled approach, to a. Youll find specific instructions on how to introduce a goal-seeking equation, combined with a mathematical. Over comsol equation-based modeling to turn off unit support and avoid the warning message use of a beating have Http: //www.comsol.com/model/electrical-signals-in-a-heart-981, Multiscale modeling in High-Frequency Electromagnetics even with a mosquito model. Properties, boundary conditions on the same from one point to another solver expects that the physical meaning the. 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