The resulting temperature snapshots are then feed into a structural model, where inertia effect are neglected. Hence, static structural solutions are needed.<\/li><\/ul>\n\n\n\nTo emphasize, capturing the exact temperature or stress time-histories is not mandatory: the figure of merit here will be stress range, i.e. extreme values.<\/p>\n\n\n\n
Now, the regular approach would be to solve in the time domain for the temperature distribution in the structure T(x,t) at a series of instants t1<\/sub>..tn<\/sub>. From this, a series of n structural solutions would be performed, producing the structural displacement U(x,t), and derived quantities such as stress\/strain. Note that, the structural displacements could be estimated \u201con the fly\u201d, as structural displacement at time \u201ct\u201d obviously doesn\u2019t depend on past or future values.<\/p>\n\n\n\nAlternatively, the POD approach would consist in:<\/p>\n\n\n\n
Evaluating the temperature distributions at all time points, i.e. the snapshot matrix ( Compressing those snapshots using SVD method: this produces a set of nodal amplitudes ( Evaluate the individual structural response for each of the thermal POD vectors .<\/li> Combine the individual structural responses to obtain the physical response at all locations and time points.<\/li><\/ol>\n\n\n\nThe expected computational gain is clearly related to the ratio between the number of time points and that of the required POD vectors, and we should expect the solution time reduction to be close to that value (keeping in mind that the SVD decomposition by itself will only require an additional solution time comparable to one additional time point).<\/p>\n\n\n\n\n\n\n\n
Thermal stress induced by thermal shock in a tee-shaped junction<\/h2>\n\n\n\n Let\u2019s apply our nice principle to a typical example that is: a tee-junction. Whenever experiencing sudden temperature change, the inner and outer regions of the pipe will momentarily have different temperature (the diffusion process will limit the through-thickness heat transport), and compressive mechanical stress will develop mostly in the inner region, with a time constant that can easily be estimated by hand. Additionally, in the case of fluid flow being stopped in either the branch or run, each part will also experience different temperatures, and again differential dilatation and mechanical stress will develop, this time at the junction. <\/p>\n\n\n\n
The FEM is shown hereafter, comprising 96 000 elements and 116 160 nodes:<\/p>\n\n\n\nA finite elements model dedicated at capturing the through-thickness bi-metal effect<\/figcaption><\/figure>\n\n\n\nIn this example, we assume a generic, 48×6 tee, made of steel (CTE = 15ppm\/K, density of 7.85, thermal conductivity of 15W\/m\/K, specific thermal capacity of 500J\/g\/K, elastic modulus of 200 GPa). This gives a thermal diffusivity of 3.8mm\u00b2\/s, to be compared with the thickness of 6mm. The time required for heat to travel a distance of 6mm can therefore be estimated to be:
Now, let\u2019s assume a \u201ccold plug\u201d suddenly travels through the run (here: the horizontal part), while the fluid in the branch remains standing. The fluid bulk temperature in the flow would assume a (purposedly) very sharp temperature change:<\/p>\n\n\n\n
Linear decrease of 200K between t=0 and t=1s<\/li> Plateau for 1s (t=1 to t=2)<\/li> Slower, linear return to initial temperature in 3s (t=2 to t=5s)<\/li> Second decrease, with half the initial amplitude, and twice the duration (i.e. 50K\/s) (t=5 to t=7s)<\/li> Return to initial temperature between t=7 and t=10s<\/li><\/ul>\n\n\n\nThe total simulated duration is 100s (but only the first 20s are shown for the sake of clarity).<\/p>\n\n\n\nThe coefficient of heat exchange in the region of fluid flow will obviously be much higher than in the stagnant region. To obtain a high contrast, we select values for the film coefficient value of 1000 and 25 000W\/m\u00b2\/K, respectively. Elsewhere, we assume negligible heat fluxes (i.e. adiabatic boundaries).<\/p>\n\n\n\n
In terms of boundary conditions, we assume that radial dilatation is free, but axial direction is blocked on the upper pipe.<\/p>\n\n\n\nConvection film coefficient distribution (fluid is reamains stagnant in the vertical pipe)<\/figcaption><\/figure>\n\n\n\n<\/p>\n\n\n\n
Now, we can put our method to the test of this highly dynamic thermal event. First of all, we let ANSYS solve for the temperature fields. Following the rules of the trade, we select initial time-step size based on thermal diffusivity and element size (here: 0.2mm radially, so initial time step is a bit less than 0.01s), and then let auto-time stepping algorithm do its job. We end up with a set of 157 results, all defined on 116kDOF, so approximately 18 M nodal temperatures values.<\/p>\n\n\n\n
Temperature distribution at a few key points in time are given:<\/p>\n\n\n\n
At the end of fluid temperature decrease (i.e. at t=1s)<\/li> At the end of the plateau (i.e. at t=2s)<\/li><\/ul>\n\n\n\nTemperature distribution at the end of temperature jump.<\/figcaption><\/figure>\n\n\n\nTemperature distribution at the end of temperature plateau<\/figcaption><\/figure>\n\n\n\nNow, we can apply the SVD procedure to this rather huge volume of data. Let\u2019s import this, using the snapshot matrix into the APDL Math workspace, as a dense matrix, using the command *IMPORT.<\/p>\n\n\n\n
<\/p>\n\n\n\n
<\/p>\n\n\n\n
*DMAT,X,D,IMPORT,RST,myResultsFileName.RTH,1,nbSnapshots<\/code><\/pre>\n\n\n\nTo this, the code answers nicely:<\/p>\n\n\n\n
Import Data From File RUN_THERMAL_FULL.RTH (Format=RST)\n\n IMPORT SOLUTION : Nodal Solution\n DATA SETS : [1,157]<\/code><\/pre>\n\n\n\nWell, I don\u2019t know about you, but reading this rather large file on a small machine (16Gb intel i3) only requires 0.14s. OK, that\u2019s not real processing power but mere I\/O, however this step alone could have been excruciatingly slow, completely ruining the total process efficiency. As it shows, it doesn\u2019t!<\/p>\n\n\n\n
Going one step further, we can now do the actual compression. We specify a tolerance of 0.001, to be compared with the defaults (see previous post for explanations).<\/p>\n\n\n\n
Allocate a [157][157] Dense Matrix : VCONJ\n (TOLERANCE = 0.001)\n MATRIX SIZE AFTER COMPRESSION : 7<\/code><\/pre>\n\n\n\nAnd this is where the beauty of this appears. Firstly, the POD basis only includes 7 vectors, instead of 157, so that the time required for structural analysis can be decreased by a factor of about 20. And, secondly, the required CPU for SVD reduction is 1.73s. I don\u2019t know about you, but I have had difficulty believing this, so I repeated the operation with various settings, and of course the results change (I\/O can play a role) but the basics are demonstrated: SVD reduction is numerically both very fast and efficient. As a side note, performing the same operation with Matlab requires about 4.1s.<\/p>\n\n\n\n
As a matter of fact, I did a sensitivity study to check the convergence rate, as shown in the table below:<\/p>\n\n\n\nTrial<\/strong><\/td>Threshold<\/strong> [-<\/strong>]<\/td>CPU<\/strong> [s]<\/strong><\/td>POD vectors<\/strong><\/td>Error<\/strong> L2 \/ LINF<\/strong><\/td><\/tr>1<\/strong><\/td>10-1<\/sup><\/td>2.3<\/td> 2<\/td> 1.3 104<\/sup><\/td>1.2 103<\/sup><\/td><\/tr>2<\/strong><\/td>10-2<\/sup><\/td>2.2<\/td> 4<\/td> 1.5 103<\/sup><\/td>3.7 102<\/sup><\/td><\/tr>3<\/strong><\/td>10-3<\/sup><\/td>2.4<\/td> 7<\/td> 1.3 102<\/sup><\/td>2.2 101<\/sup><\/td><\/tr>4<\/strong><\/td>10-4<\/sup><\/td>2.3<\/td> 11<\/td> 7.4 100<\/sup><\/td>1.6 100<\/sup><\/td><\/tr><\/tbody><\/table>The nothing-less-than-astounding convergence rate of SVD applied to thermal shock<\/figcaption><\/figure>\n\n\n\nAs expected the reconstruction error decreases with a rate similar to the threshold value. Surprisingly, the reconstruction error (estimated using the *NRM function, applied to the reconstruction error matrix) gives L2 values that are larger<\/em> than the Linf<\/sub> norm. The documentation is not really helpful in order to understand this behavior.<\/p>\n\n\n\nFor the time being, I assume that a threshold of 10-3<\/sup> is largely acceptable for engineering purpose. What do these magnificents 7 POD vectors, worth 157 ordinary results look like? Here are they (I removed one quadrant for better readability, since obviously, the results are symmetric with respect to the X=0 and Y=0 plans).<\/p>\n\n\n\nPOD vectors 1 in-phase temperature change in branch and run, while vector 2 and 3 capture the through-thickness temperature gradients. Vector 4 corresponds to heat exchange between branch and run (out-of-phase temperature variations). Modes 5,6, and 7 are all local modes at the junction.<\/p>\n\n\n\nThermal POD vectors<\/figcaption><\/figure>\n\n\n\nNow, we can also estimate the reconstruction error induced by the POD method. Here, using a threshold value of 10-3<\/sup> (i.e. 7 POD vectors), the error has been mapped on the structure at t=1 and 2s, to get an estimate of the error amplitude and localization. Clearly, the reconstruction error is really small (of the order of 10-1<\/sup>K, to be compared with a temperature jump of the order of 2.102<\/sup>K), so a relative error of less than 10-3<\/sup><\/p>\n\n\n\nReconstruction error of temperature distribution at the end of temperature jump.<\/figcaption><\/figure>\n\n\n\nReconstruction error of temperature distribution at the end of temperature plateau.<\/figcaption><\/figure>\n\n\n\n\n\n\n\nFrom thermal to structural<\/h2>\n\n\n\n As long as thermal quantities are concerned, we can largely accept to work with a set of 7 POD vectors. Going further, we now need to solve for the structural response of each of these 7 temperature patterns, and linearly combine the results to obtain actual physical quantities (in practice, we just need to multiply each resulting vector by the corresponding amplitude, and do the sum. This is performed using a series of LCFACT\/LCADD commands).<\/p>\n\n\n\n
Namely, for each time points, we will define a now \u201cload case\u201d in POST1, like so<\/p>\n\n\n\n
lczero\nlcdef,erase\n*do,ind,1,nbSingularValues\n! LCDEF, LCNO, LSTEP, SBSTEP, KIMG\nlcdef,ind,ind,1\n*enddo<\/code><\/pre>\n\n\n\nAnd then we do the linear combination of results<\/p>\n\n\n\n
lcfact,1,SigmaVconj(1,ind_step)\nlcase,1\n*do,indVec,2,nbSingularValues\nlcfact,indVec,SigmaVconj(indVec,ind_step)\nlcoper,add,indVec\n*enddo<\/code><\/pre>\n\n\n\nNow, we should compare results obtained using both approaches. Again, what we are looking after local stress levels, so let\u2019s see how the results compare. We begin with the earliest phase of the transient, where the diffusion process starts. Although this is not the time point where we would expect the largest mechanical stress, this is clearly at those early stages that the largest deviation between both methods should exist (because the local, short lived temperature patterns will be \u201cdiscarded\u201d in the POD process as being non-significant).<\/p>\n\n\n\n
For example, at time=0.01s, the peak stress intensity if found to be equal to 1.30 and 1.33Mpa for the POD and the full method, respectively. Needless to say, this is well below engineering accuracy. What happens at later stages of the transient?<\/p>\n\n\n\nStress distribution estimated using the POD thermal-elastic method – early stage<\/figcaption><\/figure>\n\n\n\nStress distribution estimated using the FULL thermal-elastic method – early stage<\/figcaption><\/figure>\n\n\n\nBetween t=1 and 2s, the largest stress intensity values are observed. Namely, using the POD reduction, max stress level is obtained at time=1.6s, and the stress (SEQV) level is equal to 605MPa (above the elastic limit, by the way). With the FULL method, this value is also estimated to be equal 605MPa. So, both methods do provide indistinguishable results.<\/p>\n\n\n\nStress distribution estimated using the POD thermal-elastic method – peak stress time point<\/figcaption><\/figure>\n\n\n\nStress distribution estimated using the FULL thermal-elastic method – peak stress time point<\/figcaption><\/figure>\n\n\n\nAnd, by the way, in case you are suspecting by nature-which is definitely a healthy disease to have if you are an engineer working for safety related components- this is not coincidental: the results also do perfectly match for the preceding and following time points.<\/p>\n\n\n\n\n\n\n\n
Summary and Conclusion<\/h2>\n\n\n\n This first test case is really promising. We have seen that it is possible to compress a long thermal transient using SVD, achieving a reduction factor of about 20, with completely negligible differences in structural results obtain with either the regular, \u201cfull\u201d approach, and the reduced one. Actually, the structural pass is so economical that is can be done \u201con the fly\u201d, hence there is no need for storage (only pre-calculated results need to be linearly combined, hence the limiting factor for solution time becomes I\/O and not CPU. For example, in my case the gain factor on elapsed time was 7 and not 22). Needless to say, the method is only applicable as long as the structure remains elastic,<\/p>\n\n\n\n
Importantly, here we applied this method to the structural pass only of the thermal elastic simulation. However, once a first thermal transient has been solved, and POD vectors are obtained, there is no reason not to use that vector basis to speed up the thermal transient analysis (i.e. search for thermal solutions in this vector space, that is, solve for less than 10 unknowns instead of 105<\/sup>. This will be covered in a next post.<\/p>\n\n\n\nThat\u2019s all for today!<\/p>\n\n\n\n
WWAM-Ep04-InputFiles<\/a> Download<\/a><\/div>\n\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"
Data reduction for real-world applications In the previous post of this series (here: Working Wonders with ADPL Math \u2013 Ep 03: Data Reduction Fundamentals), we have seen that using a bunch of APDL Math commands, it is possible to reduce large volume of data (i.e. snaphots) very efficiently. Now, data reduction method like POD is obviously […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[4,5],"class_list":["post-492","post","type-post","status-publish","format-standard","hentry","category-technical-literature","tag-ansys-apdl-math","tag-model-reduction"],"yoast_head":"\n
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Now, data reduction method like POD is obviously […]","og_url":"https:\/\/alma-consulting.eu\/index.php\/2021\/12\/14\/working-wonders-with-adpl-math-illustrated-ep04-data-reduction-applied-to-thermal-elastic-problems\/","og_site_name":"Alma Consulting","article_published_time":"2021-12-14T19:48:34+00:00","article_modified_time":"2022-05-28T11:49:16+00:00","og_image":[{"url":"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/12\/image-1-1024x770.png","type":"","width":"","height":""}],"author":"admin9661","twitter_card":"summary_large_image","twitter_misc":{"Written by":"admin9661","Est. reading time":"11 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/alma-consulting.eu\/index.php\/2021\/12\/14\/working-wonders-with-adpl-math-illustrated-ep04-data-reduction-applied-to-thermal-elastic-problems\/#article","isPartOf":{"@id":"https:\/\/alma-consulting.eu\/index.php\/2021\/12\/14\/working-wonders-with-adpl-math-illustrated-ep04-data-reduction-applied-to-thermal-elastic-problems\/"},"author":{"name":"admin9661","@id":"https:\/\/alma-consulting.eu\/#\/schema\/person\/67349dff7b3613d00b9310b853e1544c"},"headline":"Working Wonders with ADPL Math\u00a0– Ep 04: Data Reduction Applied To Thermal-Elastic problems","datePublished":"2021-12-14T19:48:34+00:00","dateModified":"2022-05-28T11:49:16+00:00","mainEntityOfPage":{"@id":"https:\/\/alma-consulting.eu\/index.php\/2021\/12\/14\/working-wonders-with-adpl-math-illustrated-ep04-data-reduction-applied-to-thermal-elastic-problems\/"},"wordCount":2145,"publisher":{"@id":"https:\/\/alma-consulting.eu\/#organization"},"image":{"@id":"https:\/\/alma-consulting.eu\/index.php\/2021\/12\/14\/working-wonders-with-adpl-math-illustrated-ep04-data-reduction-applied-to-thermal-elastic-problems\/#primaryimage"},"thumbnailUrl":"https:\/\/alma-consulting.eu\/wp-content\/uploads\/2021\/12\/image-1-1024x770.png","keywords":["ANSYS APDL Math","Model Reduction"],"articleSection":["Technical Literature"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/alma-consulting.eu\/index.php\/2021\/12\/14\/working-wonders-with-adpl-math-illustrated-ep04-data-reduction-applied-to-thermal-elastic-problems\/","url":"https:\/\/alma-consulting.eu\/index.php\/2021\/12\/14\/working-wonders-with-adpl-math-illustrated-ep04-data-reduction-applied-to-thermal-elastic-problems\/","name":"Working Wonders with ADPL Math\u00a0- Ep 04: Data Reduction Applied To Thermal-Elastic problems - 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