{"id":5721,"date":"2023-08-29T19:04:18","date_gmt":"2023-08-29T18:04:18","guid":{"rendered":"http:\/\/mosaic-modeling.de\/?page_id=5721"},"modified":"2024-07-14T14:56:36","modified_gmt":"2024-07-14T13:56:36","slug":"use-of-basic-elements-iii-orthogonal-collocation-on-finite-elements","status":"publish","type":"page","link":"https:\/\/mosaic-modeling.de\/?page_id=5721","title":{"rendered":"Use of Basic Elements III &#8211; Orthogonal Collocation on Finite Elements"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">On this page<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5721\/#Model-description\" >Model description<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5721\/#Model-workflow\" >Model workflow<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5721\/#Notation\" >Notation<\/a><ul class='ez-toc-list-level-5' ><li class='ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5721\/#Base-names\" >Base names<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5721\/#Indices\" >Indices<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5721\/#Equations\" >Equations<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5721\/#Equation-systems\" >Equation systems<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5721\/#Simulation-workflow\" >Simulation workflow<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5721\/#Equation-systems-and-Indexing\" >Equation systems and Indexing<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5721\/#Specifications-and-solution\" >Specifications and solution<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<p>This example shows how to implement orthogonal collocation by hand in MOSAICmodeling.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Model-description\"><\/span>Model description<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The model is fairly simple as it consists of only one differential equation:<\/p>\n<p><p class=\"ql-left-displayed-equation\" style=\"line-height: 36px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-438afcbd511b3ed6bad04c375c8d559c_l3.png\" height=\"36\" width=\"147\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#100;&#125;&#122;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#100;&#125;&#116;&#125;&#32;&#61;&#32;&#122;&#94;&#50;&#32;&#45;&#32;&#98;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#122;&#32;&#43;&#32;&#49;&#46;&#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n<p>Therein, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-ec5583fa081a1e03212c151e3c222412_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"\/> is the differentiated variable, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-ad69adf868bc701e561aa555db995f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"\/> is a parameter, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-fd9cb27edab3f0a8a249bc80cc9c6ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"\/> is time. If we apply orthogonal collocation on this differential equation, we obtain<\/p>\n<p><p class=\"ql-left-displayed-equation\" style=\"line-height: 55px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-361f91fcb9a2ddb561535509ee0e564e_l3.png\" height=\"55\" width=\"231\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;&#92;&#115;&#117;&#109;&#95;&#123;&#106;&#61;&#48;&#125;&#94;&#123;&#78;&#74;&#125;&#97;&#95;&#123;&#106;&#44;&#99;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#122;&#95;&#123;&#99;&#61;&#106;&#125;&#32;&#61;&#32;&#122;&#95;&#123;&#99;&#125;&#94;&#50;&#32;&#45;&#32;&#98;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#122;&#95;&#99;&#32;&#43;&#32;&#49;&#46;&#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n<p>Finally, if we divide the time horizon into several finite elements, we have the following discretization. In addition, we need a equation that enforces continuity between the finite elements:<\/p>\n<p><p class=\"ql-left-displayed-equation\" style=\"line-height: 55px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-46641579626bcead2c765e95e4dd17c4_l3.png\" height=\"55\" width=\"449\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;&#92;&#115;&#117;&#109;&#95;&#123;&#106;&#61;&#48;&#125;&#94;&#123;&#78;&#74;&#125;&#123;&#97;&#95;&#123;&#106;&#44;&#99;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#122;&#95;&#123;&#99;&#61;&#106;&#44;&#105;&#125;&#125;&#61;&#32;&#40;&#40;&#122;&#95;&#123;&#99;&#44;&#105;&#125;&#41;&#94;&#123;&#50;&#125;&#32;&#45;&#32;&#98;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#122;&#95;&#123;&#99;&#44;&#105;&#125;&#32;&#43;&#32;&#49;&#41;&#92;&#99;&#100;&#111;&#116;&#32;&#104;&#95;&#123;&#105;&#125;&#44;&#32;&#92;&#92;&#91;&#50;&#101;&#120;&#93;&#122;&#95;&#123;&#99;&#61;&#51;&#44;&#105;&#125;&#61;&#122;&#95;&#123;&#99;&#61;&#48;&#44;&#105;&#43;&#49;&#125;&#46;&#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n<p>In the discretized equations, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-607b0fe3e387a1c7f89d53e7c8dbb0d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#106;&#44;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"25\" style=\"vertical-align: -6px;\"\/> is a collocation coefficient equal to the derivative of the <em>j<\/em>-th lagrange basis polynomial at the <em>c<\/em>-th collocation point. The index <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-31318c5dcb226c69e0818e5f7d2422b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"\/> marks the finite elements and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-ec762950a25d611e8cb122d8b5b223f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#95;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -3px;\"\/> is the length of the <em>i<\/em>-th finite element. Optionally, the time can also be discretized although it usually does not explicitly appear in chemical engineering applications:<\/p>\n<p><p class=\"ql-left-displayed-equation\" style=\"line-height: 17px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-688bf07ae7882a302e041c8ce85d5aaa_l3.png\" height=\"17\" width=\"187\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;&#116;&#95;&#123;&#105;&#44;&#99;&#125;&#61;&#116;&#95;&#123;&#105;&#45;&#49;&#44;&#99;&#61;&#78;&#67;&#125;&#43;&#92;&#116;&#97;&#117;&#95;&#123;&#99;&#125;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#104;&#95;&#123;&#105;&#125;&#46;&#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Model-workflow\"><\/span>Model workflow<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h4><span class=\"ez-toc-section\" id=\"Notation\"><\/span>Notation<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>Once again, we begin by setting up the notation. Therefore, perform the following steps:<\/p>\n<ol>\n<li>Go to the Notation tab in the &#8220;Model&#8221; section<\/li>\n<li>Add a description of your notation<\/li>\n<li>Add the base names below\u00a0by clicking on &#8220;+ Name&#8221; at the bottom, entering these names, adding a description of the variable and confirming it<\/li>\n<li>Go to the Indicies tab and add the index below<\/li>\n<li>Once you have added all variables and symbols, click on &#8220;Save&#8221; at the top of the editor and save your notation at a suitable location in your folder structure<\/li>\n<\/ol>\n<h5><span class=\"ez-toc-section\" id=\"Base-names\"><\/span>Base names<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-2d0f4e922bf6aa03f0b4a3128b5a72d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>, collocation point<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-0e55b0b3943237ccfc96979505679274_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"\/>, collocation coefficient, i.e., derivative of lagrange basis polynomial<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-ad69adf868bc701e561aa555db995f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"\/>, parameter<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-2ce27f7d2d82e3b238176ec7e7ee9118_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\"\/>, length of finite element<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-fd9cb27edab3f0a8a249bc80cc9c6ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"\/>, time<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-ec5583fa081a1e03212c151e3c222412_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"\/>, state variable<\/li>\n<\/ul>\n<h5><span class=\"ez-toc-section\" id=\"Indices\"><\/span>Indices<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-276a76eafbebc4494deafceec7cc4ddd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"\/>, Index of collocation points 1..NC<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-31318c5dcb226c69e0818e5f7d2422b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"\/>, Index of finite elements 1..NI<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-b09880662630fc49b25d42badb906d51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"9\" style=\"vertical-align: -4px;\"\/>, Index of lagrange basis polynomials 1..NJ<\/li>\n<\/ul>\n<p>The notation is available with ID 8480.<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Equations\"><\/span>Equations<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>Go to the Equation tab. Proceed as follows:<\/p>\n<ol>\n<li>Load the notation you just created<\/li>\n<li>Add an informative description of the current equation<\/li>\n<li>Type the first model equation in LaTeX style into the window MosaicTex. For the differential operator, you must use \\diff{Var}{IndependentVariable}<\/li>\n<li>Save the equation at a suitable location in your folder structure<\/li>\n<li>Click on &#8220;New&#8221; at the top of the editor and repeat the steps 1 and 2<\/li>\n<li>Enter the remaining model equations for the discretized forms and save them as well<\/li>\n<\/ol>\n<p>The equations have IDs 8481, 8486, 8490, 8497, and 8492.<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Equation-systems\"><\/span>Equation systems<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>Now, we will combine our equations to three equation system. To this end, go to the Equation System tab and carry on as described below:<\/p>\n<ol>\n<li>Load your notation from above<\/li>\n<li>Add a suitable description<\/li>\n<li>Go to the tab Connected Elements and click on &#8220;+ EQU\/EQS&#8221; at the bottom of the editor<\/li>\n<li>In the opening popup window, select your differential equation and confirm<\/li>\n<li>Save your equation system with this single differential equation<\/li>\n<li>Repeat the steps 1-5 with the single discretized equation and the discretized form with finite elements in combination with the continuity equation<\/li>\n<\/ol>\n<p>The resulting three equation systems have the IDs 8482, 8487, and 8491.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Simulation-workflow\"><\/span>Simulation workflow<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Start by going to the &#8220;Simulation&#8221; section of MOSAICmodeling.<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Equation-systems-and-Indexing\"><\/span>Equation systems and Indexing<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>Here, we describe the following steps in a simultaneous way. In MOSAICmodeling, you would rather set up the simulations sequentially: go to the tab Equation System and select the first of your newly saved equation system from above. For this system, you do not have to specify and indices.<\/p>\n<p>For the discretized version of the equation system, you need to specify the number of collocation points (NC = 3) and the number of lagrange basis polynomials (NJ = 3). For the discretized version on multiple finite elements, you also need to set an appropriate number of finite elements. Here, we choose NI = 5. For the latter two systems, click on Confirm Index Data to instantiate the systems.<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Specifications-and-solution\"><\/span>Specifications and solution<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>Go to the tab Specifications and then select the tab Variables. For the ODE system, proceed as follows:<\/p>\n<ol>\n<li>Add a helpful description of the simulation<\/li>\n<li>In the specification window, you can see all possible variable classifications. Select ALL VARIABLES at the top. You should now see the four variables that are part of your model<\/li>\n<li>Change the entry in the column Type for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-ad69adf868bc701e561aa555db995f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"\/> to Design Value<\/li>\n<li>Change the type classification of the variables <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-ec5583fa081a1e03212c151e3c222412_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"\/> to state variable.\u00a0 The variable <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-fd9cb27edab3f0a8a249bc80cc9c6ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"\/> should already be classified as differential value. Your degree of freedom should now be zero<\/li>\n<li>Assign the values given in Table 1 to the three variables<\/li>\n<li>Save your variable specification by clicking on the &#8220;Save&#8221; button above the calculation of the degrees of freedom<\/li>\n<li>Save your simulation by clicking on &#8220;Save&#8221; at the top of the editor<\/li>\n<\/ol>\n<p>For the discretized ODE, do as follows:<\/p>\n<ol>\n<li>Add a helpful description of the simulation<\/li>\n<li>In the specification window, you can see all possible variable classifications. Select ALL VARIABLES at the top. You should now see the four variables that are part of your model<\/li>\n<li>Change the entry in the column Type for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-ad69adf868bc701e561aa555db995f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-557e48759d6a14996a13967fc3a43375_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#106;&#61;&#48;&#44;&#99;&#61;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -6px;\"\/>&#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-e1d8e1c3efafc3434aa503b2eb61cd3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#106;&#61;&#51;&#44;&#99;&#61;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"62\" style=\"vertical-align: -6px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-ad69adf868bc701e561aa555db995f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-be26908d62a5038f1e5f7f1f8a4e5159_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#95;&#123;&#105;&#61;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"32\" style=\"vertical-align: -3px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-0aef9983a319d46487c39b170059617e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#95;&#123;&#105;&#61;&#48;&#44;&#99;&#61;&#78;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"72\" style=\"vertical-align: -6px;\"\/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-37c20495a10199fae311eb324bfacf82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#95;&#123;&#99;&#61;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"32\" style=\"vertical-align: -3px;\"\/> to Design Value<\/li>\n<li>Assign the other six variables <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-60d65cf892851a29ed35ace5d4dfb033_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#95;&#123;&#105;&#61;&#49;&#44;&#99;&#61;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -6px;\"\/>&#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-7622c155474f054ec0e4e871491c012a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#95;&#123;&#105;&#61;&#49;&#44;&#99;&#61;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"57\" style=\"vertical-align: -6px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-269345fde62dac62ae30d668c3b5b88c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#95;&#123;&#99;&#61;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"31\" style=\"vertical-align: -3px;\"\/>&#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-70783d5d0b218fc9a7df1267e187e5e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#95;&#123;&#99;&#61;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"32\" style=\"vertical-align: -3px;\"\/> as ITERATION_VALUE<\/li>\n<li>Assign the values given in Table 1 to the variables<\/li>\n<li>Save your variable specification by clicking on the &#8220;Save&#8221; button above the calculation of the degrees of freedom<\/li>\n<li>Save your simulation by clicking on &#8220;Save&#8221; at the top of the editor<\/li>\n<\/ol>\n<p>For the discretized ODE on finite elements, take the following steps:<\/p>\n<ol>\n<li>Add a helpful description of the simulation<\/li>\n<li>In the specification window, you can see all possible variable classifications. Select ALL VARIABLES at the top. You should now see the four variables that are part of your model<\/li>\n<li>In addition to the same <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-607b0fe3e387a1c7f89d53e7c8dbb0d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#106;&#44;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"25\" style=\"vertical-align: -6px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-ad69adf868bc701e561aa555db995f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"\/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-0aef9983a319d46487c39b170059617e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#95;&#123;&#105;&#61;&#48;&#44;&#99;&#61;&#78;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"72\" style=\"vertical-align: -6px;\"\/> from the previous simulation, change the entry in the column Type for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-26ccfd1f5cf6fe3d50df8e005885a16f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#95;&#123;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -3px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-7f15824e54bc9f9c3ed217fce3d50c38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#95;&#123;&#99;&#61;&#48;&#44;&#105;&#61;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"58\" style=\"vertical-align: -6px;\"\/> to Design Value<\/li>\n<li>Assign the remaining <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-9324134302269a28d8d830c91606622e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#95;&#123;&#105;&#44;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"21\" style=\"vertical-align: -6px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-9f9aaf99209ea4195009cd196d39f254_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#95;&#123;&#99;&#44;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"23\" style=\"vertical-align: -6px;\"\/>&#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-70783d5d0b218fc9a7df1267e187e5e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#95;&#123;&#99;&#61;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"32\" style=\"vertical-align: -3px;\"\/> as ITERATION_VALUE<\/li>\n<li>Assign the values given in Table 1 to the variables<\/li>\n<li>Save your variable specification by clicking on the &#8220;Save&#8221; button above the calculation of the degrees of freedom<\/li>\n<li>Save your simulation by clicking on &#8220;Save&#8221; at the top of the editor<\/li>\n<\/ol>\n<p>The simulations are available with IDs 8484, 8489, and 8494; the respective variable specifications have IDs 8483, 8488, and 8493. The solutions are compared in Figure 1. They were obtained by plotting the ODE solution against the solutions of the other two cases. Note that MOSAICmodeling does not automatically generate plot functions for discretized systems. These must be written by the user.<\/p>\n\n\n<figure class=\"wp-block-table is-style-stripes has-small-font-size\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Name<\/th><th class=\"has-text-align-left\" data-align=\"left\">Description<\/th><th class=\"has-text-align-left\" data-align=\"left\">Value \/ Initial guess<\/th><th class=\"has-text-align-left\" data-align=\"left\">Solution<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-ad69adf868bc701e561aa555db995f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Parameter<\/td><td class=\"has-text-align-left\" data-align=\"left\">2.0<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-ec5583fa081a1e03212c151e3c222412_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">State variable<\/td><td class=\"has-text-align-left\" data-align=\"left\">-3.0<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-72de385b53a37f6800a0ee68a8a957bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#95;&#123;&#99;&#61;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"31\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Collocation point 1<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.15505<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-1ef23d6ee999f5d9afa2e0288c8b7fde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#95;&#123;&#99;&#61;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"32\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Collocation point 2<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.6449<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-5c644737b5e97083e1ada3da9c22fc1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#97;&#117;&#95;&#123;&#99;&#61;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"32\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Collocation point 3<\/td><td class=\"has-text-align-left\" data-align=\"left\">1.0<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-557e48759d6a14996a13967fc3a43375_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#106;&#61;&#48;&#44;&#99;&#61;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Derivative of lagrange basis polynomial<\/td><td class=\"has-text-align-left\" data-align=\"left\">-4.1394<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-fd239a25487746d409656630967ecd20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#106;&#61;&#49;&#44;&#99;&#61;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Derivative of lagrange basis polynomial<\/td><td class=\"has-text-align-left\" data-align=\"left\">3.2247<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-092d761ea5b31ebb14c6de508544f81b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#106;&#61;&#50;&#44;&#99;&#61;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Derivative of lagrange basis polynomial<\/td><td class=\"has-text-align-left\" data-align=\"left\">1.1678<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-536e4bcc64d6e8daee1286e4f6ca6d22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#106;&#61;&#51;&#44;&#99;&#61;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Derivative of lagrange basis polynomial<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.2532<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-ef14d46a00ee65f9a827f994c3a33535_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#106;&#61;&#48;&#44;&#99;&#61;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"62\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Derivative of lagrange basis polynomial<\/td><td class=\"has-text-align-left\" data-align=\"left\">1.7394<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-8eb248d7e03ff41056e42fa951bfadf3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#106;&#61;&#49;&#44;&#99;&#61;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"62\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Derivative of lagrange basis polynomial<\/td><td class=\"has-text-align-left\" data-align=\"left\">-3.5678<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-417b63f578a32d311f7ac2def88c8510_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#106;&#61;&#50;&#44;&#99;&#61;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"62\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Derivative of lagrange basis polynomial<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.7753<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-2001d1cdadaa7e7cfd844aec52b78d53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#106;&#61;&#51;&#44;&#99;&#61;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"62\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Derivative of lagrange basis polynomial<\/td><td class=\"has-text-align-left\" data-align=\"left\">1.0532<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-14922f40397995d1edfa82a80ee507a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#106;&#61;&#48;&#44;&#99;&#61;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"62\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Derivative of lagrange basis polynomial<\/td><td class=\"has-text-align-left\" data-align=\"left\">-3.0<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-0147b990e53a6ac9d2b34aa45a256b15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#106;&#61;&#49;&#44;&#99;&#61;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"62\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Derivative of lagrange basis polynomial<\/td><td class=\"has-text-align-left\" data-align=\"left\">5.532<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-39f767829ab6f47f4edf2c2f04273ac1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#106;&#61;&#50;&#44;&#99;&#61;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"62\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Derivative of lagrange basis polynomial<\/td><td class=\"has-text-align-left\" data-align=\"left\">-7.532<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-e1d8e1c3efafc3434aa503b2eb61cd3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#95;&#123;&#106;&#61;&#51;&#44;&#99;&#61;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"62\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Derivative of lagrange basis polynomial<\/td><td class=\"has-text-align-left\" data-align=\"left\">5.0<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-be26908d62a5038f1e5f7f1f8a4e5159_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#95;&#123;&#105;&#61;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"32\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Length of finite element (for case 2)<\/td><td class=\"has-text-align-left\" data-align=\"left\">1.0<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-0aef9983a319d46487c39b170059617e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#95;&#123;&#105;&#61;&#48;&#44;&#99;&#61;&#78;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"72\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Initial time (for case 2)<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.0<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-37c20495a10199fae311eb324bfacf82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#95;&#123;&#99;&#61;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"32\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Initial condition of discretized state variable (for case 2)<\/td><td class=\"has-text-align-left\" data-align=\"left\">-3.0<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-9324134302269a28d8d830c91606622e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#95;&#123;&#105;&#44;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"21\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Times at other collocation points<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.5<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.15505, 0.6449, 1.0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-aef4db79611396af66b71853f08d4b29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#95;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Solution at other collocation points<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.0<\/td><td class=\"has-text-align-left\" data-align=\"left\">-1.657, 0.032, 0.207<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-26ccfd1f5cf6fe3d50df8e005885a16f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#95;&#123;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Length of finite element (for case 3)<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.2<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-8b79556e50d73fb2c3bb9daae3f80f2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#95;&#123;&#99;&#61;&#48;&#44;&#99;&#61;&#78;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"76\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Initial time (for case 3)<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.0<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-7f15824e54bc9f9c3ed217fce3d50c38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#95;&#123;&#99;&#61;&#48;&#44;&#105;&#61;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"58\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Initial condition of discretized state variable (for case 3)<\/td><td class=\"has-text-align-left\" data-align=\"left\">-3.0<\/td><td class=\"has-text-align-left\" data-align=\"left\"><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-9324134302269a28d8d830c91606622e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#95;&#123;&#105;&#44;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"21\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Times at other collocation points<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.5<\/td><td class=\"has-text-align-left\" data-align=\"left\">See results after code generation<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-9f9aaf99209ea4195009cd196d39f254_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#95;&#123;&#99;&#44;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"23\" style=\"vertical-align: -6px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Solution at other collocation points<\/td><td class=\"has-text-align-left\" data-align=\"left\">-1.5<\/td><td class=\"has-text-align-left\" data-align=\"left\">See results after code generation<\/td><\/tr><\/tbody><\/table><figcaption class=\"wp-element-caption\">Table 1: Overview of design values and initial conditions.<\/figcaption><\/figure>\n\n\n<p><a style=\"display: inline !important;\" href=\"http:\/\/mosaic-modeling.de\/wp-content\/uploads\/2023\/08\/Comparison_ODE_OCFE.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5742\" src=\"http:\/\/mosaic-modeling.de\/wp-content\/uploads\/2023\/08\/Comparison_ODE_OCFE.png\" alt=\"\" width=\"700\" height=\"525\" srcset=\"https:\/\/mosaic-modeling.de\/wp-content\/uploads\/2023\/08\/Comparison_ODE_OCFE.png 700w, https:\/\/mosaic-modeling.de\/wp-content\/uploads\/2023\/08\/Comparison_ODE_OCFE-300x225.png 300w\" sizes=\"auto, (max-width: 700px) 100vw, 700px\" \/><\/a><\/p>\n<p>Figure 1: Comparison between the three different cases. The solution with one finite element is stable, but differs significantly from the ODE solution. With more finite elements, the solution is approximated much better.<\/p>","protected":false},"excerpt":{"rendered":"<p>This example shows how to implement orthogonal collocation by hand in MOSAICmodeling. Model description The model is fairly simple as it consists of only one differential equation: &nbsp; &nbsp; Therein, is the differentiated variable, is a parameter, and is time. If we apply orthogonal collocation on this differential equation, we obtain &nbsp; &nbsp; Finally, if [&hellip;]<\/p>\n","protected":false},"author":252,"featured_media":0,"parent":5463,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-5721","page","type-page","status-publish","czr-hentry"],"_links":{"self":[{"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=\/wp\/v2\/pages\/5721","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=\/wp\/v2\/users\/252"}],"replies":[{"embeddable":true,"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5721"}],"version-history":[{"count":13,"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=\/wp\/v2\/pages\/5721\/revisions"}],"predecessor-version":[{"id":6148,"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=\/wp\/v2\/pages\/5721\/revisions\/6148"}],"up":[{"embeddable":true,"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=\/wp\/v2\/pages\/5463"}],"wp:attachment":[{"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5721"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}