Latest Discussions - COMSOL Forums https://www.comsol.com/forum/ Most recent forum discussions Tue, 26 May 2020 17:33:59 +0000 COMSOL Forum: Latest Discussions https://www.comsol.com/shared/images/logos/comsol_logo.gif https://www.comsol.com/forum/ Electrostatic charge dissipation https://www.comsol.com/forum/thread/260422/electrostatic-charge-dissipation?last=2020-05-26T17:33:59Z <p>I want to simulate electrostatic charge disspation over the time on polymer surface (Flat surface or patterned surface). Are there any resources to refer to implement charge dissipation simulation while varying the time?</p> <p>Also, I want to know the triboelectrification simulation using COMSOL.</p> <p>Please give me some link or idea for that.</p> <p>Thanks,</p> Tue, 26 May 2020 17:33:59 +0000 4.2020-05-26 17:33:59.260422 PDE as boundary condition https://www.comsol.com/forum/thread/260421/pde-as-boundary-condition?last=2020-05-26T16:05:42Z <p>Hello,</p> <p>I would like to solve a diffision / heat transfer problem where one of the boundary conditions is a PDE:</p> <p><img class="latexImg" src="data:image/png;base64,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" /></p> <p>Boundary conditions:</p> <p><img class="latexImg" src="data:image/png;base64,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" /></p> <p><img class="latexImg" src="data:image/png;base64,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" /> at x=l <img class="latexImg" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAARCAQAAACcev82AAAJJWlDQ1BpY2MAAEjHlZVnUJNZF8fv8zzphUASQodQQ5EqJYCUEFoo0quoQOidUEVsiLgCK4qINEWQRQEXXJUia0UUC4uCAhZ0gywCyrpxFVFBeUHfGZ33nf2w/5l7z2/+c+bec8/5cAEgiINlwct7YlK6wNvJjhkYFMwE3ymMn5bC8fR0A/+odyMAWon3dMG/FyEiMo2/HBeXVy4/RZAOAJS9zJpZ6SkrfHSZ6eHxX/jsCguWC1zmGysc/ZXHvuR8ZdGXHF9v7vKrUADgSNHfcfh3/N97V6TCEaTHRkVmM32So9KzwgSRzLSVTvC4XKanIDkqNiHyu4L/VfI/KD0yO30lcpNTNglio2PSmf93qJGBoSH4Nos3Xl96DDH6/3c+K/rmJdcDwJ4DANn3zQuvBKBzFwDSj755ast9peQD0HGHnyHI/OqhVjY0IAAKoAMZoAhUgSbQBUbADFgCW+AAXIAH8AVBYAPggxiQCAQgC+SCHaAAFIF94CCoArWgATSBVnAadILz4Aq4Dm6Du2AYPAZCMAleAhF4BxYgCMJCZIgGyUBKkDqkAxlBbMgacoDcIG8oCAqFoqEkKAPKhXZCRVApVAXVQU3QL9A56Ap0ExqEHkLj0Az0N/QRRmASTIcVYA1YH2bDHNgV9oXXw9FwKpwD58N74Qq4Hj4Jd8BX4NvwMCyEX8JzCECICANRRnQRNsJFPJBgJAoRIFuRQqQcqUdakW6kD7mHCJFZ5AMKg6KhmChdlCXKGeWH4qNSUVtRxagq1AlUB6oXdQ81jhKhPqPJaHm0DtoCzUMHoqPRWegCdDm6Ed2OvoYeRk+i32EwGAaGhTHDOGOCMHGYzZhizGFMG+YyZhAzgZnDYrEyWB2sFdYDG4ZNxxZgK7EnsZewQ9hJ7HscEaeEM8I54oJxSbg8XDmuGXcRN4Sbwi3gxfHqeAu8Bz4Cvwlfgm/Ad+Pv4CfxCwQJAotgRfAlxBF2ECoIrYRrhDHCGyKRqEI0J3oRY4nbiRXEU8QbxHHiBxKVpE3ikkJIGaS9pOOky6SHpDdkMlmDbEsOJqeT95KbyFfJT8nvxWhiemI8sQixbWLVYh1iQ2KvKHiKOoVD2UDJoZRTzlDuUGbF8eIa4lzxMPGt4tXi58RHxeckaBKGEh4SiRLFEs0SNyWmqViqBtWBGkHNpx6jXqVO0BCaKo1L49N20hpo12iTdAydRefR4+hF9J/pA3SRJFXSWNJfMluyWvKCpJCBMDQYPEYCo4RxmjHC+CilIMWRipTaI9UqNSQ1Ly0nbSsdKV0o3SY9LP1RhinjIBMvs1+mU+aJLEpWW9ZLNkv2iOw12Vk5upylHF+uUO603CN5WF5b3lt+s/wx+X75OQVFBSeFFIVKhasKs4oMRVvFOMUyxYuKM0o0JWulWKUypUtKL5iSTA4zgVnB7GWKlOWVnZUzlOuUB5QXVFgqfip5Km0qT1QJqmzVKNUy1R5VkZqSmrtarlqL2iN1vDpbPUb9kHqf+rwGSyNAY7dGp8Y0S5rFY+WwWlhjmmRNG81UzXrN+1oYLbZWvNZhrbvasLaJdox2tfYdHVjHVCdW57DO4Cr0KvNVSavqV43qknQ5upm6Lbrjegw9N708vU69V/pq+sH6+/X79D8bmBgkGDQYPDakGroY5hl2G/5tpG3EN6o2ur+avNpx9bbVXatfG+sYRxofMX5gQjNxN9lt0mPyydTMVGDaajpjpmYWalZjNsqmsz3Zxewb5mhzO/Nt5ufNP1iYWqRbnLb4y1LXMt6y2XJ6DWtN5JqGNRNWKlZhVnVWQmumdaj1UWuhjbJNmE29zTNbVdsI20bbKY4WJ45zkvPKzsBOYNduN8+14G7hXrZH7J3sC+0HHKgOfg5VDk8dVRyjHVscRU4mTpudLjujnV2d9zuP8hR4fF4TT+Ri5rLFpdeV5OrjWuX6zE3bTeDW7Q67u7gfcB9bq742aW2nB/DgeRzweOLJ8kz1/NUL4+XpVe313NvQO9e7z4fms9Gn2eedr51vie9jP02/DL8ef4p/iH+T/3yAfUBpgDBQP3BL4O0g2aDYoK5gbLB/cGPw3DqHdQfXTYaYhBSEjKxnrc9ef3OD7IaEDRc2UjaGbTwTig4NCG0OXQzzCKsPmwvnhdeEi/hc/iH+ywjbiLKImUiryNLIqSirqNKo6Wir6APRMzE2MeUxs7Hc2KrY13HOcbVx8/Ee8cfjlxICEtoScYmhieeSqEnxSb3JisnZyYMpOikFKcJUi9SDqSKBq6AxDUpbn9aVTl/+FPszNDN2ZYxnWmdWZ77P8s86ky2RnZTdv0l7055NUzmOOT9tRm3mb+7JVc7dkTu+hbOlbiu0NXxrzzbVbfnbJrc7bT+xg7AjfsdveQZ5pXlvdwbs7M5XyN+eP7HLaVdLgViBoGB0t+Xu2h9QP8T+MLBn9Z7KPZ8LIwpvFRkUlRctFvOLb/1o+GPFj0t7o/YOlJiWHNmH2Ze0b2S/zf4TpRKlOaUTB9wPdJQxywrL3h7cePBmuXF57SHCoYxDwgq3iq5Ktcp9lYtVMVXD1XbVbTXyNXtq5g9HHB46YnuktVahtqj249HYow/qnOo66jXqy49hjmUee97g39D3E/unpkbZxqLGT8eTjgtPeJ/obTJramqWby5pgVsyWmZOhpy8+7P9z12tuq11bYy2olPgVMapF7+E/jJy2vV0zxn2mdaz6mdr2mnthR1Qx6YOUWdMp7ArqGvwnMu5nm7L7vZf9X49fl75fPUFyQslFwkX8y8uXcq5NHc55fLslegrEz0bex5fDbx6v9erd+Ca67Ub1x2vX+3j9F26YXXj/E2Lm+dusW913ja93dFv0t/+m8lv7QOmAx13zO503TW/2z24ZvDikM3QlXv2967f592/Pbx2eHDEb+TBaMio8EHEg+mHCQ9fP8p8tPB4+xh6rPCJ+JPyp/JP63/X+r1NaCq8MG4/3v/M59njCf7Eyz/S/liczH9Ofl4+pTTVNG00fX7Gcebui3UvJl+mvFyYLfhT4s+aV5qvzv5l+1e/KFA0+Vrweunv4jcyb46/NX7bM+c59/Rd4ruF+cL3Mu9PfGB/6PsY8HFqIWsRu1jxSetT92fXz2NLiUtL/wFCLJC+DRlcgAAAACBjSFJNAAB6JgAAgIQAAPoAAACA6AAAdTAAAOpgAAA6mAAAF3CculE8AAAAAmJLR0QA/4ePzL8AAAAJcEhZcwAAAHgAAAB4AJ31WmAAAAAHdElNRQfkBRoRECI0A1FNAAAA1UlEQVQoz5WSSwGEMAxEX3CAhVqoBVYCFmoBC2sBLCABJFAJYKESsgd+aZcL7SW/TibTiPLmVK+qqUD6//BT7ESvxRXFjiMitbRl+UgoQALjYbV8s4yiMCn2slxWT28z+6jRtpSGeDnWBkR3rosJez4apaEBOgYSo0ZDRmGhPqzaUAmsOc1T9+Eat73GBM9ciFBi3n0UVsIjuiaieBDPpsmoX6DfS7Crb6k0bLo9kjnVvse8NbeE7IrNMmXNHQnEk57RHYozfqCnpcv+O3MKHXD2uaL8ADqXoPTV9dWcAAAAJXRFWHRkYXRlOmNyZWF0ZQAyMDIwLTA1LTI2VDE3OjE2OjM0KzAwOjAw45d5MgAAACV0RVh0ZGF0ZTptb2RpZnkAMjAyMC0wNS0yNlQxNzoxNjozNCswMDowMJLKwY4AAAARdEVYdHBkZjpTcG90Q29sb3ItMAArzvERWAAAABF0RVh0cGRmOlNwb3RDb2xvci0xACvPM3tvAAAAIXRFWHRwczpIaVJlc0JvdW5kaW5nQm94ADE0eDEwKzI5OSs2MzkMJDOGAAAAH3RFWHRwczpMZXZlbAAhUFMtQWRvYmUtMi4wIEVQU0YtMi4wYCQiHwAAAABJRU5ErkJggg==" /> &lt;-----</p> <p>(this is the boudary condition for a well-stirred fluid or an ideal heat conductor)</p> <p>Initial condition:</p> <p><img class="latexImg" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARUAAAAXCAQAAACPWBPWAAAJJWlDQ1BpY2MAAEjHlZVnUJNZF8fv8zzphUASQodQQ5EqJYCUEFoo0quoQOidUEVsiLgCK4qINEWQRQEXXJUia0UUC4uCAhZ0gywCyrpxFVFBeUHfGZ33nf2w/5l7z2/+c+bec8/5cAEgiINlwct7YlK6wNvJjhkYFMwE3ymMn5bC8fR0A/+odyMAWon3dMG/FyEiMo2/HBeXVy4/RZAOAJS9zJpZ6SkrfHSZ6eHxX/jsCguWC1zmGysc/ZXHvuR8ZdGXHF9v7vKrUADgSNHfcfh3/N97V6TCEaTHRkVmM32So9KzwgSRzLSVTvC4XKanIDkqNiHyu4L/VfI/KD0yO30lcpNTNglio2PSmf93qJGBoSH4Nos3Xl96DDH6/3c+K/rmJdcDwJ4DANn3zQuvBKBzFwDSj755ast9peQD0HGHnyHI/OqhVjY0IAAKoAMZoAhUgSbQBUbADFgCW+AAXIAH8AVBYAPggxiQCAQgC+SCHaAAFIF94CCoArWgATSBVnAadILz4Aq4Dm6Du2AYPAZCMAleAhF4BxYgCMJCZIgGyUBKkDqkAxlBbMgacoDcIG8oCAqFoqEkKAPKhXZCRVApVAXVQU3QL9A56Ap0ExqEHkLj0Az0N/QRRmASTIcVYA1YH2bDHNgV9oXXw9FwKpwD58N74Qq4Hj4Jd8BX4NvwMCyEX8JzCECICANRRnQRNsJFPJBgJAoRIFuRQqQcqUdakW6kD7mHCJFZ5AMKg6KhmChdlCXKGeWH4qNSUVtRxagq1AlUB6oXdQ81jhKhPqPJaHm0DtoCzUMHoqPRWegCdDm6Ed2OvoYeRk+i32EwGAaGhTHDOGOCMHGYzZhizGFMG+YyZhAzgZnDYrEyWB2sFdYDG4ZNxxZgK7EnsZewQ9hJ7HscEaeEM8I54oJxSbg8XDmuGXcRN4Sbwi3gxfHqeAu8Bz4Cvwlfgm/Ad+Pv4CfxCwQJAotgRfAlxBF2ECoIrYRrhDHCGyKRqEI0J3oRY4nbiRXEU8QbxHHiBxKVpE3ikkJIGaS9pOOky6SHpDdkMlmDbEsOJqeT95KbyFfJT8nvxWhiemI8sQixbWLVYh1iQ2KvKHiKOoVD2UDJoZRTzlDuUGbF8eIa4lzxMPGt4tXi58RHxeckaBKGEh4SiRLFEs0SNyWmqViqBtWBGkHNpx6jXqVO0BCaKo1L49N20hpo12iTdAydRefR4+hF9J/pA3SRJFXSWNJfMluyWvKCpJCBMDQYPEYCo4RxmjHC+CilIMWRipTaI9UqNSQ1Ly0nbSsdKV0o3SY9LP1RhinjIBMvs1+mU+aJLEpWW9ZLNkv2iOw12Vk5upylHF+uUO603CN5WF5b3lt+s/wx+X75OQVFBSeFFIVKhasKs4oMRVvFOMUyxYuKM0o0JWulWKUypUtKL5iSTA4zgVnB7GWKlOWVnZUzlOuUB5QXVFgqfip5Km0qT1QJqmzVKNUy1R5VkZqSmrtarlqL2iN1vDpbPUb9kHqf+rwGSyNAY7dGp8Y0S5rFY+WwWlhjmmRNG81UzXrN+1oYLbZWvNZhrbvasLaJdox2tfYdHVjHVCdW57DO4Cr0KvNVSavqV43qknQ5upm6Lbrjegw9N708vU69V/pq+sH6+/X79D8bmBgkGDQYPDakGroY5hl2G/5tpG3EN6o2ur+avNpx9bbVXatfG+sYRxofMX5gQjNxN9lt0mPyydTMVGDaajpjpmYWalZjNsqmsz3Zxewb5mhzO/Nt5ufNP1iYWqRbnLb4y1LXMt6y2XJ6DWtN5JqGNRNWKlZhVnVWQmumdaj1UWuhjbJNmE29zTNbVdsI20bbKY4WJ45zkvPKzsBOYNduN8+14G7hXrZH7J3sC+0HHKgOfg5VDk8dVRyjHVscRU4mTpudLjujnV2d9zuP8hR4fF4TT+Ri5rLFpdeV5OrjWuX6zE3bTeDW7Q67u7gfcB9bq742aW2nB/DgeRzweOLJ8kz1/NUL4+XpVe313NvQO9e7z4fms9Gn2eedr51vie9jP02/DL8ef4p/iH+T/3yAfUBpgDBQP3BL4O0g2aDYoK5gbLB/cGPw3DqHdQfXTYaYhBSEjKxnrc9ef3OD7IaEDRc2UjaGbTwTig4NCG0OXQzzCKsPmwvnhdeEi/hc/iH+ywjbiLKImUiryNLIqSirqNKo6Wir6APRMzE2MeUxs7Hc2KrY13HOcbVx8/Ee8cfjlxICEtoScYmhieeSqEnxSb3JisnZyYMpOikFKcJUi9SDqSKBq6AxDUpbn9aVTl/+FPszNDN2ZYxnWmdWZ77P8s86ky2RnZTdv0l7055NUzmOOT9tRm3mb+7JVc7dkTu+hbOlbiu0NXxrzzbVbfnbJrc7bT+xg7AjfsdveQZ5pXlvdwbs7M5XyN+eP7HLaVdLgViBoGB0t+Xu2h9QP8T+MLBn9Z7KPZ8LIwpvFRkUlRctFvOLb/1o+GPFj0t7o/YOlJiWHNmH2Ze0b2S/zf4TpRKlOaUTB9wPdJQxywrL3h7cePBmuXF57SHCoYxDwgq3iq5Ktcp9lYtVMVXD1XbVbTXyNXtq5g9HHB46YnuktVahtqj249HYow/qnOo66jXqy49hjmUee97g39D3E/unpkbZxqLGT8eTjgtPeJ/obTJramqWby5pgVsyWmZOhpy8+7P9z12tuq11bYy2olPgVMapF7+E/jJy2vV0zxn2mdaz6mdr2mnthR1Qx6YOUWdMp7ArqGvwnMu5nm7L7vZf9X49fl75fPUFyQslFwkX8y8uXcq5NHc55fLslegrEz0bex5fDbx6v9erd+Ca67Ub1x2vX+3j9F26YXXj/E2Lm+dusW913ja93dFv0t/+m8lv7QOmAx13zO503TW/2z24ZvDikM3QlXv2967f592/Pbx2eHDEb+TBaMio8EHEg+mHCQ9fP8p8tPB4+xh6rPCJ+JPyp/JP63/X+r1NaCq8MG4/3v/M59njCf7Eyz/S/liczH9Ofl4+pTTVNG00fX7Gcebui3UvJl+mvFyYLfhT4s+aV5qvzv5l+1e/KFA0+Vrweunv4jcyb46/NX7bM+c59/Rd4ruF+cL3Mu9PfGB/6PsY8HFqIWsRu1jxSetT92fXz2NLiUtL/wFCLJC+DRlcgAAAACBjSFJNAAB6JgAAgIQAAPoAAACA6AAAdTAAAOpgAAA6mAAAF3CculE8AAAAAmJLR0QA/4ePzL8AAAAJcEhZcwAAAHgAAAB4AJ31WmAAAAAHdElNRQfkBRoRECI0A1FNAAAD+0lEQVRo3u2a3XmrOBCG39knDagFWtCWoJRASuCUgEsgJdglJCXEJYQSohIOJcxeYBPxjxHYu2cz3BgQHzPfDKPRyKL8yI8skb8ercCP/FdkNlQkE/toJSUX82gd/l2yv1f6nM+EijiMlg8hw0rSnJwoNsNN4jEeLX2v7GBVn3OdODC8KbceGNLbn2ohpBQ4Uo4klyuOIg5TUUgo+IjHeezR9cpeVnU5nx58xK14RcZXpEMvz2P5bK5+XMNmJaqlIOfzXqES+7ks88q+VrU5nxpo1rmcI8dIKppo5vNKOWkcamP8vUJlp/cMeWUvq9qcT9UqKe+rZjlHXHWTBr8r3GWifG9d/+NFzEgpv9YrK6TN+VOoHBkGKLVWxoVKiSGv70mOwetpwDyHAxKsFLyvLocN5+Dsu2Arxen5drgtZQkLG6HkwGHguts+VCb0CThvQkUsR35pKYZCvJZAQhWAZXoA+RDHAceblP1Q0DNnyfD6a0Ady/BXUnVqede6W/J97rGc16FuRukCFmJRJKHSCvD1ykZ96/HAK3ewKuD8qRn+wYuWQErGKwA2+IYzrpFWaSWewyhFPXdeZHz6mCb7u39QDYTFWtR1spyFGJSMTF7weEkpONAOFbt5Zp3SJ+D8mlUKfK2CnhhKh2etY9lyAi0nHOEugdYRPbBO/NTN1agtEcNwQ6ubm2ZYaGVEE5613DuDogc5kZEAnr91dQ7ZyqqrXEMlZXLWrcHFduqIvnoJCXFR7yHIHnY6VDYSgxu83qFtloVwOgwxq3D8Ai4rwFBF2r6VVRd5grr7t8jBjnImyh1eBw2UfKSq6GQG9QI2KN0m37cUdVrUc0t2GmVBg3wqdkaDURTJyHmlpMLyJYd1pfN2Vl2lziqtIWIw6gHfW3Fc6hAxZDo4zTQjsp6BZ5bu44QVfhjCZiA1LkfdTuZZiEOpeFYvBV5fpf8B+53WgcP6hJxfmi1fTavLcMQoCm/NNYdiMfyum2PklxEJX+SdZlChYGN6lbimW5vwu37TBds+sgU3xsJtLbilKBTDWxnfXrmHVSHn11rlhVzqhlfVLHXLYCIoMeQ8U4gjaQoh6JZO71hJSSK+NvQsJzlywHDkOUiKNm4BLEWtrRypOOmaOmCMhT1QTqNP21a/a1+rQs6DCLLtr5Yk2IEx9T1Md1eoG+MkcXs1DUpO1orvTRr7kVqNsrA8q9yGMsjM5+1PrbOqzfk0yILtwi12fBcZdNwiBO933Ge78J6cT/9f5ZVsOneJjVwaLxSxVKuS68NE99upmfXKNtLlfDJU1HOWabXutSuTEVH9/FmywCvbSJfz+UQav+6IToSTa43/47G/V/qcy88/9n9kmfwD9Zmh/P2dSFMAAAAldEVYdGRhdGU6Y3JlYXRlADIwMjAtMDUtMjZUMTc6MTY6MzQrMDA6MDDjl3kyAAAAJXRFWHRkYXRlOm1vZGlmeQAyMDIwLTA1LTI2VDE3OjE2OjM0KzAwOjAwksrBjgAAABF0RVh0cGRmOlNwb3RDb2xvci0wACvO8RFYAAAAEXRFWHRwZGY6U3BvdENvbG9yLTEAK88ze28AAAAidEVYdHBzOkhpUmVzQm91bmRpbmdCb3gAMTY2eDE0KzIyMis2MzapTLaKAAAAH3RFWHRwczpMZXZlbAAhUFMtQWRvYmUtMi4wIEVQU0YtMi4wYCQiHwAAAABJRU5ErkJggg==" /></p> <p>How do I implement this boundary condition in COMSOL (mathematics package)?</p> <p>Best, Tobias</p> <p>EDIT: Modified the equations such that they fit to the solution by Jeff.</p> Tue, 26 May 2020 14:13:47 +0000 4.2020-05-26 14:13:47.260421 COMSOL CFD Capability https://www.comsol.com/forum/thread/260412/comsol-cfd-capability?last=2020-05-26T14:03:05Z <p>Hi, my name is Julia and I am currently an undergraduate Mechanical Engineering student at Georgia Southern University. I am working on a project that requires complex FSI simulations and I am looking for guidance if COMSOL is capable of such a task or if someone could point me in the right direction to a software capable of what I need.</p> <p>The issue at hand involves two parallel plates with a compressible flow between them. The top plate is flexible while the bottom one is fixed. I would expect the top plate to bend downward and touch the bottom one at some point. Do you know any software that could do collision detection in such a case?</p> <p>Thank you for your time, Julia</p> Tue, 26 May 2020 14:03:05 +0000 4.2020-05-26 14:03:05.260412 How to simulate a point-wise concentrate load traveling along a structure? https://www.comsol.com/forum/thread/260403/how-to-simulate-a-point-wise-concentrate-load-traveling-along-a-structure?last=2020-05-26T13:09:10Z <p>Now I'm working on the moving-load problems of the beams and plates. I want to use comsol to simulate a point-wise force or mass traveling along a beam, or traveling along a path on the plate surface, and explore the deformation of the beam and plate. I have read the <a href="https://www.comsol.com/blogs/3-approaches-to-modeling-moving-loads-and-constraints-in-comsol/">post concerning this topic</a>, but I still have no idea on how to deal with the concentrate moving load. Should I use the delta function? Is is possible to use comsol to consider the moving mass problem in which there exist the inertial interaction between the supporting structure and moving mass?</p> Tue, 26 May 2020 13:09:10 +0000 4.2020-05-26 13:09:10.260403 evaluate statistical quantities over all particle https://www.comsol.com/forum/thread/260401/evaluate-statistical-quantities-over-all-particle?last=2020-05-26T07:16:59Z <p>running a charged particle tracing simulation with 100(say) input electrons gives a final quantitity qx(x-coordinates) of al particles. How do I evalute say the mean of qx( over all the elctrons) at any time step?</p> Tue, 26 May 2020 07:16:59 +0000 4.2020-05-26 07:16:59.260401 Pressure transmission between 2 objects that are stacked on top of each other https://www.comsol.com/forum/thread/260393/pressure-transmission-between-2-objects-that-are-stacked-on-top-of-each-other?last=2020-05-26T06:56:10Z <p>Hello! I'm analysing a structure which contains two domes stacked on top of each other, with the curved surfaces in touch. However, when I applied a pressure on the topmost surface of the upper dome, there was minimal deformation produced in the bottom dome. I suspected that there was a problem with the contact, so I carried out two similar simulations using two rectangular blocks and two conical frustra respectively. I maintained the same object thickness and material properties throughout for fair comparison.</p> <p>The deformation produced in the bottom rectangular block was a lot more significant than that produced in the bottom conical frustrum and bottom dome. Even when I increased the pressure tenfold, to the point where the top dome almost completely collapsed, there was still little to no deformation in the bottom dome.</p> <p>Are the problems that I am facing due to inherent software limitations? Or am I doing something wrong?</p> <p>Details of my simulations can be found on the google site accessible via this link: <a href="https://sites.google.com/view/comsoltroubleshooting2/home?authuser=2">https://sites.google.com/view/comsoltroubleshooting2/home?authuser=2</a></p> <p>Can someone with expertise in this area kindly offer some guidance? Thank you so much! :)</p> Tue, 26 May 2020 02:14:32 +0000 4.2020-05-26 02:14:32.260393 Modeling incident beam with finite spot size https://www.comsol.com/forum/thread/260392/modeling-incident-beam-with-finite-spot-size?last=2020-05-26T01:11:32Z <p>Is it possible to model an incident light beam with a spot size of, say, 20 microns? In particular, is it possible to do this for a beam incident on an infinite array (modeled using floquet boundary conditions)?</p> <p>Essentially, I want to model an array of resonators on a substrate with an incident beam that only covers an area of radius 20 microns, similar to what would be the case in a physical measurement, and measure the transmittance through the substrate.</p> Tue, 26 May 2020 01:10:34 +0000 4.2020-05-26 01:10:34.260392 Cell Periodicity Equation https://www.comsol.com/forum/thread/260391/cell-periodicity-equation?last=2020-05-25T19:34:31Z <p>Hi everyone, I'd like to use the cell periodicity boundary condition for my problem, but I have some issues understanding the equation and unfortunately, it is not specified in the User's manual:</p> <p>u_dst = u_src + e_avg * (r_dst - r_src)</p> <p>The first part is clear, but what does the second part after the plus sign stand for. I think, it must be about preventing rigid body motion, but I don't understand, what e_avg stands for. Does anyone know, what e_avg stands for here?</p> <p>Thanks Simon</p> Mon, 25 May 2020 19:20:26 +0000 4.2020-05-25 19:20:26.260391 Modelling Current Flow with AC/DC Module https://www.comsol.com/forum/thread/260382/modelling-current-flow-with-ac-dc-module?last=2020-05-25T16:26:27Z <p>Hi!</p> <p>I'm new to Comsol, and I'm trying to model current flow and resistance through a resistive medium (eventually, the resistance will fluctuate, and the study will be time-dependent). However, my study only outputs electric potential values. Is there a way to change the output variables/ make an equation-based output to solve for current based on resistance and electric potential, for example?</p> <p>Thank you for reading my question!</p> Mon, 25 May 2020 16:13:07 +0000 4.2020-05-25 16:13:07.260382 Propagating light in an optical fiber https://www.comsol.com/forum/thread/260381/propagating-light-in-an-optical-fiber?last=2020-05-25T15:55:13Z <p>Hi all,</p> <p>I am trying to simulate light propagation (Gaussian or any arbitrary field) inside an optical fiber. I am unable to find the proper way to simulate it correctly using the wave optics module. Should it be using the beam envelope method? Where does the field is inserted analitically? using ports or maybe by the scattered boundaries?! If someone has a working simulation i would be happy to hear how it works.</p> <p>Thanks,</p> <p>Shlomi.</p> Mon, 25 May 2020 15:55:13 +0000 4.2020-05-25 15:55:13.260381 Difference between inflow and flux boundary conditions in TDS module. https://www.comsol.com/forum/thread/260372/difference-between-inflow-and-flux-boundary-conditions-in-tds-module?last=2020-05-25T15:37:05Z <p>What is exactly the difference between the conditions of inflow and flux in transport of dilute species? Is there any particular reason one would prefer either of them for some case?</p> Mon, 25 May 2020 15:37:05 +0000 4.2020-05-25 15:37:05.260372 Average function to turn frequency into time domain https://www.comsol.com/forum/thread/260371/average-function-to-turn-frequency-into-time-domain?last=2020-05-25T14:53:43Z <p>Hello, I have reached out to support about this and am still waiting for a response, but just in case someone here can help me: I have run a frequency domain study on a dielectric with two excitation voltages of two different frequencies to calculate the capacitance of the dielectric, Support then taught me how to use this equation to convert my voltage and current signals from frequency to time domain 'by mapping their phase information': <strong>aveop1(with(1,p))<em>cos(2</em>pi<em>f1</em>t)+aveop1(with(2,p))cos(2<em>pi</em>f2* t)</strong> with p as acoustic point sources in their example. I have managed to plot my voltage and current signals by using this equation, however I don't quite get how this equation means? I know that aveop is the average, but what does it have to do with the phase? Does someone maybe have an idea? I tried comparing the signals to my signals simulated using IFT and the current signals are not identical in terms of their phase, so I'm not quite sure which one is correct. It would definitely help my understanding if someone would kindly explain what the equation (in bold) above does. Jun</p> Mon, 25 May 2020 14:53:43 +0000 4.2020-05-25 14:53:43.260371 okada type, friction between two surface https://www.comsol.com/forum/thread/260351/okada-type-friction-between-two-surface?last=2020-05-25T19:06:11Z <p>Hi all.</p> <p>I need some suggestions about how to model a dislocation source (Okada type). The source is modeled as a couple of contact boundaries. I want to put some friction on the surfaces and I add the friction node (under the contact node in Solid mechanics). After, I planned to made a comparison with the results from a similar model to see if they are comparable. From Comsol, gotthe same results only if I choose a value friction = 4. But from what I understood the friction coefficient should between 0 and 1 in Comsol, as 1 should be the maximum friction. Can somebody explain this to me? I tried to see something more in the knowledge base but coudn't find anything. Maybe I am not considering some general conditions??</p> <p>Thanks to all who can help me. Chiara</p> Mon, 25 May 2020 11:02:09 +0000 4.2020-05-25 11:02:09.260351 Create a hole on a surface during simulation https://www.comsol.com/forum/thread/260331/create-a-hole-on-a-surface-during-simulation?last=2020-05-25T08:34:58Z <p>Hello dear,</p> <p>is it possible to create holes in the model during the simulation? I would like to simulate the temperature input during laser drilling and it would be very helpful to create the model with the corresponding hole after drilling so that this is taken into account for the second hole.</p> <p>Is there an application for this in Comsol? Or is there an alternative how I can display the borehole after drilling?</p> <p>Thank you and greetings Marcel Jung</p> Mon, 25 May 2020 08:34:58 +0000 4.2020-05-25 08:34:58.260331 Pressure transmission between microdomes that are tangent to one another https://www.comsol.com/forum/thread/260323/pressure-transmission-between-microdomes-that-are-tangent-to-one-another?last=2020-05-25T19:22:06Z <p>Hello! I've encountered some problems with pressure transmission between microdomes that are tangent to one another. There seems to be a problem with the meshing at the contact points. I've tried form assembly and contact pair generation but the solution will not converge. The details of my problem are summarised in this google site: <a href="https://sites.google.com/view/comsol-troubleshooting/home?authuser=2">https://sites.google.com/view/comsol-troubleshooting/home?authuser=2</a></p> Mon, 25 May 2020 06:46:21 +0000 4.2020-05-25 06:46:21.260323 External Current Density in MF module and MEF module https://www.comsol.com/forum/thread/260313/external-current-density-in-mf-module-and-mef-module?last=2020-05-25T03:38:38Z <p>Hello!</p> <p>Different magnetic fields are obtained using External Current Density in MF module and MEF module.</p> <p>I tried to add some currents (which come from MATLAB codes ) in a conductor sphere to obtain the induced magnetic field in space. I used MF module and MEF module ,respectively. But the magnitudes of the two magnetic fields are quite different.</p> <p>So does anybody know why ?</p> <p>How does COMSOL process external current density?</p> <p>Thanks for reading.</p> Mon, 25 May 2020 03:37:54 +0000 4.2020-05-25 03:37:54.260313 Problem With Simple Conductive Heat Transfer Simulation https://www.comsol.com/forum/thread/260312/problem-with-simple-conductive-heat-transfer-simulation?last=2020-05-25T04:54:43Z <p>Hello, COMSOL noob here attempting a senior design project for school.</p> <p>I am attempting to model a very simple block of Mo (at room temp) placed onto a heated stage (at 250degC). However, my solution becomes a uniform temperature of 250degC immediately.</p> <p>To combat consistent intialization I have a ramp starting at 0 for the temperature boundary (<a href="https://www.comsol.com/forum/thread/108101/time-dependent-temperature?last=2016-03-31T06:37:28Z">as suggested here</a>). This just yields the entire model heating uniformly along the ramp.</p> <p>Any pointers would be greatly appreciated!</p> <p>I've attached my program. I have tried different types of boundaries and added/removed geometries but to no avail. I have also tried changing the heat convection parameters but they have no effect. I have also tried a swept mesh but it yields a gradient in the wrong axis. Excuse any dumb mistakes, my team and I are learning this program on the fly.</p> <p>Thank you so much in advance!</p> Mon, 25 May 2020 00:49:58 +0000 4.2020-05-25 00:49:58.260312 summation of a matrix (vector) https://www.comsol.com/forum/thread/260311/summation-of-a-matrix-vector?last=2020-05-24T21:25:20Z <p>The below post is related to an <a href="//www.comsol.com//forum/thread/106131/cummulative-sum-in-comsol">archived discussion</a></p> <hr /> <p>[I want to calculate sum(A.B) where both A and B are matrix (vector) and also they are coefficient of a PDE (sum(A(p,t).<em>B(p,t))</em>dp/dt=f(p,t) Thank you for your help and time]</p> Sun, 24 May 2020 21:25:20 +0000 4.2020-05-24 21:25:20.260311 Evaluate value in every single mesh element https://www.comsol.com/forum/thread/260301/evaluate-value-in-every-single-mesh-element?last=2020-05-24T16:59:16Z <p>Hello everyone!</p> <p>I would like to know if it is possible to evaluate value (for example surface velocity) in every single element that composes a mesh of a 2D structure?</p> <p>Thank you in advance.</p> Sun, 24 May 2020 16:59:16 +0000 4.2020-05-24 16:59:16.260301 Mismatch about strain-stress and Young's modulus https://www.comsol.com/forum/thread/260293/mismatch-about-strain-stress-and-young-s-modulus?last=2020-05-25T19:37:30Z <p>Hi everyone!</p> <p>I am doing structural analysis and I got a problem about the stress and strain. For a simple solid cube, the input Young's modulus is 2.8GPa while the obtained ratio of stress and strain is 3.2GPa. The stress is integration of tensor of the loading surface; the strain is ratio of the displacement and original length.</p> <p>Is my extraction about stress and strain wrong or any missing about the model ?</p> <p>Thanks in advance.</p> Sun, 24 May 2020 15:35:26 +0000 4.2020-05-24 15:35:26.260293