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With stabilization (mesh b) 2. With stabilization (mesh a) 1. No stabilization (mesh a) F-bar method PERFORMANCE OF ASSUMED STRAIN ANALYSES ½ ÆÁÆÌ Â ´ µØÖ Æ ´ µ ½ ´ µ Æ Ú ¯ ¿ ØÖ Â Â Æ · ´½ ¯µ ½ Â Æ ½ ¿ ½ Æ · Ë ÒØ ª È Ö´ ª Ú ÈÖ ´ Ì µ¡ µÄÒ ¡ Æ ÆÚ ÖÒ ÖÒ ÎÒ ª ÎÒ È Ö ´ Ú µ ¡ ÎÒ 32 Materials Process Design and Control Group Cornell University ÖÒ¡ ÄÌÒ Modified sensitivity weak form ( stabilized F-bar method ) ½ ¯ ¿ ¾ Æ Sensitivity of the assumed deformation gradient TREATMENT OF INCOMPRESSIBILITY Å ØØÑ Ò Ûµ Æ ½ Ë Å Ò Ûµ º ´ º ËÓÐÚ Ø ØÓ Ñ × × Ò× Ø Ú ØÝ Ë Ð × ØÓ Ì × × Ø × ÒÓÛÒ ÓÖÖ ¹ ´ µ º Ø Ñ Ò Ö Ñ ÒØ ØÒ ØÒ·½ Ñ × 33 Materials Process Design and Control Group Cornell University ÓÒ Ø ÒÓÖÑ Ð Ò Ø Ò ÒØ Ð ÓÒØ Ø ØÖ Ø ÓÒ× × Ò× Ø Ú Ø × Å ÓÐ ØÒº ÍÄ × Ò× ¹ Ö Ø ÖÞ ×Ø ÒØ ÒÓ × Ó Ñ × Ò ÓÒ ÑÙ×Ø ØÖ Ò× Ö Ø × Ú Ö ÓÖÑ Ø ÓÒ ÔÖÓ Ð Ñ ÓÖ Ø ÆÒ ÌÒ µ Ö ÔÖ × ÒØ Ø ÉÒ µ × Ö Ø Þ Ø ÓÒ ÓÐ µ ´ ÓÖÑ Ø ÓÒ × Ò× Ø Ú ØÝ Ø Ø Ñ Ð ×Ø Ø ÓÑÔÙØ Ø ÓÒ Ó ×× ÖÝ × Ø Ó Ú Ö ÒØ ´ ÄÒÒ Ûµ ØÒ ØÒ·½ º × Ò× Ø Ú Ø × Ó Ø Ø Ñ Ò Ö Ñ ÒØ ÔÔÖÓÔÖ Ø ØÖ Ò× Ö ÓÔ Ö ØÓÖ× Æ ´ÓÐ Ò ÓÖÖ ×ÔÓÒ Ò Æ ´Ò Ûµ Æ ´ÓÐ µ ¿ Ò Ò Æ Æ Ò ´ Æ ØØÑ º Ä Ø ¾ Ë ÉÒ Æ ´Ò Û µ Ò Û Ñ × Ù× Ò Ñ × ×ÔÓÒ Ò ØÓ Ø Ø Ñ Ø Ö Ð ×ØÓÖÝ Ó Ø Ò Å × ØÖ Ò× Ö × Ù× × Ò× Ø Ú ØÝ Ó Ø ÓÐ µ ´ ÜÆ Ò ¿º Ä Ø ÉÒ Ö ÔÖ × ÒØ Ø ´ ÜÆ Ò ØÒ º Ì ÓÖÑ Ø ÓÒ ÔÖÓ Ð Ñ ÓÖ Ø Ò ÔÔÖÓÔÖ Ø ØÖ Ò× Ö ÓÔ Ö ØÓÖ ØÓ ÓÑÔÙØ Ø Ò Ûµ Ö Ø Ø Ú ØÝ Ò ÐÝ× ×º ¾º Í× ´ ½º ËÓÐÚ Ø DATA TRANSFER FOR SENSITIVITY PROBLEM É ¬ É ´ ´ Ó ¬ ØÓ ¬ Ø Ó µ µ ¬ Ü´ Ü Ø · Æ ¬ · ¡¬ ¬ µ ¼ µ 34 Materials Process Design and Control Group Cornell University Ø Ü· Ü ¬ ¬ Æ Ò Ö ÓÖÑ Ò ×Ø Ü· ÜÆ Ü´ · ¡¬ ¬ Ü Design sensitivity of the current forming stage due to variations in parameters of the current forming stage MULTISTAGE SENSITIVITY ANALYSIS É · · É Æ Æ ¬ É É ´ ´ Ó Ø · µ µ · ¼ Ó Æ · ¡¬ µ ¬ · ¡¬ µ Æ Á · ÄÓ Ø Ó ¬ ØÓ ¬ Ó ¬ Ø · ¡¬ ¬ É ´ ´ Ó ¬ ¬ · Ü´ Ü Ø ¬ ¬ Æ Ø ¼ ¬ ¬ µ · ¡¬ µ 35 Materials Process Design and Control Group Cornell University · Æ Ü· Ü Ü· ÜÆ Ü´ Æ Ò Ö ÓÖÑ Ò ×Ø Ü Design sensitivity of the current forming stage due to variations in design parameters of previous forming stages MULTISTAGE SENSITIVITY ANALYSIS t @ 2 B o Q) 64 @ (@ B o) @ (@ B o ) @ Y (Y Y ]+ 3 7 5 :: :: :: 0 :: 0 0 0 :: :: :: :: q q M q M q 9 > > > > > > > > > > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > > > > > > > > > > = 8 9 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < = > > > > > :: > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > : (Y @ Y Y ] 3 7 5 th q stage Materials Process Design and Control Group Cornell University 36 and properties after q stage βq is the design space of the th Λ represents preform shape q 2 B o Q) 64 @ Qi @ Qi t @ q = [ 1 ..

With stabilization (mesh b) 2. With stabilization (mesh a) 1. No stabilization (mesh a) F-bar method PERFORMANCE OF ASSUMED STRAIN ANALYSES ½ ÆÁÆÌ Â ´ µØÖ Æ ´ µ ½ ´ µ Æ Ú ¯ ¿ ØÖ Â Â Æ · ´½ ¯µ ½ Â Æ ½ ¿ ½ Æ · Ë ÒØ ª È Ö´ ª Ú ÈÖ ´ Ì µ¡ µÄÒ ¡ Æ ÆÚ ÖÒ ÖÒ ÎÒ ª ÎÒ È Ö ´ Ú µ ¡ ÎÒ 32 Materials Process Design and Control Group Cornell University ÖÒ¡ ÄÌÒ Modified sensitivity weak form ( stabilized F-bar method ) ½ ¯ ¿ ¾ Æ Sensitivity of the assumed deformation gradient TREATMENT OF INCOMPRESSIBILITY Å ØØÑ Ò Ûµ Æ ½ Ë Å Ò Ûµ º ´ º ËÓÐÚ Ø ØÓ Ñ × × Ò× Ø Ú ØÝ Ë Ð × ØÓ Ì × × Ø × ÒÓÛÒ ÓÖÖ ¹ ´ µ º Ø Ñ Ò Ö Ñ ÒØ ØÒ ØÒ·½ Ñ × 33 Materials Process Design and Control Group Cornell University ÓÒ Ø ÒÓÖÑ Ð Ò Ø Ò ÒØ Ð ÓÒØ Ø ØÖ Ø ÓÒ× × Ò× Ø Ú Ø × Å ÓÐ ØÒº ÍÄ × Ò× ¹ Ö Ø ÖÞ ×Ø ÒØ ÒÓ × Ó Ñ × Ò ÓÒ ÑÙ×Ø ØÖ Ò× Ö Ø × Ú Ö ÓÖÑ Ø ÓÒ ÔÖÓ Ð Ñ ÓÖ Ø ÆÒ ÌÒ µ Ö ÔÖ × ÒØ Ø ÉÒ µ × Ö Ø Þ Ø ÓÒ ÓÐ µ ´ ÓÖÑ Ø ÓÒ × Ò× Ø Ú ØÝ Ø Ø Ñ Ð ×Ø Ø ÓÑÔÙØ Ø ÓÒ Ó ×× ÖÝ × Ø Ó Ú Ö ÒØ ´ ÄÒÒ Ûµ ØÒ ØÒ·½ º × Ò× Ø Ú Ø × Ó Ø Ø Ñ Ò Ö Ñ ÒØ ÔÔÖÓÔÖ Ø ØÖ Ò× Ö ÓÔ Ö ØÓÖ× Æ ´ÓÐ Ò ÓÖÖ ×ÔÓÒ Ò Æ ´Ò Ûµ Æ ´ÓÐ µ ¿ Ò Ò Æ Æ Ò ´ Æ ØØÑ º Ä Ø ¾ Ë ÉÒ Æ ´Ò Û µ Ò Û Ñ × Ù× Ò Ñ × ×ÔÓÒ Ò ØÓ Ø Ø Ñ Ø Ö Ð ×ØÓÖÝ Ó Ø Ò Å × ØÖ Ò× Ö × Ù× × Ò× Ø Ú ØÝ Ó Ø ÓÐ µ ´ ÜÆ Ò ¿º Ä Ø ÉÒ Ö ÔÖ × ÒØ Ø ´ ÜÆ Ò ØÒ º Ì ÓÖÑ Ø ÓÒ ÔÖÓ Ð Ñ ÓÖ Ø Ò ÔÔÖÓÔÖ Ø ØÖ Ò× Ö ÓÔ Ö ØÓÖ ØÓ ÓÑÔÙØ Ø Ò Ûµ Ö Ø Ø Ú ØÝ Ò ÐÝ× ×º ¾º Í× ´ ½º ËÓÐÚ Ø DATA TRANSFER FOR SENSITIVITY PROBLEM É ¬ É ´ ´ Ó ¬ ØÓ ¬ Ø Ó µ µ ¬ Ü´ Ü Ø · Æ ¬ · ¡¬ ¬ µ ¼ µ 34 Materials Process Design and Control Group Cornell University Ø Ü· Ü ¬ ¬ Æ Ò Ö ÓÖÑ Ò ×Ø Ü· ÜÆ Ü´ · ¡¬ ¬ Ü Design sensitivity of the current forming stage due to variations in parameters of the current forming stage MULTISTAGE SENSITIVITY ANALYSIS É · · É Æ Æ ¬ É É ´ ´ Ó Ø · µ µ · ¼ Ó Æ · ¡¬ µ ¬ · ¡¬ µ Æ Á · ÄÓ Ø Ó ¬ ØÓ ¬ Ó ¬ Ø · ¡¬ ¬ É ´ ´ Ó ¬ ¬ · Ü´ Ü Ø ¬ ¬ Æ Ø ¼ ¬ ¬ µ · ¡¬ µ 35 Materials Process Design and Control Group Cornell University · Æ Ü· Ü Ü· ÜÆ Ü´ Æ Ò Ö ÓÖÑ Ò ×Ø Ü Design sensitivity of the current forming stage due to variations in design parameters of previous forming stages MULTISTAGE SENSITIVITY ANALYSIS t @ 2 B o Q) 64 @ (@ B o) @ (@ B o ) @ Y (Y Y ]+ 3 7 5 :: :: :: 0 :: 0 0 0 :: :: :: :: q q M q M q 9 > > > > > > > > > > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > > > > > > > > > > = 8 9 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < = > > > > > :: > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > : (Y @ Y Y ] 3 7 5 th q stage Materials Process Design and Control Group Cornell University 36 and properties after q stage βq is the design space of the th Λ represents preform shape q 2 B o Q) 64 @ Qi @ Qi t @ q = [ 1 ..

28 Materials Process Design and Control Group Cornell University ¼ ¬Ô · ¡¬ Ô µ ¬Ôµ Ø Æ Ü· Ü Æ Ü´ÜÒ · Ü Ò Ü Ü Parameter sensitivity in the time increment [n,n+1] UL SENSITIVITY FORMULATION · Æ Ê · Ê ´ ¬× · ¡¬×µ Æ Æ Á · ÄÓ Ê· Ê ´ ¬× µ Ó Ó ¼ Æ ÜÒ Ò ØÒ Ø Æ Ò ÜÒ· ÜÒ Á · ÄÒ É ´ Ü´ · ´ · É Ò · ÉÒ Æ Ò ¼ Ò ¬× µ ¬× µ É ÜÒ · ÜÆ Ò Ü´ Ò· Ò É Ò ÜÒ Æ Ü Ü Ü´ÜÒ Ü· ÜÆ ¬ × · ¡¬ × µ ØÒ ¬ × · ¡¬ × µ Ò Ø Ö ¼ Ø Ø ¬× · ¡¬× µ ¬× µ 29 Materials Process Design and Control Group Cornell University Æ Æ Ö· Ü· ÜÆ Ü´ÜÒ · ÜÆ Ò Ö Shape sensitivity in the time increment [n,n+1] UL SENSITIVITY FORMULATION Ò´ Æ Æ Æ Ö Æ Æ Ö ÖÒ Ü Ò Ò · Ö Æ Ò Ö ÄÒ È ÖÄÌÒ ¡ ÖÒ ÎÒ 30 Materials Process Design and Control Group Cornell University Ò ÖÒ ÜÆ µ ÄÒ¡ ´ÒªÒµ ¡ Kinematic relationships Ò ÎÒ È Ö ÖÒ¡ ÄÌÒ ¡ ÎÒ Æ È Ö ¡ ÖÒ ¼ ÖÒ¡ È Ö · Ö Æ Æ Weak form for shape and parameter sensitivities UL SENSITIVITY FORMULATION Ò Ú ÚÓÐ ÚÓÐ µ ÓÖÑ ´ÙÒ×Ø Ð µ ÓÒ¬ ÙÖ Ø ÓÒ Extension of the F-bar method by Owen (96) Ê ÖÒ ÓÒ¬ ÙÖ Ø ÓÒ ÓÖÑ ´×Ø «»ÐÓ ÓÒ¬ ÙÖ Ø ÓÒ B-bar method 31 Materials Process Design and Control Group Cornell University 3.

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A Continuum Sensivity Analysis of Large Deformations with Applications to Metal Form Design by Akkaram S.


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