AIAA-2006-2066-173

18
47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Confere 1 - 4 May 2006, Newport, Rhode Island AIAA 2006-2066 Copyright © 2006 by Jeroen A.S. Witteveen, Hester Bijl. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

description

Paper about Polynomial Chaos

Transcript of AIAA-2006-2066-173

Page 1: AIAA-2006-2066-173

��������� � ������ � ����������������������� �"!#�%$%���&�'$(�)+* ���,$-��./�0�1$-�&�� ��� 2�3�4 �&!5� ��67��$-� 89 � �&����!5���3��$%�&!#�3:

;=<&>@?A<&B�C�DFEGDIHKJML@LN<�O�<�<&BQPSRTBIUWV0<&XYLN<&>[Z\J^]`_bacedYf gSheikjmlonpdYqSrSlohts0uvgxwmdSySz�j=u{f|u~}{sN�%w�zmd��ed�h�z�dYq�f|�{j=�Nr

���1�S�����%�����A�{�-���T���&���������@�����F�������Q�����|�=�=���x�F���Q�Y�=�������S�x�����@�{���S�&���m���1�T�T�������| ��@�����F�������¡������¢T£�|�&�[�¤����¥����F¥T���=¦§�=���������=�{���`�|���F�������©¨=�����N�����S��¨kª¤���©���¤«%��¨=���F�0���Y¬T�=��«-�F��¨=¦&��¨0���=���­�=�������F�=�{�&�Y�F���F���®�@����`�����S�¯¨��±°¯�N���F�S�F���0�M���"�&�����F�=�F�=¦²�����F�T�=���x�F�&�­�`�=�����@ª�³m���0�¯�=�������F�=�{�&�Y�F���F���"�������=�F�"���´���S�S�����=¦��{��`���=�µ�F��¨ �S���e���©���T�T�&���F�����@¶·�����=�{�§���T�����e�&���T���������T�����Y�F���F���0�[��¬��x¨��F�����@�������µ�����`�&���F���¸�&�®�����|�=�=��£�[�F���­�Y�=�&���"�{¥����'�o�[���������F¢µ�|��ª,¹º�=���������Y�=���0������¨������@�����S��¨��M���x�����µ�F�T���=¦ �����F�T�=���x�F�&���Y�=�&���0�S��S�{¥��{���������m�A���#���\�=�������F�=�{�&�Y�F���F�����S�=�&�1�@�&��¢A�´�o�[�������S�&�m�¡��� �������@�S���@���3������¢µ�|�&�[�@ª¼»\�������F�=�{�&��~�Y�&���S�M���`�[�&���|�����\���¤�����@�{���S���F�´���=���=�e�=�&�����[�{�S�{�Y�@¶k�A���F�=�T���x�F�&�(�=���=�����=�{�&�Y�F�S�F��������¨½�=���T�A���F�=�T�&£�[�F���%�T���������T�����Y�F���F���1���m¥����F¥=�F�=¦'�����`�����S�&��� ¥����Y�F�&¢µ�|���5����� �S�S��¨��F��¨kª¾³���¥��{�����®�S���S�5������¢µ�|�&�[�5�&����`��������¨=�{����¨������@����¨����T¦��&¨=¥����@���F���T£v¨��|¿À�����F���Á�&��¨W�����[�µ�F���S�S�Y���@�S�=�����º�[���`�T�������@�5�S���S�x������¢µ�|�&�[�@ªÂ �=�����������F�S�3�����§¥����Y�| ���¨1¢T�1�@���[���&�Y���S����«-�F����à ���m�~�[Ä®�&�Y�F�¯���o�"���F�����F���=�@ªÅmÆ Å�ÇGÈ�É�Ê©Ë®Ì�Í=È�Î�ÊÏÇ

ÐÒÑ5Ó�Ô�Õ�Ö�×�Ø~ÙpØ~Ú|Ô�ÑmÙ{ÛµÜ~Ó�ÚFÝYÑmÓYÝ`ÜGØ�ÞmÝYß~Ý�Ú�ܤÙ{ÑxÚ|ÑTÓ�ß~Ý`Ù�Ü�ÚFÑ�àºÚFÑ&Ø�ÝYß~Ý`Ü�Ø©Ú|ÑxÕ�Ô�á�ÝYÛFÚ|ÑmàeØ~Þ�Ý%Ú|Ñ�âm×�ÝYÑTÓ�Ý%Ô{ãÀÖ�Þ�ä�Ü�Ú�ÓYÙ{Û�×�ÑTÓ�ÝYß�Ø~Ù�Ú|Ñ�åØ�ÚFÝ`Ü\Ø~Ô�Ú|ÑmÓYß�Ý@Ù�Ü�Ý"Ø~Þ�Ý5ß�Ý`Û|Ú�Ù{æ�ÚFÛFÚ±Øvä²Ô�ã�Ó�Ô�Õ�Ö�×�Ø~Ù{Ø�ÚFÔ�ÑmÙ�Û¤Ö�ß�Ý@á�Ú�Ó�Ø�ÚFÔ�ÑTÜYç´è­×�Õ�ÝYß~Ô�×TÜeÕ�ÝYØ�Þ�Ô�á�Ü�ÞmÙNé�Ý[æ=ÝYÝ`Ñ´á�ÝYé�Ý`Û|Ô�ÖTÝ@á²Ø~Ôê ×mÙ{Ñ&Ø~Ú±ã�ä#Ø~Þ�Ý`Ü�Ý�×�ÑmÓYÝYß�Ø~Ù{ÚFÑ&Ø�ÚFÝ`Ü%ÚFÑ�ÓYÔ�Õ�Ö�×�ØSÙpØ~Ú|Ô�ÑmÙ{ÛIÖ�ß~Ô�æ�ÛFÝYÕ§Ü`ë�Ý�ç àmçFëµØ~Þ�ݧì#Ô�Ñ&Ø�Ý0í(Ù�ß�ÛFÔxÙ�Ö�Ö�ß~Ô�Ù�ÓSÞ�ë=Ø�Þ�ݧÖ=ÝYß�Ø�×�ß~æmÙ{Ø�ÚFÔ�ÑÕ�Ý�Ø~Þ�Ô�áµë^î~î~ïðî�ñ"Ø~Þ�Ý"Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛÏÓSÞmÙ�Ô�Ü%ÝYò�ÖTÙ{ÑmÜ�Ú|Ô�Ñ�ë ñSïtîvó"Ý�Ø~Ó�ç#ô¤Þ�Ý"Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛÏÓSÞmÙ�Ô�Ü%Ý�ò�ÖmÙ�ÑmÜ�Ú|Ô�Ñ�ÞTÙ�Ü-Ö�ß~Ôpé�Ý`Ñ1Ø~Ô¯æ=ÝÙ5Ü�×TÓYÓ�Ý@Ü�Ü�ã�×�ÛQÙ{ÖmÖ�ß�Ô&Ù�ÓSÞ#Ø~Ô#Ü�Ô�Û|é�Ýe×�ÑTÓ�ÝYß�Ø~Ù�Ú|Ñ&Øvä ê ×mÙ�Ñ�Ø~Ú±õ=ÓYÙpØ~Ú|Ô�Ñ1Ö�ß~Ô�æmÛ|Ý`Õ�Ü`çeô¤ÞmÝ\Ô�ß�ÚFà�ÚFÑmÙ�ÛÀÞmÔ�Õ�Ô�à�ÝYÑ�Ý`Ô�×mÜ�ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛÓSÞmÙ�Ô�ÜQÝYò�ÖTÙ{ÑmÜ�Ú|Ô�ѧÚFÜÏÖ�ß~Ô�Ö=Ô�Ü�Ý`á�æ�ä�ö�ÞTÙ{Ñ�Ý`Õ÷Ù�Ñmá"ø�ÖmÙ�Ñ�Ô�Ü`ç ñ0ÐbØ(ÚFÜÏæmÙ�Ü�Ý`á�Ô�ѧÞ�Ô�Õ�Ô�à�ÝYÑ�Ý`Ô�×mÜÏÓSÞmÙ{Ô&ÜIØ�Þ�Ý`Ô�ß~äeÔ�ã�ù'Ú|Ý`Ñ�ÝYß�î�úÙ{ÑTá1ÓYÙ�Ñ¡Ù�ÓSÞ�Ú|Ý`é�ÝeÝ�ò�ÖTÔ�Ñ�ÝYÑ&Ø~ÚFÙ�ÛIÓYÔ�Ñ�é�Ý`ß�à�ÝYÑmÓYÝ%ã�Ô�ß3ö%Ù{×TÜ�Ü�ÚFÙ�Ñ¡á�Ú�Ü�Ø�ß~Ú|æ�×�Ø�Ý`á1×�ÑmÓYÝYß�Ø~Ù{ÚFÑ1ÚFÑ�Ö�×�Ø%ÖTÙ{ßSÙ{Õ�Ý�Ø~ÝYßSÜYç%ÐbØ-ÝYÕ�Ö�ÛFÔpä�Üû­Ý`ß�Õ�Ú|Ø�Ý-Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�ÜGÚFÑxØ�Þ�Ý3ÓYÔ�Ñ&Ø�ÝYò�ؤÔ�ãÀÙ§ö%Ù{ÛFÝYß~ü�Ú|ÑxÜvØ~Ô�ÓSÞmÙ�Ü�Ø�Ú�Ó�õmÑ�Ú|Ø�Ý-ÝYÛFÝYÕ�Ý`Ñ�Ø®Ù�Ö�Ö�ß~Ô�Ù�ÓSÞ�çkô¤Þ�Ý-Õ�ÝYØ�Þ�Ô�áxÞmÙ�ܤæ=ÝYÝ`ÑÙ{ÖmÖ�Û|ÚFÝ`á�ÚFѽÓ�Ô�Õ�Ö�×�Ø~Ù{Ø�ÚFÔ�ÑmÙ�ÛÏÕ�Ý`ÓSÞTÙ{Ñ�Ú�ÓYÜ-æ�ä²ö�ÞmÙ{ÑmÝYÕýÙ{Ñmá½ø�ÖTÙ{Ñ�Ô&Ü�ñvþSú"ã�Ô�ßeö%Ù�×mÜ�Ü�Ú�Ù{ѲÙ{Ñmá�ÑmÝ`Ù{ß�åvö%Ù{×mÜ~Ü�Ú�Ù{Ñ�á�Ú�ÜvØ~ß�ÚFæ�×�åØ�ÚFÔ�ÑTÜYç·ù�Ù{Û|Ø�Ý`ß~Ü©Ù�Ñmá5û­×�ä�Ü�ÝNî�ÿSïtî��-ÞmÙNé�Ý%Ù{Ö�ÖmÛ|ÚFÝ`áxÖ=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛAÓSÞmÙ�Ô�ÜGØ�Ô§Ö�ß~Ô�æmÛ|Ý`Õ�ܤÚFÑxâm×�Ú�á#á�ä�ÑmÙ{Õ�Ú�ÓYÜ`ç��­Ñ5ÚFÕ\Ö=Ô�ß�Ø~Ù�Ñ&ØÝ�ò�Ø~ÝYÑmÜ�Ú|Ô�Ñ�Ô{ã(Ø�Þ�Ý"Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛÏÓSÞTÙ{Ô�Ü%Õ�ÝYØ�Þ�Ô�á�Ø�Ô#ß~Ý`Ù�ÓSÞ�Ý�ò�ÖTÔ�Ñ�ÝYÑ&Ø~ÚFÙ�ÛÏÓ�Ô�Ñ&é�ÝYß~à�Ý`ÑmÓ�Ýeã�Ô�ߺÜ�ÝYé�ÝYßSÙ{ÛÀØvä�ÖTÝ@Ü3Ô�ã¤Ü�Ø~Ù�Ñmá�Ù{ßSáá�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�ÑmÜ%ÞmÙ�Ü%æTÝ`ÝYÑ�Ú|Ñ&Ø~ß�Ô�á�×mÓYÝ`á�æ�ä��®ÚFײÙ{Ñmá��-Ù{ß~Ñ�Ú�Ù�á�Ù�ü�ÚFÜ`ç^î�ó¸ÐbØ3ÞTÙ�Ü%æ=ÝYÝYѲÙ�Ö�Ö�ÛFÚ|Ý@á¯Ø~Ô5âmÔ�/Ö�ß~Ô�æmÛ|Ý`Õ�Ü`ëµâm×mÚFá�åÜ�Ø�ß~×mÓ�Ø~×�ß�Ý®Ú|Ñ&Ø�Ý`ß~Ù�Ó�Ø~Ú|Ô�ÑmÜYë&á�Ú�=×mÜ�ÚFÔ�Ñ�ë&Þ�Ý`Ù{ØGÓ�Ô�Ñmá�×mÓ�Ø�ÚFÔ�Ñ"Ù�Ñmá§Ù�á�é�Ý`Ó�Ø�ÚFÔ�Ñ�Ö�ß~Ô�æ�ÛFÝYÕ§Ü`ç ó�ï ��ïtî��vþSñ~ñ��­Ý`Ó�Ý`Ñ&Ø�ÛFä�ë�Ø�Þ�Ý�Ý�ò�ÖTÔ�Ñ�ÝYÑ&Ø~ÚFÙ�ÛÓ�Ô�Ñ�é�ÝYß~à�Ý`ÑmÓ�ÝÏÔ{ã�Ø~Þ�ݤÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û�ÓSÞmÙ�Ô�ÜkÕ�Ý�Ø�ÞmÔ�áeÞmÙ�Ükæ=ÝYÝ`ÑeÝYò�Ø�ÝYÑTá�Ý`áºØ�Ô�×�ÑmÓYÝYß�Ø~Ù{ÚFѺÖTÙ{ßSÙ{Õ�Ý�Ø~ÝYßSÜ��©Ú|Ø�Þ�Ù{ß~æ�Ú|Ø�ßSÙ{ß~ä�ÚFÑ�Ö�×�Øá�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�ÑmÜ`ç^î��Sïtî��/ô¤Þ�Ý"Û�ÙpØ�Ø�Ý`ߺÖ=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛÏÓSÞmÙ�Ô�Ü%Ý�ò�Ø~ÝYÑmÜ�Ú|Ô�ÑmÜ3Ù�ß�ݧæmÙ�Ü�Ý@á�Ô�Ñ�Ø�ÞmÝ"Ñ�×�Õ�ÝYß~ÚFÓ`Ù{ÛÏÓYÔ�Õ�Ö�×�ØSÙpØ�ÚFÔ�ѲÔ�ã¤Ù�ÑÔ�Ö�Ø�ÚFÕ�Ù�Û(Ü�Ý�Ø\Ô{ã­Ô�ß�Ø~Þ�Ô�à�Ô�ÑmÙ�Û·æmÙ�Ü�Ú�Ü3Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛFÜe×mÜ�Ú|Ñmà¡Ù{ѲÔ�ß�Ø~Þ�Ô�à�Ô�ÑmÙ�Û|Ú��`Ù{Ø�ÚFÔ�ѲÙ{ÛFà�Ô�ß�Ú|Ø�ÞmÕxç¡è­Ô�Ñ�åtÚFÑ&Ø�ß~×mÜ�ÚFé�Ý0ÖTÔ�Û|ä�Ñ�Ô�åÕ�ÚFÙ�Û·ÓSÞmÙ{Ô&Ü-Ù{Ö�Ömß�Ô&Ù�ÓSÞ�Ý@Ü��ºÞmÙNé�Ý�Ù{Û�Ü�Ô5æ=ÝYÝ`Ñ�á�ÝYé�Ý`Û|Ô�ÖTÝ@áµç§ô¤Þ�Ý"Ó�Ô�Õ�Ö�×�Ø~Ù{Ø�ÚFÔ�ÑmÙ�Û·Ó�Ô&ÜvØSÜ%Ô{ãÏØ~Þ�Ý"Ü�Ø~Ù{ÑTá�Ù{ßSá1ÑmÔ�Ñ�åbÚ|Ñ&Ø~ß�×mÜ�ÚFé�ÝÙ{ÖmÖ�ß�Ô&Ù�ÓSÞ¯Ó`Ù{ѯÚFÑmÓ�ß~Ý`Ù�Ü�Ýeß~Ù�Ö�Ú�á�Û|ä��©Ú|Ø�Þ1Ø�Þ�Ý\Ñ�×�Õºæ=ÝYß%Ô{ã·×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ1ÚFÑ�Ö�×�Ø%ÖmÙ{ßSÙ{Õ�ÝYØ�ÝYßSÜ­×mÖ1Ø~Ô"Ø�ÞmÝeÖ=Ô�ÚFÑ&Ø�Ø~ÞmÙpØ-ì#Ô�Ñ&Ø~Ýí(Ù{ß~ÛFÔ5Ü�ÚFÕº×�Û�ÙpØ~Ú|Ô�Ñ�æ=Ý`ÓYÔ�Õ�Ý`ܺÙ#Õ�Ô�ß~Ý\Ý��"Ó�ÚFÝYÑ&ØeÙ{ÖmÖ�ß�Ô&Ù�ÓSÞ�ç î~ï�ñ~ÿ û­Ý`ß�Ý�Ø~Þ�Ý[ö%Ù{ÛFÝYß~ü�Ú|Ñ�Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û·ÓSÞmÙ�Ô�Ü3Ù�Ö�Ö�ß~Ô�Ù�ÓSÞ¡ÚFÜ�������! #"%$� &$�'("&)*�+�! , &-/.�.102"43&$�5�6�783%�/$9�:$�)*�/;/02<!02=(7�>02-/;/5'�3%�!02;�?@�BADCB'(./.�<!�!$�5� &)��!$�;/)�$95/� E@�! &�!02;+0(F�G�A�HI'(;�5�3%�/$J3&$�)*��;/02<!02=(7./"%02=2"*'(KLKL$M0(F:3%�/$�NJ�!;/�! 43&"47,0(FPOQ)�02;/02KL�!)SR�TU'(�!"% �VW�X �UV YZV� %3%-/5/$�;[3�C\YM$�.'("43&KL$�;[3]0(F^R_$�"%02 &.'()�$`Oa;/=2�!;/$�$�"&�!;/=/C X V HZVcbd0�egf2h2f2i�C\j2k2h2hmlnboYM$�< F�3�Cp�q�/$rG_$s3%�/$�"&<t'(;/5� �Cu V '�V �V v�� 3%3%$sE[$�$�;w_3&-/5�$�< Fx3�V ;/<yC1?�3%-/5/$�;[3nNJ$�Kz6{$�"qRM|}R_RZV~ >-/<!< X "%0(F�$� & %02"�C1YM$�.'("43&KL$�;[3n0(FQRM$�"&02 %.'()�$zOQ;�=2�!;/$�$�"%�!;/=/C X V HZV1bd0�e9f2h2f2i�CUj2k2h2h�lnb�YM$�< Fx3�C1�q�/$ZG_$s3%�/$�"&<t'(;/5/ �C1�UV 6/� u <xw_3&-/5�$�< Fx3�V ;/<yCNJ$�Kz6{$�"BRM|}R_RZV

� 0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�

47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Confere1 - 4 May 2006, Newport, Rhode Island

AIAA 2006-2066

Copyright © 2006 by Jeroen A.S. Witteveen, Hester Bijl. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Page 2: AIAA-2006-2066-173

Ó�Ô�ÑmÜ�ÚFá�Ý`ß�Ý@á\Ú|Ñ0ÓYÔ�Õeæ�Ú|ÑTÙpØ�ÚFÔ�Ñ��©Ú|Ø�Þ[Ù{ѧÔ�ß�Ø�ÞmÔ�à�Ô�ÑmÙ{ÛFÚ��`ÙpØ~Ú|Ô�Ñ\Ù�Û|à�Ô�ß~Ú±Ø~Þ�ÕÁã�Ô�ß·Ø~Þ�Ý®Ñ�×�Õ�ÝYß~Ú�ÓYÙ{ÛTÓ�Ô�Õ�Ö�×�Ø~ÙpØ~Ú|Ô�Ñ"Ô{ã�Ù�ѧÔ�Ö�Ø~Ú|Õ§Ù{ÛÜ�Ý�Ø­Ô{ãÀÔ�ß�Ø�Þ�Ô�à�Ô�ÑmÙ{Û�æmÙ�Ü�Ú�Ü(ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û�Ü`çô¤Þ�Ý5Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�Û©ÓSÞmÙ{Ô&ÜeÙ�Ö�Ö�ß~Ô�Ù�ÓSÞ�ÞmÙ�Ü�æ=Ý`ÓYÔ�Õ�ÝxÙ��(ÝYÛFÛ±åbü�Ñ�Ô�©Ñ Ù�Ñmá��©Ú�á�Ý`Û|ä�Ù�Ö�Ö�ÛFÚ|Ý@á�×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ�Øvä ê ×mÙ�Ñ&Ø�Ú|õTÓYÙ{åØ�ÚFÔ�Ñ´Õ�Ý�Ø~Þ�Ô�á ÚFÑ,Õ§Ù{Ñ�ä½æ�ßSÙ{ÑmÓSÞmÝ`Ü\Ô{ã%Ó�Ô�Õ�Ö�×�Ø~ÙpØ~Ú|Ô�ÑmÙ{Û­Ü~Ó�ÚFÝYÑTÓ�Ý#Ù{Ñmá ÝYÑmà�ÚFÑ�ÝYÝ`ß�ÚFÑ�àmç´ÐbØ"Ú�Ü�Ñ�Ô�¼ÚFÕ�ÖTÔ�ß�ØSÙ{Ñ&Ø�Ø�Ô²ÜvØ~×má�ä

Ü�ÖTÝ@Ó�Ú|õTÓ§Ù{ß~Ý`Ù�Ü�Ô�ã¤Ù�Ö�Ö�ÛFÚFÓ`ÙpØ~Ú|Ô�Ñ�Ô�ãGÖ=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛÏÓSÞmÙ�Ô�Ü��©Þ�Ú�ÓSÞ�ß�Ý ê ×mÚ|ß~ݧÓ�ÛFÔ�Ü�ÝYß3ÙpØ�Ø�Ý`Ñ�Ø~Ú|Ô�Ñ�ç+�mÔ�ß-Ý�ò�Ù�Õ\ÖmÛ|Ý�ëAÚ|Ø3ÚFÜ9�©Ú�á�Ý`Û|äÙ�ÓSü�Ñ�Ô�©ÛFÝ`á�à�Ý`á1Ø�ÞTÙpØeÑmÔ�Ñ�ÛFÚ|ÑmÝ`Ù{ß~Ú|Ø�ÚFÝ`Ü3Ó`Ù{ѽÓYÙ�×mÜ�Ý"á�Úx�"ÓY×�Û±Ø~Ú|Ý@Ü3ã�Ô�ßeÙ�Ö�Ö�ÛFä&ÚFÑ�à#Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛGÓSÞTÙ{Ô�Ü`ç î�ïtîs� ô¤Þ�Ú�Ü3Ú�ܺÙ#Ö=Ô�ÚFÑ&ØÔ{ã·ÚFÕ�ÖTÔ�ß�ØSÙ{ÑmÓYÝ�ë=Ü�ÚFÑmÓ�Ý\Ü�Ý`é�Ý`ß~Ù�Û�Øvä&Ö=Ý`Ü�Ô{ã·ÑmÔ�Ñ�ÛFÚ|ÑmÝ`Ù{ß~Ú|Ø�ÚFÝ`Ü­Ó`Ù{Ñ5ã�ß~Ý ê ×�ÝYÑ&Ø~Û|ä5æ=ݺÝYÑTÓ�Ô�×mÑ�Ø~ÝYß~Ý`á5ÚFÑ#Ö�ßSÙ�Ó�Ø~ÚFÓ`Ù{ÛÀÙ{Ö�Ö�ÛFÚ�ÓYÙpØ~Ú|Ô�ÑmÜ`çø�Ô�Õ\ݤÑ�Ô�Ñ�ÛFÚ|Ñ�Ý@Ù{ß~Ú±Ø~Ú|Ý@ÜÀß�Ý@Ü�×�Û|ØQÚ|Ñ�Ù�ÜvØ~ß�Ô�Ñ�à�Ý`ßIÓYÔ�×�ÖmÛ|Ý@áºÜ�ä�Ü�Ø�Ý`Õ/ÙpãoØ~ÝYßIØ~Þ�Ý©Ù{Ö�Ö�ÛFÚ�ÓYÙpØ~Ú|Ô�ÑeÔ�ãTÖ=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�ÓSÞmÙ{Ô&ÜYëpÔ�Ø�Þ�Ý`ßÀØvä&Ö=Ý`ÜÔ{ãµÑ�Ô�Ñ�ÛFÚ|Ñ�Ý@Ù{ß~Ú±Ø~Ú|Ý@ÜQÕ§Ù{ü�Ý­Ú±ØSÜÏá�ÚFß~Ý`Ó�Ø(Ù{Ö�Ö�ÛFÚ�ÓYÙpØ~Ú|Ô�ѧÝYé�ÝYÑ�Ú|Õ�Ö=Ô�Ü~Ü�ÚFæ�ÛFÝ�çS�©Ý@Ó�ÝYÑ&Ø~Û|ä�ë&Ü�ÝYé�ÝYßSÙ{Û�Õ�Ý�Ø~Þ�Ô�á�ÜÏÞTÙNé�Ý©æ=ÝYÝYÑ[á�ÝYé�ÝYÛFÔ�Ö=Ý`áã�Ô�ßeÝYépÙ{ÛF×mÙ{Ø�ÚFÑ�à¯Ñ�Ô�Ñ�ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�Û·ã�×�ÑmÓ�Ø�ÚFÔ�ÑmÜeÔ{ã©épÙ{ß~Ú�Ù{æ�ÛFÝ`Ü-ß~ÝYÖmß�Ý@Ü�Ý`Ñ�Ø~Ý`á�æ�ä�ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û(ÓSÞmÙ{Ô&Ü-Ý�ò�ÖmÙ�ÑmÜ�Ú|Ô�ÑmÜ�� î'ôÀÙNä�Û|Ô�ßÜ�ÝYß~Ú|Ý@Ü­Ý�ò�ÖmÙ�ÑmÜ�Ú|Ô�Ñ#Ô{ãQØ�ÞmÝeÑ�Ô�Ñ�Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�Û�ã�×�ÑTÓ�Ø�ÚFÔ�ÑAë=ÚFÑ&Ø�Ý`à�ßSÙpØ�ÚFÔ�Ñ#Ô�ã·Ú±ØSÜ�á�ÝYß~Ú|épÙpØ~Ú|é�ݺÙ{ÑTá¯Ù0Ü~Ù{Õ�Ö�ÛFÚ|Ñmà{åbæmÙ�Ü�Ý`á#Ü�ÖTÝ@Ó�Ø~ß~Ù�ÛÖ�ß~Ô{�vÝ@Ó�Ø�ÚFÔ�Ñ#Õ�ÝYØ�Þ�Ô�áµç%û­Ý`ß�Ý�ëmØ��(Ô"Ô{Ø~Þ�ÝYß�Ù{Ö�Ömß�Ô&Ù�ÓSÞ�Ý@ܤã�Ô�ß%Ù{Ö�ÖmÛ|ä�ÚFÑ�à"Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛkÓSÞmÙ�Ô�Ü­Ø�Ô"Ø~Þ�ß~ÝYݺÚFÕ�ÖTÔ�ß�ØSÙ{Ñ&Ø­Øvä�Ö=Ý`Ü�Ô{ãÑ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß~Ú±Ø~Ú|Ý@Ü©Ù{ß~Ý�á�Ú�Ü~Ó�×mÜ~Ü�Ý@áB�� ç®�ºupjTf loj=dS�{q-hðq��{j�r�g�upq2�§�phtlMupjmrºuvg9��j�y~dYq�hÒ�ploj¯loj{�Q��hB�m�pq~��§d�hÒd�qSr

è­Ô�ÑmÛ|ÚFÑ�Ý`Ù�ß­Ø~ß~Ù�ÑmÜvã�Ô�ß�Õ§Ù{Ø�ÚFÔ�ÑmÜ®Ô{ãÏÚFÑ�Ö�×�Ø%×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ&Øvä#á�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�ÑmÜ�ß�Ý@Ü�×�Û|Ø�ÚFÑ1à�ÝYÑ�Ý`ß~Ù�ÛAÚFÑ1Ñ�Ô�Ñ�åÒÜvØSÙ{Ñmá�Ù�ß~á1á�ÚFÜ�Ø�ß~Ú|åæ�×�Ø�ÚFÔ�ÑTÜYçS�IÔ�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�ÓSÞmÙ{Ô&Ü(Ý�ò�ÖmÙ{ÑTÜ�ÚFÔ�ÑmÜGæmÙ�Ü�Ý@á"Ô�Ñ5Ó�Û�Ù�Ü~Ü�ÚFÓ`Ù{Û=ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û�ܤÓYÙ�Ñ[ÚFÑ[à�ÝYÑ�Ý`ß~Ù�ÛTÑmÔ{Ø­Ù�ÓSÞ�ÚFÝYé�Ý�ÝYò�ÖTÔ�åÑ�ÝYÑ&Ø�Ú�Ù{Û�Ó�Ô�Ñ�é�ÝYß~à�Ý`ÑmÓ�Ý­ã�Ô�ßGØ�Þ�Ý@Ü�Ý�Ñ�Ô�Ñ�åbÜ�Ø~Ù�Ñmá�Ù{ßSá§á�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�ÑmÜ`çn�·ò�ÖTÔ�Ñ�ÝYÑ&Ø�Ú�Ù{Û�Ó�Ô�Ñ�é�ÝYß~à�Ý`ÑmÓ�Ý�ÓYÙ{Ñ0æTÝ�ß�ÝYØ~Ù�Ú|Ñ�Ý@á§ã�Ô�ßØ�Þ�Ú�Ü%Øvä�ÖTÝ�Ô�ã(Ñ�Ô�Ñ�ÛFÚ|Ñ�Ý@Ù{ß~Ú±Ø~Ú|Ý@Ü%æ�ä1Ó�Ô�Õ\Öm×�Ø�ÚFÑ�à#Ñ�×�Õ�ÝYß~ÚFÓ`Ù{ÛFÛ|ä¯Ù�Ñ�Ô�Ö�Ø~Ú|Õ§Ù{Û·Ü�Ý�Ø3Ô�ã(Ô�ß�Ø~Þ�Ô�à�Ô�ÑmÙ�ÛkæTÙ�Ü�ÚFÜ%Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�Üã�Ô�ßGØ�Þ�Ý%Ñ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ßGØ�ßSÙ{ÑmÜ�ã�Ô�ß~Õ§ÙpØ�ÚFÔ�ѧÔ�ãAØ~Þ�Ý�×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ0ÚFÑ�Ö�×�Ø©ÖmÙ�ß~Ù�Õ\ÝYØ�Ý`ß`çIô¤Þ�Ú�Ü(Ù�Ö�Ö�ß~Ô�Ù�ÓSÞ§ÚFܤá�Ý`Õ\Ô�ÑmÜ�Ø�ßSÙpØ�Ý@á§ã�Ô�ßÙeÕ§Ù�Ü�Ü�åbÜ�Ö�ß~Ú|ÑmàeÜ�ä�Ü�Ø�Ý`Õ Ù�Ñmá"Ø~Þ�Ý-Þ�Ý@ÙpØ©Ø�ßSÙ{ÑTÜvã�Ý`ß(Ömß�Ô�æ�Û|Ý`Õ ÚFÑ5ÙeØ��(Ô{åÒá�ÚFÕ\Ý`ÑmÜ�Ú|Ô�ÑmÙ{ÛµÓSÞmÙ�Ñ�Ñ�ÝYÛ�âTÔ�9�� ç��©upf spj�u�\lM�pfÀj=u{jmf loj�d~�{q�lohðlMd�r3lojmnpupf±n`loj�}+��j�ySd�q�hÒ�ploj¯np�pq�lM����fFd�r�IÔ�ÛFä&ÑmÔ�Õ�ÚFÙ�Û®Ñ�Ô�Ñ�ÛFÚ|Ñ�Ý@Ù{ß~Ú±Ø~Ú|Ý@Ü0Ô�ãeÙ{Ñ ×�ÑmÓYÝYß�Ø~Ù{ÚFÑ�épÙ�ß�Ú�Ù{æ�ÛFÝ1Ô�Ó`Ó�×�ßxÚ|ÑWÕ§Ù{Ñ�ä'Ñ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ßxÝ ê ×mÙpØ~Ú|Ô�ÑmÜ[Ô{ãºÖ�ßSÙ�Ó�Ø�Ú�ÓYÙ�ÛÚ|Õ�Ö=Ô�ß�Ø~Ù{ÑTÓ�Ý�ç©û®ÝYß~Ý�ë�Ù{Ñ#×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ#épÙ{ß~Ú�Ù{æ�ÛFÝ-ÚFÜ�Ù§×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ ê ×mÙ�Ñ�Ø~Ú±ØväxÔ{ãS�©Þ�Ú�ÓSÞ5Ø�Þmݺ×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ&Øvä0Ú�Ü�×�Ñ�ü�Ñ�Ô�©ÑÙ�Ü�Ô�Ö�Ö=Ô�Ü�Ý`á¯Ø�Ô5Ù{Ñ1×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ1ÖmÙ�ß~Ù�Õ\ÝYØ�Ý`ß��©Ú|Ø�Þ�Ù0ü�Ñ�Ô�©Ñ1×�ÑmÓ�Ý`ß�ØSÙ{ÚFÑ&Øvä¯á�ÚFÜ�Ø�ß~ÚFæ�×�Ø�ÚFÔ�ÑAç��IÔ�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛIÓSÞmÙ{Ô&Ü®ÓYÙ�Ñß�Ý@Ù�á�ÚFÛ|ä[æTÝ�Ù{ÖmÖ�Û|ÚFÝ`áxØ~Ô0Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛAÑmÔ�Ñ�ÛFÚ|ÑmÝ`Ù{ß~Ú|Ø�ÚFÝ`Ü­ÚFÑ�é�Ô�Û|é�ÚFÑ�à"Ù{ѯ×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ#éNÙ�ß�Ú�Ù{æmÛ|Ý�ç­û­Ô�(ÝYé�Ý`ß`ëTÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛÑ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ß�Ú|Ø�ÚFÝ`Ü(ÓYÙ�Ñ0Ù�Ý`Ó�Ø(Ø~Þ�Ý%Ó�Ô�×mÖ�Û|ÚFÑ�à\Ô{ã�Ø~Þ�Ý%Ü�ÝYØ(Ô�ãAÝ ê ×mÙ{Ø�ÚFÔ�ÑmܤÙ{ÑTá§Ø�ÞmÝ�ß~ÝYÛ�ÙpØ~Ú|é�Ý�Ù{Õ�Ô�×�Ñ&Ø(Ô�ãkÓ�Ô�Õ\Öm×�Ø~Ù{Ø�ÚFÔ�ÑmÙ�Û�GÔ�ß�ü¯Ô�ã¤Ø�Þ�Ý"Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛÏÓSÞTÙ{Ô�Ü%ÝYò�ÖmÙ{ÑmÜ�ÚFÔ�Ñ�ç5ô¤Þ�Ý`Ü�ݧÝ[�Ý`Ó�Ø~ÜeÙ{ß~Ý"ÜvØ~×má�ÚFÝ`á�æ�ä�ÓYÔ�Õ�ÖmÙ�ß�ÚFÑ�àxØ�Þ�Ý[Ù{Ö�Ö�ÛFÚ�ÓYÙpØ~Ú|Ô�Ñ�Ô{ãÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û�ÓSÞTÙ{Ô�ÜGØ~Ô�Ù\ÛFÚ|ÑmÝ`Ù{ß­Ù�ÑmáxÑ�Ô�ÑmÛ|ÚFÑ�Ý`Ù�ß©Ù�á�é�Ý`Ó�Ø~Ú|Ô�Ñ�åÒá�Úx�×mÜ�Ú|Ô�Ñ[Ömß�Ô�æ�Û|Ý`Õ��� ç®�ºupj{��u{f spj=u{�elM�{fAj�upjmf±loj=dS�pq�lohtlMd�r3lojTnNu{f n@loj&}���j=ySdYq�hb�{loj1np�pq�lM�1�Yf|d�r�IÔ�ÛFä&ÑmÔ�Õ�ÚFÙ�Û�ÓSÞmÙ{Ô&ÜQÓYÙ{ÑmÑ�Ô{Ø·á�ÚFß�Ý@Ó�Ø~Û|äeæTÝ®Ù�Ö�Ö�ÛFÚ|Ý@áeØ�Ô-Ñ�Ô�Ñ�ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�Û�ã�×�ÑmÓ�Ø�ÚFÔ�ÑmÜQÔ�ã�Ù�Ñ�×�ÑmÓ�Ý`ß�ØSÙ{ÚFÑ�éNÙ�ß�Ú�Ù{æmÛ|Ý�ë�Ü�ÚFÑmÓYÝØ�Þ�ݧö%Ù{ÛFÝYß~ü�Ú|ѯÖmß�Ô1�vÝ`Ó�Ø~Ú|Ô�Ñ1á�Ô�Ý`Ü�Ñ�Ô�Ø�ß~Ý`á�×mÓ�ݺØ�ÔxÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{ÛÀÚFÑ�Ñ�ÝYß%Ö�ß~Ô�á�×mÓ�Ø~Ü�ã�Ô�ß�Ø�Þ�Ý@Ü�Ý\Ñ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß�Ø�ÝYß~Õ§ÜYç��®ÑÙ{Ö�Ö�ß~Ô�Ù�ÓSÞ®Ø~Ô­Ø~ß�Ý@ÙpØAÝ ê ×TÙpØ�ÚFÔ�ÑTÜq�©Ú|Ø�Þ3Ñ�Ô�Ñ�Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�Ûpã�×�ÑmÓ�Ø�ÚFÔ�ÑmÜkÚ�Ü�Ø~Ô­Ñ�×�Õ�ÝYß~ÚFÓ`Ù{ÛFÛ|ä�Ú|Ñ&Ø~ÝYà�ßSÙpØ~ÝIØ�Þ�ÝGÚ|ÑmÑ�ÝYßkÖ�ß�Ô�á�×TÓ�Ø~ÜÔ{ã­Ø~Þ�Ý5Ñ�Ô�Ñ�Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛGã�×�ÑmÓ�Ø�ÚFÔ�ÑmÜ��©Ú|Ø�Þ,Ù1æTÙ�Ü�ÚFÜ\Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�Û¤ß�Ý@Ü�×�Û|Ø�ÚFÑ�à¡ã�ß�Ô�ÕýØ�Þ�ݯö%Ù�Û|Ý`ß�ü�ÚFѽÖ�ß~Ô{�vÝ`Ó�Ø�ÚFÔ�Ñ�ç��Ñ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ß(Ö=ÝYÑTá�×�ÛF×�Õ Ömß�Ô�æ�Û|Ý`Õ Ú�Ü©Ó�Ô�ÑTÜ�Ú�á�ÝYß~Ý`á0Ø�Ô"Ü�Ø�×Tá�ä§Ø�ÞmÝ3Ù�ÓYÓY×�ßSÙ�Ó�ä"Ù�Ñmá[Ø�Þ�Ý-Ý��"Ó�ÚFÝYÑmÓYä"Ô{ãÀØ�Þ�Ú�Ü©Ù{ÖmÖ�ß�Ô&Ù�ÓSÞ�ç

ô¤Þ�Ý-Ñ�×�Õ�ÝYß~Ú�ÓYÙ{Û�ß~Ý`Ü�×�Û|Ø~Ü(ã�Ô�ß­Ø�Þ�Ý%Ø~Ý`Ü�ؤÖmß�Ô�æ�Û|Ý`էܩÙ{ß~Ý�é�ÝYß~Ú±õTÝ`á[æ&ä[Ó�Ô�Õ�ÖmÙ�ß�Ú�Ü�Ô�Ñ+�©Ú|Ø�Þ#ì#Ô�Ñ&Ø�ݺí(Ù{ß~Û|Ô�Ü�Ú|Õe×�ÛFÙ{Ø�ÚFÔ�ÑmÜ`çÐÒÑ Ü�Ý`Ó�Ø�ÚFÔ�ѽÐ�Ð-Ø�ÞmÝ[Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û©ÓSÞmÙ{Ô&Ü3ÝYò�ÖTÙ{ÑmÜ�Ú|Ô�ѽÚ�Üeß~ÝYé�ÚFÝ��(Ý`áµç�ÐÒÑ Ü�Ý`Ó�Ø�ÚFÔ�Ñ Ð�Ð�кÑ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ߺØ~ß~Ù�ÑmÜvã�Ô�ß�Õ§Ù{Ø�ÚFÔ�ÑmܺÔ{ã

×�ÑmÓYÝYß�Ø~Ù�Ú|ѯÖTÙ{ßSÙ{Õ�Ý�Ø~ÝYßSÜ­Ù{ß~ÝeÓYÔ�ÑmÜ�ÚFá�ÝYß~Ý`áµç#�IÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛAÑ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß~Ú±Ø~Ú|Ý@Ü­Ô{ã·×mÑmÓ�Ý`ß�ØSÙ{ÚFѯépÙ{ß~ÚFÙ�æ�Û|Ý@Ü®Ù�ß�ݺá�ÚFÜ~Ó�×mÜ~Ü�Ý`á#ÚFÑ¡Ü�Ý`Ó�åØ�ÚFÔ�Ñ�Ðs�eçn�ÀÚFÑmÙ{ÛFÛFä�ë�ÚFÑ"Ü�Ý`Ó�Ø~Ú|Ô�Ñ��'Ñ�Ô�Ñ�Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�Ñ�Ô�Ñ�ÛFÚ|Ñ�Ý@Ù{ß~Ú±Ø~Ú|Ý@ÜQÚFÑ&é�Ô�ÛFé�Ú|Ñ�à%×�ÑTÓ�ÝYß�Ø~Ù�Ú|Ñ\épÙ{ß~ÚFÙ�æ�ÛFÝ`ÜQÙ{ß~Ý­Ó�Ô�ÑmÜ�ÚFá�Ý`ß�Ý@áµçkô¤ÞmÝÖmÙ�ÖTÝ`ß©ÚFÜ­Ü�×�Õ�Õ§Ù{ß~Ú��`Ý`á[Ú|Ñ5Ü�Ý`Ó�Ø~Ú|Ô�Ñ��®Ð�ç

Å�ÅmÆ ���,�I�-Ê��s�­Ç­ÊZ  Î�¡S�eÍa�,¡IÊ�¢��q£¤��¡QÇ#¢�ÎvÊÏÇô¤Þ�Ý[ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û¤ÓSÞmÙ�Ô�Ü3ÝYò�ÖmÙ{ÑmÜ�ÚFÔ�Ñmñ�ïðî�ó§Ú�ÜeÙ1ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛGÝYò�ÖTÙ{ÑmÜ�Ú|Ô�ѲÔ{ã®Ô�ß�Ø~Þ�Ô�à�Ô�ÑmÙ�Û·Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛFܺÚ|Ñ�Ø�Ý`ß�Õ§ÜeÔ{ã

ßSÙ{Ñmá�Ô�Õ épÙ{ß~ÚFÙ�æ�ÛFÝ`ÜGØ�Ô"Ù�Ö�Ö�ß~ÔNò�ÚFÕ§ÙpØ~Ý®Ø~Þ�Ý3×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ&Øvä0á�Ú�Ü�Ø�ß~Ú|æ�×�Ø�ÚFÔ�Ñ5Ô{ãkØ�Þ�Ý3Ô�×�Ø�Öm×�Ø`çÏô¤ÞmÝ-Ý�ò�ÖmÙ{ÑTÜ�ÚFÔ�Ñ5Ó�Ô�Ý[�"ÓYÚ|Ý`Ñ�ØSÜ©Ù{ß~ÝÜ�Ô�ÛFé�Ý`á¯ã�Ô�ß%ÚFÑ1Ø~Þ�Ý"Ó�Ô�Ñ�Ø~Ý�ò�Ø%Ô{ã(ÙxÜ�Ø�Ô�ÓSÞmÙ�Ü�Ø�Ú�ÓºõmÑ�Ú|Ø�ݧÝYÛFÝYÕ�Ý`Ñ�Ø3Ù�Ö�Ö�ß~Ô�Ù�ÓSÞ1æ&ä#×TÜ�ÚFÑ�à#Ùxö%Ù�Û|Ý`ß�ü�ÚFÑ1Ö�ß~Ô{�vÝ@Ó�Ø~Ú|Ô�Ñ�ç�íGÔ�ÑmÜ�Ú�á�Ý`ßØ�ÞmÝ%ã�Ô�ÛFÛ|Ô�©ÚFÑ�à§ÜvØ~Ô�ÓSÞmÙ�Ü�Ø�Ú�Ó�á�Úx�ÝYß~ÝYÑ&Ø�Ú�Ù{ÛAÝ ê ×mÙpØ~Ú|Ô�Ñ��©Ú|Ø�Þ5Ô�ÖTÝ`ß~Ù{Ø�Ô�ߤ¥¸Ù�Ñmá[Ü�Ô�×mß~ÓYÝ®Ø~ÝYß~Õ§¦¥�¨4©Sª�«2ª�¬­��®B¯�°^¦,¨ ©Sª�«2ª�¬�¯2ª ¨ � ¯

jz0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�

Page 3: AIAA-2006-2066-173

�©Ú|Ø�Þ\×�ÑmÓ�Ý`ß�ØSÙ{ÚFÑ3ÚFÑ�Ö�×�ØQÖmÙ�ß~Ù�Õ\ÝYØ�Ý`ß~ÜkÙ{ÑmáeÙ{Ö�Ö�ß~Ô�Ömß�Ú�ÙpØ~ÝÏÚFÑ�Ú±Ø~ÚFÙ�Û�Ù{ÑmáºæTÔ�×�Ñmá�Ù�ß�ä-Ó�Ô�Ñmá�Ú|Ø�ÚFÔ�ÑmÜ`çIô¤Þ�Ý(Ü�Ô�ÛF×�Ø~Ú|Ô�Ñ�®±°²®n¨ ©Sª�«2ª�¬�¯Ù{ÑTá"Ø~Þ�Ý-×�ÑmÓYÝYß�Ø~Ù{ÚFÑ5Ú|ÑmÖ�×�Ø­ÖmÙ�ß~Ù�Õ\ÝYØ�Ý`ß~ܤÓYÙ�Ñ[æ=Ý-Ý�ò�ÖmÙ�Ñmá�Ý@á0ÚFÑ&Ø�Ô"Ù\ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�Û�ÓSÞmÙ{Ô&ÜGÝYò�ÖmÙ{ÑmÜ�ÚFÔ�Ñ

®M¨4©Sª�«2ª�¬U¯�°o³´µ�¶ � ® µ ¨ ©Sª�«�¯�· µ ¨}¸�¨4¬�¯�¯[ª ¨ � ¯�©Þ�Ý`ß�Ý�¹�® µ ¨ ©Sª�«�¯(º[Ù�ß�Ý�Ø~Þ�Ý"Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛÏÓSÞmÙ�Ô�Ü%ÝYò�ÖTÙ{ÑmÜ�Ú|Ô�ѲÓ�Ô�Ý[�"Ó�ÚFÝYÑ&ØSÜYëZ¹· µ ¨}¸�¨4¬�¯�¯2º[á�Ý`Ñ�Ô{Ø~Ý"Õº×�Û|Ø�Ú�á�ÚFÕ�ÝYÑmÜ�Ú|Ô�ÑmÙ{ÛÏÔ�ß�åØ�ÞmÔ�à�Ô�ÑmÙ{ÛµÖ=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�ܤÚ|Ñ5Ø~ÝYß~էܩÔ{ãQÙ{Ñ�»�åÒá�Ú|Õ�Ý`ÑmÜ�ÚFÔ�ÑTÙ{Û�é�Ý`Ó�Ø~Ô�ß©Ô�ãÀßSÙ{Ñmá�Ô�Õ épÙ{ß~Ú�Ù{æ�ÛFÝ`Üz¸�¨4¬�¯�ëa�©Þ�ÝYß~Ý9»¡Ú�Ü©Ø~Þ�Ý-Ñ�×�Õºæ=ÝYßÔ{ã·×mÑmÓ�Ý`ß�ØSÙ{ÚFѯÖmÙ{ßSÙ{Õ�ÝYØ�ÝYßSÜ`ç®ô¤Þ�Ý\ß~Ù�Ñmá�Ô�Õ ÝYé�ÝYÑ&Ø�¬[ÚFÜ�×mÜ�Ý`á¯Ù�Ü�Ù�ß�à�×�Õ�ÝYÑ&Ø­Ø~Ô0ÚFÑmá�Ú�ÓYÙpØ~ݺØ~Þ�ݺßSÙ{ÑTá�Ô�Õ�Ñ�Ý@Ü�Ü®Ô{ãQØ�ÞmÝeépÙ{ß~Ú|åÙ{æmÛ|Ý@ÜYç·ô¤Þ�Ý%Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�Ü�¹· µ ¨}¸a¯2º-Ü~ÙpØ~ÚFÜ�ã�ä"Ø~Þ�Ý-Ô�ß�Ø�ÞmÔ�à�Ô�ÑmÙ{ÛFÚ|Øvä\ß~ÝYÛ�ÙpØ~Ú|Ô�Ñ�¼*·�½�¨ ¸a¯2ª(· µ ¨ ¸a¯�¾�°\¼&· ñ½ ¨}¸a¯�¾�¿�½ µ �©Ú|Ø�Þ�¿@½ µ Ø�ÞmÝ�%ß~Ô�Ñ�Ý@ÓSü�ÝYߤá�Ý`Û±ØSÙ\Ù�ÑmáÀ¼�Á�ª�Á!¾¤á�ÝYÑmÔ{Ø�Ý@Ü(Ø�Þ�Ý-ÚFÑ�Ñ�Ý`ß­Ö�ß~Ô�á�×mÓ�Ø

¼*·±½�¨ ¸a¯2ª2· µ ¨ ¸a¯�¾�°ÃÂ:Ä/ÅS·+½�¨}¸a¯�· µ ¨}¸a¯sÆǨ}¸Q¯�áȸ�ª ¨ � ¯�©Þ�Ý`ß�ݱÉ(ª&ÊË° � ª � ª�Ì�Ì@̱ëZÆ�¨}¸a¯eÚ�ܺØ~Þ�Ý��GÝ`Ú|à�Þ&Ø�ÚFÑ�à¯ã�×�ÑTÓ�Ø�ÚFÔ�Ñ,Ó�Ô�ß�ß~Ý`Ü�Ö=Ô�Ñmá�ÚFÑ�à#Ø~Ô1Ø~Þ�ÝxÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û¤æmÙ�Ü�Ú�Ü+¹· µ ¨}¸a¯2º5Ù�Ñmá¦dÍ#Ú�ÜeØ�Þ�Ý5Ü�×mÖ�ÖTÔ�ß�Ø\Ô{ã,¸�ç²ù'ÞmÝYѽØ~Þ�ݱ�GÝ`Ú|à�Þ�Ø~Ú|Ñmà¯ã�×�ÑmÓ�Ø�ÚFÔ�Ñ�Æ�¨}¸Q¯ºÚFѽØ�ÞmÝxÔ�ß�Ø~Þ�Ô�à�Ô�ÑmÙ�Û|Ú|Øvä1ß~ÝYÛ�ÙpØ~Ú|Ô�Ñ^¨ � ¯ºÓYÔ�ß~ß�Ý@Ü�Ö=Ô�ÑTá�ÜØ�Ô�Ø�Þ�Ý¡×�ÑmÓYÝYß�Ø~Ù{ÚFÑ&Øvä'á�ÚFÜ�Ø�ß~ÚFæ�×�Ø�ÚFÔ�Ñ�Ô{ã-Ø�ÞmÝ¡×�ÑTÓ�ÝYß�Ø~Ù�Ú|Ñ'ÖTÙ{ßSÙ{Õ�Ý�Ø~ÝYßSÜYë¤Ø~Þ�Ý1Ô�ß�Ø~Þ�Ô�à�Ô�ÑmÙ�Û­ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�Û�ÝYò�ÖTÙ{ÑmÜ�Ú|Ô�Ñ'æmÙ�Ü�Ú�ܹ· µ ¨ ¸a¯(º"ÚFÑΨ � ¯�Ú�Ü-Ô�Ö�Ø�ÚFÕ§Ù{ÛÏÚFÑ�Ø�Þ�Ý[Ü�Ý`ÑmÜ�Ý�Ø~ÞmÙpØeڱغÓ`Ù{ѲÙ�ÓSÞ�ÚFÝYé�ݧÝ�ò�Ö=Ô�Ñ�Ý`Ñ�Ø~ÚFÙ�Û·Ó�Ô�Ñ&é�ÝYß~à�Ý`ÑmÓ�Ý�ç ñ�ïðî���ïðî��Sïtî�ó ô¤Þ�Ý0Ô�ß~Ú|à�Ú|ÑmÙ�ÛÞ�Ô�Õ\Ô�à�Ý`Ñ�ÝYÔ�×mÜIÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛmÓSÞmÙ{Ô&ÜIÝYÕ�Ö�ÛFÔpä�ÜQÓ�Û�Ù�Ü~Ü�Ú�ÓYÙ�Û�û­ÝYß~Õ�Ú±Ø~Ý­Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛFÜQØ�Ô-Ô�æ�Ø~Ù{ÚFÑ�Ý�ò�Ö=Ô�Ñ�Ý`Ñ�Ø~ÚFÙ�ÛTÓYÔ�Ñ�é�Ý`ß�à�ÝYÑmÓYÝÏã�Ô�ßö%Ù{×TÜ�Ü�ÚFÙ�Ñ�×�ÑmÓ�Ý`ß�ØSÙ{ÚFÑ&Ø�ÚFÝ`Ü`ç ñ0ÐbØÏÞmÙ�ÜÏæTÝ`ÝYѧÝYò&Ø~ÝYÑmá�Ý`á\Ø�ÔºÔ{Ø�ÞmÝYßQØvä�Ö=Ý`Ü·Ô�ãµÔ�ß�Ø�Þ�Ô�à�Ô�ÑTÙ{Û�ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛFÜQÔ�ã�Ø�Þ�Ý#��Ü�ü�ÝYä�Ü�ÓSÞmÝYÕ�ÝØ�Ô\æ=Ý%Ù�æ�ÛFÝ®Ø~Ô\Ù�ÓSÞ�Ú|Ý`é�Ý�Ý�ò�ÖTÔ�Ñ�ÝYÑ&Ø~ÚFÙ�Û�ÓYÔ�Ñ�é�Ý`ß�à�ÝYÑmÓYÝ©ã�Ô�ß©Ü�ÝYé�ÝYßSÙ{ÛTÜ�Ø~Ù�Ñmá�Ù�ß~á"á�ÚFÜ�Ø�ß~Ú|æm×�Ø�ÚFÔ�ÑmÜ`ç îvó��Qò�Ö=Ô�Ñ�Ý`Ñ&Ø�Ú�Ù{ÛµÓ�Ô�Ñ&é�ÝYß~à�Ý`ÑmÓ�ÝÓYÙ�Ñ3æ=ÝÏÝ�ò�Ø~ÝYÑmá�Ý@á-Ø~Ô�Ù{ß~æ�Ú|Ø�ßSÙ{ß~ä®×�ÑTÓ�ÝYß�Ø~Ù�Ú|Ñ&Øvä%á�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�ÑmÜAæ�ä%Ñ�×�Õ�ÝYß~ÚFÓ`Ù{ÛFÛ|ä%Ó�Ô�Õ�Ö�×�Ø�ÚFÑ�à�Ù{Ñ3Ô�Ö�Ø�ÚFÕ§Ù{Û�Ü�ÝYØkÔ{ã�Ô�ß�Ø~Þ�Ô�à�Ô�ÑmÙ�ÛæmÙ�Ü�Ú�ܤÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛFÜ�¹{· µ ¨4Ï1¯(º�е�¶ � ×mÜ�Ú|Ñ�à§Ù�ÑxÔ�ß�Ø~Þ�Ô�à�Ô�ÑmÙ�Û|Ú��`Ù{Ø�ÚFÔ�Ñ0Ù�Û|à�Ô�ß~Ú±Ø~Þ�Õ5ç^î���ïðî����Ô�ß"Ø~Þ�Ý¡Ñ�×�Õ�Ý`ß�Ú�ÓYÙ�Û�ÚFÕ�Ö�Û|Ý`Õ�ÝYÑ&Ø~Ù{Ø�ÚFÔ�ѸØ~Þ�Ý�Ú|Ñ�õTÑ�Ú±Ø~Ý�Ü�×�Õ�Õ§ÙpØ~Ú|Ô�ѸÚFÜ[Ø~ß�×�ÑTÓYÙpØ~Ý`á¸Ù{Ø0Ø~Þ�ÝѨ%Ò]Ó � ¯bØ~Þ¸Ø~ÝYß~Õxë��©Ú±Ø~Þ¨4Ò\Ó � ¯+°Ô¨ »ÕÓ�ÖQ¯[×!Ø»S× Ö_×|ëZ�©Þ�ÝYß~Ý�Ö,ÚFÜ\Ø~Þ�Ý#Þ�ÚFà�Þ�Ý@ÜvاÔ�ßSá�Ý`ß\Ô�ã®Ø~Þ�Ý#ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛFÜ�¹· µ ¨ ¸a¯(º е�¶ � ç¸ø�×�æmÜ�Ø�Ú|Ø�×�Ø�ÚFÑ�à²Ø�ÞmÝØ�ß~×�ÑmÓ`ÙpØ~Ý`á"Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�ÓSÞmÙ{Ô&ÜÏÝYò�ÖmÙ{ÑmÜ�ÚFÔ�ÑmÜ�¨ � ¯·Ô�ã�Ø�Þ�Ý�×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ[éNÙ�ß�Ú�Ù{æmÛ|Ý@ÜGÙ�Ñmá§Ø~Þ�Ý�×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ0ÖmÙ�ß~Ù�Õ\ÝYØ�Ý`ß~ÜÏÚ|Ñ&Ø�ÔeØ�ÞmÝà�Ôpé�ÝYß~Ñ�ÚFÑ�àºÝ ê ×TÙpØ�ÚFÔ�ÑÀ¨ � ¯(ß�Ý@Ü�×mÛ±ØSܤÚ|Ñ

¥ÚÙÛq©Sª�«2ª�¬P� дµ�¶ � ® µ · µ�ÜÝßÞ ¦L¨ ©Sª�«2ª�¬�¯[Ì ¨ àȯ�Áö%Ù{ÛFÝYß~ü�Ú|ÑxÖ�ß~Ô{�vÝ@Ó�Ø�ÚFÔ�Ñ[Ô{ãL¨ àȯ(Ô�Ñ&Ø�Ô\Ý`Ù�ÓSÞ[Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛµæmÙ�Ü�Ú�Ü�¹· µ ¨}¸a¯2º�е�¶ � ß~Ý`Ü�×�Û|Ø~ܤÚFÑ#Ù\Ü�Ý�Ø­Ô�㤨%Ò²Ó � ¯(Ý ê ×mÙpØ~Ú|Ô�ÑmÜá ¥ ÙÛ ©Sª�«2ª�¬­� дµ�¶ � ® µ · µ�ÜÝ ª(·±âPãΰä¼&¦nª(·�â1¾2ª å�°ÎæPª � ª�Ì@Ì�Ì[ª�Ò¤Ì ¨%ç�¯ô¤Þ�Ú�Ü%Ó�Ô�×mÖ�Û|Ý@á¯Ü�Ý�Ø%Ô{ãGá�ÝYØ�Ý`ß�Õ�ÚFÑ�ÚFÜ�Ø�Ú�ÓeÝ ê ×mÙpØ~Ú|Ô�ÑmÜ�ÓYÙ{Ñ1æTÝ�Ü�Ô�Û|é�Ý`á#×TÜ�ÚFÑ�à[Ü�Ø~Ù�Ñmá�Ù�ß~á¯á�ÚFÜ~Ó�ß~Ý�Ø~Ú��@ÙpØ�ÚFÔ�Ñ¡Ù{Ñmá#Ú|Ø�Ý`ß~Ù{Ø�ÚFÔ�Ñ#Ø~Ý`ÓSÞ�åÑ�Ú ê ×�Ý`Ü`ç

Å�Å�ÅTÆ è ÊÏÇ���Î�Ç#�q¡IɽÈ�ÉP¡QÇ#¢UéSÊ·É­ ]¡ÀÈ�ÎvÊÏÇ�¢²Ê�éx̮ǮÍd�AÉ�È:¡IÎ�Ç Î�Ç,��Ì©ÈË��¡IÉP¡S Ú��ÈÈ�AÉ­¢ô¤Þ�Ý�õmßSÜvؤØvä�Ö=Ý�Ô�ãkÑ�Ô�Ñ�ÛFÚ|Ñ�Ý@Ù{ß~Ú±Øvä�Ø~ÞmÙpØ­Ú�Ü©Ó�Ô�ÑmÜ�ÚFá�Ý`ß�Ý@á"Ú�Ü©Ù\Ñ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ßGØ~ß~Ù�ÑmÜ�ã�Ô�ß~Õ�Ù{Ø�ÚFÔ�Ñ[Ô{ãk×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ0ÚFÑ�Ö�×�Ø­ÖmÙ�ß~Ù�Õ\å

Ý�Ø~ÝYßSÜYçÏíGÔ�Õ\Öm×�Ø�ÚFÑ�à§Ñ�×�Õ�ÝYß~ÚFÓ`Ù{ÛFÛ|ä"Ù�Ñ5Ô�Ö�Ø~Ú|Õ§Ù{Û�Ü�Ý�Ø©Ô�ãÀÔ�ß�Ø~Þ�Ô�à�Ô�ÑmÙ�ÛTÖ=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�Ü(ã�Ô�ߤØ�ÞmÝ-Ñ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ßGØ~ß~Ù�ÑmÜ�ã�Ô�ß~Õ�Ù{Ø�ÚFÔ�Ñ[Ô{ãØ�ÞmÝ�×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ[ÚFÑ�Ö�×�Ø©ÖmÙ�ß~Ù�Õ�Ý�Ø�Ý`ߤÖ�ß�Ôpé�Ý`ÜÏØ�Ô\æTÝ3ÙeépÙ{ÛF×mÙ{æmÛ|Ý-Ù{ÖmÖ�ß�Ô&Ù�ÓSÞ§Ø~Ô\Ü�Ô�ÛFé�Ý�ã�Ô�ߤÖ�ß~Ô�æmÛ|Ý`Õ�Ü(ÚFÑ�é�Ô�ÛFé�Ú|ÑmàºØ~Þ�ÚFÜ(Øvä�Ö=Ý%Ô{ãÑ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß~Ú±Ø~Ú|Ý@ÜYç¤ô¤Þ�Ú�Ü®Ù{ÖmÖ�ß�Ô&Ù�ÓSÞxÚFÜ�á�ÝYÕ�Ô�ÑmÜvØ~ß~Ù{Ø�Ý@á0ã�Ô�ß�Ù§Ü�Ú|Ñmà�ÛFÝ3×�ÑmÓYÝYß�Ø~Ù{ÚFÑ#ÖmÙ�ß~Ù�Õ\ÝYØ�Ý`ß©×mÜ�ÚFÑ�à0Ù�Õ§Ù�Ü�Ü�åbÜ�Ö�ß~Ú|Ñmà�Ü�ä�ÜvØ~ÝYÕÙ{ÑTá"ã�Ô�ß­Ù\ÓYÔ�Õeæ�Ú|ÑTÙpØ�ÚFÔ�ÑxÔ{ãÀÕº×�Û|Ø�ÚFÖ�ÛFÝ-×�ÑmÓYÝYß�Ø~Ù{ÚFÑxÖmÙ{ßSÙ{Õ�Ý�Ø~ÝYßSÜ(×mÜ�ÚFÑ�à§Ù\Þ�Ý`Ù{ؤØ�ßSÙ{ÑmÜ�ã�ÝYߤÖ�ß~Ô�æ�ÛFÝYÕ¼Ú|Ñ#ÙeØ��GÔ�åbá�Ú|Õ�ÝYÑTÜ�ÚFÔ�ÑmÙ�ÛÓSÞmÙ�Ñ�Ñ�ÝYÛ�âTÔ�-ç

ê 0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�

Page 4: AIAA-2006-2066-173

ë�ìîíðïañMò4ó%ñ_ôÈõ:ö�÷@öõPñ_ø�ù*ï­öú�õP÷@ó4ïQñMøûïQùLõPñÎüMñMý1ôUö/÷@õPó%ñ²þMõ:öõPúËô�÷�ôUö�·Þ�ä�Ü�ÚFÓ`Ù{Û­ÖmÙ�ß~Ù�Õ�Ý�Ø�Ý`ß~Ü"Ô{ãoØ�Ý`Ñ'ÞmÙNé�Ý¡Ù{Ñ'ÚFÑ�Þ�ÝYß~ÝYÑ&Ø0×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ�Øvä´ÚFÑ,Ø�ÝYß~Õ§Ü"Ô�ãºÙ½ÜvØSÙ{ÑmámÙ{ßSá ×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ�Øvä,á�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�ÑÜ�×mÓSÞ½Ù�Ü3Ù¯Ñ�Ô�ß~Õ§Ù{ÛÏá�Ú�Ü�Ø�ß~Ú|æ�×�Ø�ÚFÔ�Ñ�ç5ÐÒѽէÙpØ~Þ�ÝYÕ§Ù{Ø�Ú�ÓYÙ{ÛÏÕ\Ô�á�Ý`ÛFÜ3Ø�Þ�Ý@Ü�Ý[Ö�Þ�ä�Ü�Ú�ÓYÙ�Û·ÖmÙ{ßSÙ{Õ�ÝYØ�ÝYßSÜ�ã�ß~Ý ê ×�ÝYÑ&Ø~Û|ä�Ù{Ö�Ö=Ý`Ù�ß3ÚFѽÙÑ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ߤØ�ßSÙ{ÑTÜvã�Ô�ß�Õ§ÙpØ~Ú|Ô�Ñ[Ý�ç àTçÏÚFÑ�é�ÝYßSÜ�Ý�ë�Ý�ò�Ö=Ô�Ñ�Ý`Ñ&Ø�Ú�Ù{ÛtëTÜ ê ×TÙ{ß~Ý�ß~Ô�Ô{Ø@ë�Ý�ØSÓ{çZÿ�×�Ý-Ø~Ô§Ø�ÞmÝ`Ü�Ý-Ñ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ß©Ø�Ý`ß�էܩÚFÑmÜ�ÝYß�Ø�ÚFÑ�àØ�ÞmݧÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û·ÓSÞTÙ{Ô�Ü�ÝYò�ÖTÙ{ÑmÜ�Ú|Ô�Ñ1ã�Ô�ß%Ø�Þ�ݧ×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ¡ÖmÙ�ß~Ù�Õ�Ý�Ø�Ý`ß-Ù{Ñmá�Ù{ÖmÖ�Û|ä�ÚFÑ�à#Ù#ö%Ù{ÛFÝYß~ü&ÚFÑ�Ö�ß�Ô1�vÝ`Ó�Ø�ÚFÔ�Ñ�á�Ô�Ý`Ü%Ñ�Ô�Øß~Ý`á�×mÓYÝ-Ø�Ô0Ú|Ñ�ÑmÝYß�Ö�ß�Ô�á�×TÓ�Ø~Ü®Ô{ãQæmÙ�Ü�Ú�Ü­Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�ÜYç,�mÔ�ß­ÝYò�Ù�Õ�Ö�Û|ÝeÓ�Ô�ÑTÜ�Ú�á�ÝYß®Ø�Þ�Ý\Ü ê ×mÙ�ß�Ý-ß~Ô�Ô{Ø®Ô�ã·Ù{Ñ#×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ#ÚFÑ�Ö�×�ØÖmÙ�ß~Ù�Õ\ÝYØ�Ý`ß��z¨ ¬�¯�çGø�×�æmÜ�Ø�Ú|Ø�×�Ø�ÚFÑ�à�Ú±ØSÜGÖ=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�ÓSÞmÙ{Ô&ÜÏÝYò�ÖmÙ{ÑmÜ�ÚFÔ�ÑD¨ � ¯GÙ{Ñmá[Ù{Ö�Ö�ÛFä�Ú|ÑmàeÙ�ö%Ù{ÛFÝYß~ü&ÚFÑ"Ö�ß~Ô{�vÝ@Ó�Ø~Ú|Ô�Ñ"ß�Ý@Ü�×mÛ±ØSÜÚFÑ ¼�� �Z¨4¬�¯@ª�� â ¨ Ï1¯�¾�° á�� ³´ ½ ¶ � �M½���½�¨ Ï1¯�� ª�� â ¨4Ï1¯ ã ª å�° æ­ª � ª@Ì�Ì@Ì[Ì ¨��U¯ô¤Þ�Ú�ÜIÝ�ò�Ö�ß~Ý`Ü~Ü�ÚFÔ�Ñ\ÓYÙ�Ñ�Ñ�Ô{ØIß~Ý`Ù�á�Ú|ÛFä3æ=Ý(ß~Ý`á�×mÓ�Ý@á3Ø~Ô%ÓYÔ�Ñ&Ø~Ù�Ú|ÑeÚ|ÑmÑ�ÝYßQÖ�ß~Ô�á�×mÓ�Ø~ÜIÔ{ã=æmÙ�Ü�Ú�ÜkÖ=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�ÜL¼���½�¨ Ï1¯2ª�� â ¨ Ï1¯�¾ÀÔ�Ñ�ÛFä�ç�­Ñ,Ù{Û|Ø�Ý`ß�ÑTÙpØ�ÚFé�ÝxÙ{Ö�Ömß�Ô&Ù�ÓSÞ��(Ô�×�Û�á½æ=ÝxØ�Ô²Ó�Ô�ÑmÜ�ÚFá�Ý`ß\Ø�Þ�Ý#Ü ê ×TÙ{ß~Ý0ß~Ô&Ô�Ø��Z¨ ¬�¯±° � �Z¨4¬�¯�Ù�ÜeØ~Þ�Ý5×�ÑmÓ�Ý`ß�ØSÙ{ÚFÑ�ÚFÑ�Ö�×�ØÖmÙ�ß~Ù�Õ\ÝYØ�Ý`ß`çkÐÒѧØ�ÞmÙ{ØGÓYÙ�Ü�ݤØ~Þ�Ý®ÑmÔ�Ñ�ÛFÚ|ÑmÝ`Ù{ß·Ø�ÝYß~Õ ÚFÜ·Ø�ßSÙ{ÑmÜ�ã�Ô�ß~Õ�Ý`áeØ�ÔºÙ%ÛFÚFÑ�Ý`Ù�ß·Ø�Ý`ß�Õ¾ã�Ô�ß��©Þ�ÚFÓSÞ�Ø~Þ�Ý�ö%Ù�Û|Ý`ß�ü�ÚFÑ�Ö�ß�Ô1�vÝ`Ó�Ø�ÚFÔ�Ñá�Ô�Ý`Ü�ß~Ý`Ü�×�Û|Ø�ÚFÑ1ÚFÑ�Ñ�ÝYß%Ö�ß~Ô�á�×mÓ�Ø~Ü�Ô{ã·æmÙ�Ü�Ú�Ü�ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛFÜ`ç�ô¤ÞmÝ\á�Ú�Ü~Ù�á�épÙ{Ñ&ØSÙ{à�ÝeÔ{ã·Ø~Þ�Ú�Ü�Ù�Ö�Ö�ß~Ô�Ù�ÓSÞ#ÚFÜ�Ø�ÞTÙpØ�Ø�Þ�Ý\Ö�Þ�ä�Ü�Ú�ÓYÙ�ÛÖmÙ�ß~Ù�Õ\ÝYØ�Ý`ß��z¨ ¬U¯IÔ{ãoØ~ÝYÑ�ÞmÙ�ÜQÙ�Ñ�Ô�ß�Õ§Ù�Û&Ô�ß·Ù{Ñ�Ô�Ø�Þ�Ý`ß·ÜvØSÙ{ÑmámÙ{ßSáºá�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�Ñ�çn�®Ü·Ù�ß~Ý`Ü�×�Û±ØQÚ|Ø~ÜQÑmÔ�Ñ�ÛFÚ|ÑmÝ`Ù{ßIØ~ß~Ù�ÑmÜ�ã�Ô�ß~Õ�Ù{Ø�ÚFÔ�Ñ��¨4¬�¯­á�Ô�Ý@Ü©Ú|Ñ#à�ÝYÑ�Ý`ß~Ù�ÛµÑ�Ô{Ø�Ó�Ô�ß~ß~Ý`Ü�ÖTÔ�Ñmá"Ø~Ô"Ù"Ü�Ø~Ù{ÑTá�Ù{ßSá[á�ÚFÜ�Ø�ß~Ú|æm×�Ø�ÚFÔ�Ñ�ç¤û­Ô�Õ�Ô�à�Ý`Ñ�ÝYÔ�×mÜ©Ù{ÑTáð�®Ü�ü�ÝYä[ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛAÓSÞmÙ{Ô&ÜÝ�ò�ÖmÙ�ÑmÜ�Ú|Ô�ÑmÜ�ÓYÙ�Ñ-Ù�ÓSÞ�ÚFÝYé�ÝÀÝYò�ÖTÔ�Ñ�ÝYÑ&Ø�Ú�Ù{Û&Ó�Ô�Ñ&é�ÝYß~à�Ý`ÑmÓ�Ýkã�Ô�ß�×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ-Ú|Ñ�Öm×�Ø~Ü��©Ú±Ø~Þ3Ù©Ñ�Ô�ß~Õ§Ù{Û{Ù�Ñmá-Ü�Ý`é�Ý`ß~Ù�ÛpÔ{Ø�ÞmÝYßAÜ�Ø~Ù�Ñmá�Ù{ßSáá�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�ÑmÜ`ç[ô¤Þ�Ú�Ü3ÚFÕ�Ö�Û|ÚFÝ`Ü3Ø�ÞmÙ{Ø-Ø�Þ�Ý0ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{ÛGÓSÞmÙ{Ô&Ü%Ý�ò�ÖmÙ{ÑTÜ�ÚFÔ�ÑÀ�(Ô�×�Û�á�Ñ�Ô{Ø3æ=Ý0Ù{æmÛ|Ý�Ø~Ô#Ô�æ�ØSÙ{ÚFѲÝ�ò�ÖTÔ�Ñ�ÝYÑ&Ø~ÚFÙ�ÛÓ�Ô�Ñ�é�ÝYß~à�Ý`ÑmÓ�Ý®ã�Ô�ߤØ~Þ�Ý-×�ÑmÓYÝYß�Ø~Ù{ÚFÑxÖmÙ{ßSÙ{Õ�Ý�Ø~ÝYß���¨4¬�¯�ç î��Sïtîvó�©Ý@Ó�ÝYÑ&Ø~Û|ä�ë{Ø�ÞmÝ©ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{ÛTÓSÞmÙ{Ô&ÜIÕ\ÝYØ�Þ�Ô�á\ÞmÙ�ÜQæTÝ`ÝYѧÝYò&Ø~ÝYÑmá�Ý`áeØ�Ô-ß~Ý`Ù�ÓSÞeÝYò�Ö=Ô�ÑmÝYÑ&Ø�Ú�Ù{ÛmÓYÔ�Ñ�é�Ý`ß�à�ÝYÑmÓYÝ·ã�Ô�ßÏÙ{ß~æ�Ú|Ø�ßSÙ{ß~ä×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ&Øvä¡á�ÚFÜ�Ø�ß~Ú|æm×�Ø�ÚFÔ�ÑmÜ`ç î���ïðî��ÁÐbØ\ÚFܺٯÓ�Ô�Õºæ�ÚFÑmÙpØ~Ú|Ô�Ñ�Ô{ã¤Ø�Þ�Ý0ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{ÛGÓSÞmÙ{Ô&Ü-Ý�ò�ÖmÙ�ÑmÜ�ÚFÔ�Ñ�Ù{ÑTá²Ù{ѲÔ�ß�Ø�ÞmÔ�à�Ô�ÑmÙ{Û|åÚ��`ÙpØ~Ú|Ô�Ñ1Ù{ÛFà�Ô�ß�Ú|Ø�ÞmÕ Ø~Ô[Ñ�×�Õ�Ý`ß�Ú�ÓYÙ�Û|ÛFä#Ó�Ô�Õ�Ö�×�Ø�Ý�Ù{Ñ1Ô�Ö�Ø�ÚFÕ§Ù{ÛIÜ�ÝYØ�Ô�ãÏæmÙ�Ü�Ú�Ü�ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û�Ü�¹�� µ ¨ Ï1¯2º�е�¶ � Ô�ß�Ø�ÞmÔ�à�Ô�ÑmÙ{Û_�©Ú±Ø~Þß~Ý`Ü�ÖTÝ@Ó�غØ~Ô¯Ø~Þ�Ý[Ömß�Ô�æmÙ{æ�ÚFÛFÚ±Øvä²á�Ý`ÑmÜ�Ú|Øvä�ã�×�ÑmÓ�Ø�ÚFÔ�Ñ�Ô�ã©Ø�Þ�Ýx×�ÑmÓYÝYß�Ø~Ù�Ú|ѽÖTÙ{ßSÙ{Õ�Ý�Ø~ÝYß@ç��(ä�×mÜ�ÚFÑ�à1Ø�Þ�ÝxÖ�ß~Ô�æmÙ�æ�Ú|ÛFÚ|Øvä�á�Ý`ÑmÜ�Ú±Øväã�×�ÑmÓ�Ø�ÚFÔ�ѲÔ�ã���¨4¬�¯3Ù�ÜJ�GÝ`Ú|à�Þ�Ø~Ú|Ñmà[ã�×mÑmÓ�Ø~Ú|Ô�ѲÚ|Ñ�Ø�ÞmݧÔ�ß�Ø�Þ�Ô�à�Ô�ÑTÙ{ÛFÚ±Øvä#ß~ÝYÛ�ÙpØ~Ú|Ô�ÑѨ � ¯�ëÀÙ#Ü�ÝYØ3Ô{ã¤ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛFÜ%Ú�Ü3Ó�Ô�Õ�Ö�×�Ø�Ý@á�©Þ�Ú�ÓSÞ Ó`Ù{Ñ�Ù�ÓSÞ�Ú|Ý`é�Ý[Ý�ò�ÖTÔ�Ñ�ÝYÑ&Ø~ÚFÙ�Û(ÓYÔ�Ñ�é�Ý`ß�à�ÝYÑmÓYÝ�ã�Ô�ß\Ø�Þ�ÝxÑ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ßeØ�ßSÙ{ÑmÜ�ã�Ô�ß~Õ§ÙpØ�ÚFÔ�ѲÔ�ã­Ø�ÞmÝ[ÚFÑ�Ö�×�Ø�×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ&Øvä²á�Ý�åÜ~Ó�ß~Ú|æ=Ý`á�æ�ä���¨4¬�¯�ç5ÐÒÑ�Ø~Þ�ÚFÜ��(ÙNä¡Ö�ß�Ô�æ�ÛFÝYÕ§Ü%Ú|Ñ�é�Ô�ÛFé&ÚFÑ�à5Ñ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß-Ø~ß~Ù�ÑmÜvã�Ô�ß�Õ§Ù{Ø�ÚFÔ�ÑmÜ�Ô{ã©×�ÑTÓ�ÝYß�Ø~Ù�Ú|Ñ�ÚFÑ�Ö�×�ØeÖmÙ{ßSÙ{Õ�Ý�Ø~ÝYßSÜÓYÙ�Ñ�æTÝ­ÞTÙ{Ñmá�ÛFÝ`á�Ù�ÜQÝ[�"Ó�ÚFÝYÑ&Ø~Û|ä\Ù�ÜQÔ�Ø�Þ�Ý`ß·Ö�ß~Ô�æ�ÛFÝYÕ§Ü`çÀô¤Þ�Ý­ÓYÔ�Õ�Ö�×�ØSÙpØ~Ú|Ô�ÑmÙ{ÛmÓYÔ�Ü�Ø·Ô{ã=Ø�Þ�Ý®Ñ&×mÕ\Ý`ß�Ú�ÓYÙ�Û�Ó�Ô�Õ\Öm×�Ø~Ù{Ø�ÚFÔ�Ñ�Ô{ã�Ø�ÞmÝæmÙ�Ü�Ú�Ü(ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û�ÜGÚFÜ(Ñ�Ý`à�ÛFÚ|à�Ú|æ�ÛFÝ%Ó�Ô�Õ�ÖmÙ�ß�Ý@á�Ø�ÔeØ�ÞmÝ%ÓYÔ�Õ�Ö�×�ØSÙpØ~Ú|Ô�ÑmÙ{Û�Ó�Ô&ÜvؤÔ{ãIÜ�Ô�Û|é�ÚFÑ�à3Ø~Þ�Ý�×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ�Øvä ê ×mÙ�Ñ&Ø�Ú|õTÓYÙ{Ø�ÚFÔ�ÑÖ�ß~Ô�æ�ÛFÝYÕ ¨%ç�¯©Ú|Ø~Ü�ÝYÛ|ãvç#�Qò�Ö=Ô�Ñ�Ý`Ñ&Ø�Ú�Ù{ÛkÓYÔ�Ñ�é�Ý`ß�à�ÝYÑmÓYÝ%ÚFÜ�á�ÝYÕ�Ô�ÑTÜvØ~ß~Ù{Ø�Ý`áxã�Ô�ß�Ù§Õ§Ù�Ü�Ü�åbÜ�Ö�ß~Ú|Ñmà�Ø�Ý`Ü�Ø®Ö�ß~Ô�æmÛ|Ý`Õ Ü�×�æ:�vÝ@Ó�Ø®Ø~Ô"Ô�ÑmÝ×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ¡ÖmÙ{ßSÙ{Õ�Ý�Ø~ÝYß-Ù{ÑTá¯Ú|Ø%Ú�Ü-Ù{ÖmÖ�Û|ÚFÝ`á1Ø�ÔxÕº×mÛ±Ø~Ú|Ö�ÛFÝ�×�ÑmÓYÝYß�Ø~Ù{ÚFÑ�ÖmÙ{ßSÙ{Õ�Ý�Ø~ÝYßSÜ­ã�Ô�ß�Ø�Þ�Ý�Þ�Ý@ÙpØ%Ø�ßSÙ{ÑmÜ�ã�ÝYß�Ö�ß~Ô�æ�ÛFÝYÕ Ú|Ñ�ÙØ��(Ô{åÒá�Ú|Õ�Ý`ÑmÜ�ÚFÔ�ÑTÙ{ÛµÓSÞmÙ{ÑmÑ�ÝYÛ�âmÔ�-ç� ì����Mô�ú�õPø�ø���ø@þ_öó4ñ! �þMöï#"nò4ôÈúÐÒѯØ~Þ�Ú�Ü�Ü�Ý`Ó�Ø�ÚFÔ�Ñ1Ù0Û|ÚFÑ�Ý@Ù{ß�Õ�Ù�Ü�Ü�åÒÜ�Ö�ß~ÚFÑ�à�Ø�Ý@ÜvØ%Ö�ß~Ô�æ�ÛFÝYÕ �©Ú|Ø�Þ¡Ù�Ñ1×�ÑTÓ�ÝYß�Ø~Ù�Ú|Ñ1Õ�Ù�Ü�Ü�ÖmÙ�ß~Ù�Õ\ÝYØ�Ý`ß�Ü�×�æ:�vÝ`Ó�Ø®Ø~ÔxÙ0Ñ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ßØ�ßSÙ{ÑTÜvã�Ô�ß�Õ§ÙpØ~Ú|Ô�ѽÚ�Ü\ÓYÔ�ÑmÜ�Ú�á�ÝYß~Ý`áµëGÜ�Ý`ݱ�ÀÚFà�×�ß~Ý � Ù�ç�ô¤ÞmÝ[Õ§Ù�Ü~Ü�åbÜ�Ö�ß~Ú|Ñ�à¡Ö�ß~Ô�æ�ÛFÝYÕ ÞmÙ�Ü\æTÝ`ÝYÑ,ÓSÞ�Ô�Ü�ÝYÑ�Ù�Ü�Ü�ÚFÕ�Ö�Û|Ý5Õ�Ô�á�ÝYÛã�Ô�ß%Ù0ÓYÔ�Õ�Ö�×�ØSÙpØ~Ú|Ô�ÑmÙ{ÛkÖ�ß�Ô�æ�ÛFÝYÕ Ú|Ñ¡Õ�Ý`ÓSÞmÙ�Ñ�ÚFÓ`Ù{ÛkÝYÑ�à�Ú|Ñ�Ý`ÝYß~Ú|Ñmà0ÚFÑ&é�Ô�ÛFé�Ú|Ñ�à0×�ÑmÓ�Ý`ß�ØSÙ{ÚFѯէÙ�Ü~Ü®á�Ú�Ü�Ø�ß~Ú|æ�×�Ø�ÚFÔ�Ñ�ç%ÐÒѯÖ�ßSÙ�Ó�Ø�Ú�ÓYÙ�ÛÙ{ÖmÖ�Û|Ú�ÓYÙ{Ø�ÚFÔ�ÑmÜ`ë&Ø�Þ�Ý%Õ§Ù�Ü~Ü©á�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�Ñ[Ô�ãkÙ�ÜvØ~ß�×TÓ�Ø�×mß�Ý%ÓYÙ�Ñ[æ=Ý-Ù{ÑxÚ|Õ�Ö=Ô�ß�Ø~Ù{Ñ&Ø©Ü�Ô�×�ßSÓ�Ý�Ô{ãk×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ�Øvä�çQô¤Þ�Ý�Õ�Ô�Ø�ÚFÔ�ÑxÔ{ãØ�ÞmÝ-Õ�Ù�Ü�ܤÚ�ܤà�Ôpé�ÝYß~Ñ�Ý`á0æ�ä§Ø�Þ�Ý%ã�Ô�Û|ÛFÔ�©ÚFÑ�à§Ü�Ý@Ó�Ô�ÑTá�åtÔ�ß~á�ÝYß(Ô�ßSá�Ú|ÑTÙ{ß~ä"á�Úx�ÝYß~ÝYÑ&Ø~ÚFÙ�ÛµÝ ê ×mÙpØ~Ú|Ô�Ñ

á�ñ%$á:« ñ Ó'&( $�°^æPª «*)�æPª ¨�+1¯�©Þ�Ý`ß�Ý,(9Ú�Ü-Ø~Þ�Ý0Õ§Ù�Ü�Ü`ë & ÚFܺØ�ÞmÝ[Ü�Ö�ß�ÚFÑ�à1Ü�Ø�Ú��Ñ�Ý`Ü~ÜYë�$´á�ÝYÑ�Ô�Ø�Ý@Ü3Ø~Þ�Ý0Ö=Ô�Ü�Ú±Ø~Ú|Ô�ѲÔ{ã©Ø~Þ�Ý0Õ§Ù�Ü~Ü`ë�-$ � ° � Ú�ܺØ�Þ�Ý[Ú|Ñ�Ú|Ø�Ú�Ù{Û

Ö=Ô�Ü�Ú±Ø~Ú|Ô�Ñ Ù{ÑTá.-$ î °mæ¡ÚFÜeØ~Þ�Ý5Ú|Ñ�Ú|Ø�Ú�Ù{Û©é�ÝYÛFÔ�Ó�Ú|Øvä�Ô�ã®Ø~Þ�Ý[Õ§Ù�Ü�Ü`ç�ô¤Þ�Ý#Ü�Ô�ÛF×�Ø�ÚFÔ�Ñ´ÙpØ�«�° � Ú�ܧÓ�Ô�ÑmÜ�Ú�á�Ý`ß�Ý@áµç½ô¤Þ�Ý5Õ§Ù�Ü~Ü( ¨ ¬U¯®Ú�Ü%Ù�Ü~Ü�×mÕ\Ý@á5Ø�Ôxæ=Ý\×mÑmÓ�Ý`ß�ØSÙ{ÚFÑD�©Ú|Ø�Þ�Ö�ß�Ô�æmÙ{æmÚ|ÛFÚ±Øvä#á�Ý`ÑmÜ�Ú±Øvä5ã�×�ÑTÓ�Ø�ÚFÔ�Ñ0/21�¨3(ä¯�çºô¤�GÔ[Ö�ß~Ô�æmÙ�æ�Ú|ÛFÚ|Øvä5á�ÚFÜ�Ø�ß~Ú|æm×�Ø�ÚFÔ�ÑmÜÙ{ß~Ý�Ó�Ô�ÑTÜ�Ú�á�ÝYß~Ý`á�ë�Ù0×�ÑmÚ±ã�Ô�ß�Õ Ù�Ñmá�Ù0ÛFÔ�à�Ñ�Ô�ß~Õ§Ù{ÛIá�ÚFÜ�Ø�ß~ÚFæ�×�Ø�ÚFÔ�Ñ��©Ú|Ø�Þ�Õ\Ý@Ù{Ñ5461 ° � Ù�Ñmá¯épÙ{ß~Ú�Ù{ÑmÓYÝ�7µñ1 °]æ­Ì æUç�çºô¤ÞmÝÙ{ÑTÙ{ÛFä�Ø~ÚFÓ`Ù{Û�Ü�Ô�ÛF×�Ø�ÚFÔ�Ñ[ã�Ô�ߤØ~Þ�ß~ÝYÝ%épÙ{ÛF×�Ý`ܤÔ�ãkØ~Þ�Ý-Õ§Ù�Ü~Ü8( °ä¹�æPÌ �P� � � � Ì àPº®Ú�Ü­Ü�Þ�Ô�©Ñ0ÚFÑð�kÚFà�×mß�Ý � æAçô¤Þ�ݤ×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ\Õ§Ù�Ü~Ü9( ¨ ¬�¯ÀÙ{ÖmÖTÝ@Ù{ßSÜIÚ|ÑeØ�Ý`ß�Õ§ÜIÔ{ã�Ú|Ø~ÜIÚFÑ&é�ÝYßSÜ�Ý(ÚFÑeØ~Þ�Ý­Ü�Ý`Ó�Ô�Ñmá-Ø~ÝYß~Õ/Ô�ãM¨�+1¯�ç9:®ã�ÓYÔ�×�ßSÜ�Ý�ëpÕº×mÛ±Ø~Ú|Ö�ÛFä�Ú|Ñmà¨�+1¯Gæ&ä;( �(Ô�×�Û�á"æ�ß~ÚFÑ�à<( Ø�Ô\Ø�ÞmÝ%õmß~Ü�ؤØ�Ý`ß�Õ Ô�ãAØ~Þ�Ý%Ý ê ×mÙpØ~Ú|Ô�ÑxÓYÔ�Ñ&Ø~Ù�Ú|ÑmÚ|Ñ�àeØ�ÞmÝ�ÞmÚ|à�Þ�Ý`Ü�ؤØ�ÝYÕ�Ö=Ô�ßSÙ{Ûµá�Ý`ß�ÚFépÙpØ�ÚFé�Ý%Ô�ãAØ�ÞmÝ

= 0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�

Page 5: AIAA-2006-2066-173

K

Mx

> '@?qK,'( % 3A} %./"&�!;�=�)�02;CB/=2-/"*'�3%�!02;0 0.5 1 1.5 2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time t

posi

tion

x

M=0.6

M=1

M=1.4

> 62?q'(;'(< 7�3&�!)�'(<È %02<!-�3%�!02;D6EGFIHKJML�N2O,P*QRLTSTU2VWV�XV�YZJ�E\[RF�]WL%V^]8YZJW_I`bacLdS<O

Ý ê ×mÙ{Ø�ÚFÔ�Ñ�ç%ÐÒÑ1Ø�ÞmÙ{Ø%Ó`Ù�Ü�Ý3Ý�ò�Ö�ÛFÚFÓYÚ±Ø%Ø~Ú|Õ�Ý\Ú|Ñ&Ø�Ý`à�ßSÙpØ~Ú|Ô�Ñð�GÔ�×�ÛFá¯Ú|Ñ�é�Ô�Û|é�ÝeÙ�ѯÚ|Ñ�é�ÝYßSÜ�ÚFÔ�Ñ#Ô�ã·Ø�Þ�Ý�Ó�Ô�×�Ö�ÛFÝ`á#Õ§Ù{Ø�ß~Ú±ò¯ß~Ý`Ü�×�Û|Ø�ÚFÑ�àã�ß~Ô�Õ¾Ø�Þ�Ý�Ù{Ö�Ö�ÛFÚ�ÓYÙpØ~Ú|Ô�Ñ�Ô{ãµÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛTÓSÞmÙ�Ô�Ü`çkô¤Þ�Ý`ß�ÝYã�Ô�ß~Ý�ë�ÞmÝYß~Ý­Ú±Ø(ÚFÜGÓSÞ�Ô�Ü�ÝYÑ\Ø~Ôºá�Ý�õTÑ�Ý­Ø~Þ�Ý­Ñ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß·Ø~ß~Ù�ÑmÜ�ã�Ô�ß~Õ�Ù{Ø�ÚFÔ�ÑÔ{ã®Ø�Þ�Ý5Ô�ß~ÚFà�ÚFÑmÙ{ÛGÚ|Ñ�Öm×�اÖmÙ{ßSÙ{Õ�Ý�Ø~ÝYße( ¨ ¬�¯eÙ�Ü�Ù¡Ñ�Ý��¼×�ÑmÓ�Ý`ß�ØSÙ{ÚFÑ�ÖmÙ�ß~Ù�Õ�Ý�Ø�Ý`ß�( ¨4¬�¯Wf�°g( ¨ ¬�¯�hµî{ç²ÐÒÑ��ÀÚFà�×�ß~Ý � Ø�ÞmÝ×�Ñ�Ú|ã�Ô�ß~Õ Ù{Ñmá�ÛFÔ�à�Ñ�Ô�ß~Õ§Ù{Û�×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ�Øvä�á�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�ÑmÜ·Ô�ãi( ¨ ¬�¯ÏÙ{Ñmá\Ø~Þ�Ý�ÓYÔ�ß~ß�Ý@Ü�Ö=Ô�ÑTá�Ú|Ñmà-×�ÑmÓ�Ý`ß�ØSÙ{ÚFÑ&Øvä\á�Ú�Ü�Ø�ß~Ú|æ�×�Ø�ÚFÔ�ÑmÜÏÔ{ã( f ¨ ¬�¯®Ù{ß~ݺÜ�ÞmÔ�©Ñ�ç¤ÐbØ�ÓYÙ�Ñ#æTÝ\Ü�Ý`ÝYÑ5Ø~ÞmÙpØ�Ø�Þ�ÝeÑ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ß©Ø�ßSÙ{ÑmÜ�ã�Ô�ß~Õ§ÙpØ~Ú|Ô�Ñxæ=Ý�Ø��(ÝYÝ`Ñ�( ¨ ¬�¯®Ù{Ñmá�( f ¨ ¬U¯®á�ÝYã�Ô�ß~էܩØ�ÞmÝÔ�ß~ÚFà�ÚFÑmÙ{Û�×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ&Øvä0á�Ú�Ü�Ø�ß~Ú|æ�×�Ø�ÚFÔ�ÑmÜ©Ü�Ú|à�Ñ�Ú|õTÓYÙ�Ñ�Ø~Û|ä�ç

0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

M, M*=M−1

prob

abili

ty d

istri

butio

n fu

nctio

n

MM*=M−1

> '@?d-/;�� F�02"%K�./"%026'(6/�!<!� 3}7#5/�! 43&"%�!6/-�3%�!02;0 0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

1

M, M*=M−1

prob

abili

ty d

ensi

ty fu

nctio

n

MM*=M−1

> 6�?B<!02=2;�02"&K,'(<U.�"&026'(6/�!<!� 3}7,5�$�;� &� 3}7D6EGFIHKJML�jRO,k^[KYbHI]lHZ[Km�LCJ�]MUnE\[2]^oepbEGV^]MJ�E\`bHI]�EG_I[KV9qr_nJ*]�QRLTSsU2V�V�Et[u]�QRLTSsU2V�V�XV�YZJ�Et[RFTYZJM_n`bacLdSvO

ø�ÚFÑmÓ�Ý1Ø�Þ�Ý¡×�ÑmÓYÝYß�Ø~Ù{ÚFÑ&Øvä,Ô{ãs(wf1¨ ¬�¯[ÓYÙ�ѸÑ�Ô{ØxæTÝ�á�Ý`Ü~Ó�ß~ÚFæTÝ@á'æ�ä,Ü�Ø~Ù�Ñmá�Ù�ß~á'á�Ú�Ü�Ø�ß~Ú|æ�×�Ø�ÚFÔ�ÑmÜ`ë­Ø~Þ�Ý1Ô�Ö�Ø�ÚFÕ§Ù{Û%Ü�ÝYØ~Ü[Ô{ãfz0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�

Page 6: AIAA-2006-2066-173

Ô�ß�Ø�ÞmÔ�à�Ô�ÑmÙ{ÛmæmÙ�Ü�ÚFÜGÖ=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛFÜÏÞmÙNé�Ý­Ø�Ô\æTÝ�Ó�Ô�Õ�Ö�×�Ø�Ý`á0Ñ�×�Õ�ÝYß~ÚFÓ`Ù{ÛFÛ|äeØ�Ô\ß�Ý@Ù�ÓSÞ§ÝYò�Ö=Ô�ÑmÝYÑ&Ø�Ú�Ù{Û�Ó�Ô�Ñ&é�ÝYß~à�Ý`ÑmÓ�Ý�çAôAÔeÔ�æ�ØSÙ{ÚFÑÝ�ò�Ö=Ô�Ñ�Ý`Ñ&Ø�Ú�Ù{Û�Ó�Ô�Ñ&é�ÝYß~à�Ý`ÑmÓ�ÝIØ~Þ�ݤ�GÝ`Ú|à�Þ�Ø~Ú|Ñmà­ã�×�ÑmÓ�Ø�ÚFÔ�ÑÇÆ�¨ Ï1¯ÀÚFÑ�¨ � ¯AÞmÙ�ÜkØ�Ô�æTÝ¤Ý ê ×TÙ{Û�Ø�Ô�Ø�Þ�ݤÖ�ß~Ô�æmÙ�æ�ÚFÛ|Ú|Øvä-á�ÝYÑmÜ�Ú|Øvä%ã�×�ÑmÓ�Ø~Ú|Ô�Ñ/dx1¨4Ï1¯%Ô{ã¤Ï�ë��©ÞmÝYß~Ý�ÏP¨ ¬�¯%Ú�Ü�Ø�ÞmÝ"Û|ÚFÑ�Ý@Ù{ß%Ø�ßSÙ{ÑTÜvã�Ô�ß�Õ§ÙpØ~Ú|Ô�Ñ1Ô�ãGØ~Þ�ݧ×�ÑmÓYÝYß�Ø~Ù{ÚFÑ�Ú|ÑmÖ�×�غÖmÙ�ß~Ù�Õ�Ý�Ø�Ý`ß�(wf{¨4¬�¯�Ø�Ô¯Ù5Ü�Ø~Ù�Ñmá�Ù{ßSáá�Ô�Õ�Ù�Ú|ÑzyG{ � ª ��| ë8y æPª�}˯3Ô�ßð¨^{l}Ϊ�}�¯�ç½û®ÝYß~Ý0Ø~Þ�ÝxÜ�Ý�ØSÜ\Ô{ã®Ô�ß�Ø~Þ�Ô�à�Ô�ÑmÙ�ÛÏÖ=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛFÜ\Ù{ß~Ý[ÓYÔ�Õ�Ö�×�Ø~Ý`á�×mÜ�ÚFÑ�à²ö�ß~Ù�Õ\åø�ÓSÞ�Õ�Ú�á�Ø­Ô�ß�Ø�ÞmÔ�à�Ô�ÑmÙ{ÛFÚ��`ÙpØ~Ú|Ô�Ñ

� µ ¨ Ï1¯Z°�~ µ ¨ Ï1¯�{ µ hµî´â ¶ ��� µ âR�JâȨ Ï1¯2ª Êû° � ª � ª@Ì�Ì�Ì�ª4Ö�ª ¨��U¯�©Ú|Ø�Þ�� � ° � Ù�Ñmá

� µ â�° ¼�~ µ ¨ Ï1¯[ª��Jâ:¨4Ï1¯�¾¼� â ¨ Ï1¯[ª�� â ¨4Ï1¯�¾ Ì ¨��U¯�©Þ�Ý`ß�Ý\Ø�ÞmݧÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û�ÜǹC~ µ ¨ Ï1¯2º"Ù{ß~Ý\Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�Ü�Ô{ã¤Ý�ò�Ù�Ó�Ø-á�ÝYà�ß~ÝYÝ9ÊTç"ô¤Þ�Ý�Õ�Ô�á�Ú±õmÝ@á²ö�ß~Ù�Õeåvø�ÓSÞ�Õ�Ú�á�Ø3Ù{ÛFà�Ô�ß�Ú|Ø�Þ�Õ�©Ú|Ø�Þ\Ô�Ñ�ݤß~ÝYÔ�ß�Ø�ÞmÔ�à�Ô�ÑmÙ{ÛFÚ��`ÙpØ~Ú|Ô�Ñ-ÜvØ~ÝYÖ\ÚFÜIÚFÕ�Ö�Û|Ý`Õ�ÝYÑ&Ø�Ý@áµçÀô¤ÞmÝ(Ñ�×�Õ�Ý`ß�Ú�ÓYÙ�Û&ÚFÑ&Ø�Ý`à�ßSÙpØ�ÚFÔ�ÑTÜAÚFѱ¨3��¯ÀÙ{ß~Ý(ÖTÝ`ß�ã�Ô�ß�Õ�Ý`áºÝ[�"Ó�ÚFÝYÑ&Ø~Û|äæ�ä²Ó�Ô�ÑTÜ�Ú�á�ÝYß~ÚFÑ�à5Ø�Þ�ÝxÚFÑ�Ñ�ÝYß\Ö�ß~Ô�á�×mÓ�ØSܺÔ�ã©Ø�Þ�Ý[ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛFÜeÙ�Ü\Ü�×�Õ�Õ§ÙpØ~Ú|Ô�ÑmÜ3Ô�ã­Ú|Ñ&Ø~ÝYà�ßSÙ{Û�Ü3Ôpé�ÝYß3Õ�Ô�Ñ�Ô�Õ�Ú�Ù{Û�ÜYç1ô¤ÞmÝ`Ü�ÝÖ�ß~Ô�á�×mÓ�ØSÜ�Ô{ãGÕ�Ô�Ñ�Ô�Õ�Ú�Ù{Û�Ü��©Ú|Ø�Þ¡Ø�ÞmÝû�(ÝYÚFà�Þ&Ø~Ú|Ñ�à0ã�×�ÑmÓ�Ø�ÚFÔ�ѲÙ{ß~ݺÚFÑ&Ø�ÝYà�ß~Ù{Ø�Ý@á¯×mÜ�Ú|Ñmà5ö%Ù�×mÜ�Ü ê ×mÙ�á�ß~Ù{Ø�×�ß~ݺÚFÑ&Ø�Ý`à�ßSÙpØ�ÚFÔ�ÑAç3ÐÒÑØ�ÞmÚFÜ#�(ÙNä0Ø�Þ�Ý3Ô�ß�Ø~Þ�Ô�à�Ô�ÑmÙ�ÛµÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛFÜ­Ù�ß�Ý3ÓYÔ�Õ�Ö�×�Ø~Ý`á5Ý��§ÓYÚ|Ý`Ñ&Ø�ÛFä��©Ú|Ø�Þ��+¨��ûÖQ¯®Ó�Ô�Õ�Ö�×�Ø~ÙpØ~Ú|Ô�ÑmÙ{Û��GÔ�ß�ü�ëP�©Þ�Ý`ß�Ý�� ÚFÜØ�ÞmݺÑ�×�Õºæ=ÝYß�Ô{ãGö%Ù�×mÜ�Ü ê ×mÙ�á�ßSÙpØ~×�ß~Ý-ÖTÔ�Ú|Ñ&Ø~Ü`ç®ô¤Þ�ݺØ�Þ�ß~ÝYÝ3Ø�ÝYß~Õ Ø~Ù�æ�Û|ݱ¼���½�¨ Ï1¯W� µ ¨4Ï1¯W� â ¨ Ï1¯�¾­ÚFÜ�Ó�Ô�Õ�Ö�×�Ø�Ý@á#ÚFÑ1Ù"Ü�Ú|Õ�ÚFÛFÙ�ß�¤ÙNäx×TÜ�ÚFÑ�à5Ü�×�Õ�Õ§ÙpØ~Ú|Ô�ÑmÜ�Ô{ãÏÚFÑ&Ø�Ý`à�ßSÙ{Û�Ü®Ô�ãÏÕ�Ô�Ñ�Ô�Õ\Ú�Ù{Û�Ü`ç3ô¤Þ�ÝeÖ=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛIÝ�ò�ÖmÙ�ÑmÜ�Ú|Ô�ÑmÜ­ã�Ô�ß®Ø~Þ�Ý�ÖTÔ&Ü�Ú|Ø�ÚFÔ�Ñ¡Ô{ã·Ø~Þ�Ý�Õ§Ù�Ü~Ü$_¨4«2ª�¬�¯­Ù{Ñmá0Ø�Þ�Ý-×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ[ÖmÙ�ß~Ù�Õ�Ý�Ø�Ý`ß�(wf¨4¬�¯©Ù{ß~Ý

$_¨4«2ª�¬�¯z° �´ ½ ¶ � $a½�¨ «�¯M��½�¨4ÏP¨ ¬�¯�¯2ª ( f ¨ ¬�¯�° î´µ�¶ � ( fµ � µ ¨4ÏP¨ ¬�¯�¯2ª ¨ � æU¯�©Ú|Ø�ÞxØ�Þ�Ý-ÖTÙ{ßSÙ{Õ�Ý�Ø~ÝYß©Ó�Ô�Ý[�"ÓYÚ|Ý`Ñ�ØSÜ ( fµ ° ¼3(wf1¨ Ï1¯[ª�� µ ¨ Ï1¯�¾¼�� µ ¨4Ï1¯ ñ ¾ Ì ¨ �1� ¯ø�ÚFÑmÓ�Ý3Ø�Þ�ÝeÑ&×mÕºæ=ÝYß­Ô�ãI×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ#ÖmÙ{ßSÙ{Õ�Ý�Ø~ÝYßSܤ»�ÚFÜ®Ô�Ñ�Ý�ë�Ø�ÞmݺÑ�×�Õºæ=ÝYß®Ô{ãQÝ�ò�ÖmÙ�ÑmÜ�ÚFÔ�ÑxØ~ÝYß~Õ�ÜǨ4Ò^Ó � ¯(Ý ê ×mÙ{Û�Ü�¨xÖûÓ � ¯�çè­Ô�Ø�Ú�Ó�Ý�Ø�ÞTÙpØ©Ø�ÞmÝ-Ô�Ö�Ø~Ú|Õ§Ù{ÛµÖ=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛµÓSÞmÙ�Ô�Ü(Ý�ò�ÖmÙ�ÑmÜ�ÚFÔ�Ñ0ã�Ô�ߤØ~Þ�Ý-×�ÑmÓYÝYß�Ø~Ù{ÚFÑxÖmÙ{ßSÙ{Õ�Ý�Ø~ÝYß8(wf�ÚFÑ�¨ � æ�¯GÚ�ܤÝ�ò�Ù�Ó�Ø©Ú|ÑxØ�ÞmÝõmßSÜvØ%Ø��(Ô0Ø~ÝYß~Õ§Ü-á�×�Ý�Ø�ÔxØ~Þ�ݧÛ|ÚFÑ�Ý@Ù{ß�Ø�ßSÙ{ÑmÜ�ã�Ô�ß~Õ§ÙpØ~Ú|Ô�Ñ1æ=Ý�Ø��(ÝYÝYÑ�(wf1¨ ¬�¯%Ù�ÑmáDÏP¨4¬�¯�ç0ø�×�æmÜ�Ø�Ú|Ø�×�Ø~Ú|Ô�Ñ�Ô�ãÏØ~Þ�ݧÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛÓSÞmÙ�Ô�Ü(Ý�ò�ÖmÙ�ÑmÜ�ÚFÔ�ÑTÜ�¨ � æU¯(Ú|Ñ&Ø�Ôð¨+{¯¤Ù{ÑTá[Ù§ö%Ù{ÛFÝYß~ü�Ú|ÑxÖ�ß~Ô{�vÝ@Ó�Ø�ÚFÔ�ÑxÔ�Ñ&Ø~Ô\Ý@Ù�ÓSÞ[ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{ÛµæTÙ�Ü�ÚFÜ�¹2� µ ¨4Ï1¯(º �µ�¶ � ß~Ý`Ü�×�Û±ØSܤÚ|Ñ

á�ñ�$Qâá:« ñ Ó & ��� �C� â�� î�� �C�´½ ¶ ���W�C� â hµî�� � �

-( f½ â $ ½ °Îæ å�°Îæ­ª � ª@Ì�Ì@Ì[ª4Ö_ª ¨ ��� ¯�©Ú|Ø�Þ -( f½ â ° �¼�� ñ â ¨4Ï1¯�¾ î´µ�¶ � ( fµ ¼���½�¨ Ï1¯W� µ ¨4Ï1¯W� â ¨4Ï1¯�¾[ª ¨ �@� ¯�©Þ�Ú�ÓSÞ,ÚFÜ[Ù²Ü�ä�Ü�Ø�ÝYÕ Ô{ã8¨tÖðÓ � ¯�á�Ý�Ø�Ý`ß�Õ�ÚFÑ�Ú�ÜvØ~ÚFÓ#Ý ê ×mÙpØ~Ú|Ô�ÑmÜYçWô¤Þ�Ý#ß~Ý`Ü�×�Û±Ø~Ú|Ñmà½Ü�ä�Ü�Ø�ÝYÕ2Ô{ã%Ý ê ×mÙ{Ø�ÚFÔ�ÑmÜD¨ ��� ¯\ÚFÜ8�GÝ@Ù{ü�Û|äÓ�Ô�×�Ö�ÛFÝ`áµëÀÜ�ÚFÑmÓYÝ�Ø�Þ�Ý"Ü�×�Õ�Õ§ÙpØ�ÚFÔ�Ñ�ÚFÑ�Ø�Þ�Ý"Ü�Ý`ÓYÔ�Ñmá1Ø�Ý`ß�Õ Ô{ã9¨ ��� ¯�ß�Ý@Ü�×�Û|Ø~Ü3Ú|ѲÙ[Ø�ß~ÚFá�ÚFÙ�à�Ô�ÑTÙ{ÛIÕ§ÙpØ�ß~Ú|ò�ç"ô¤ÞmÝ"Ü�×�Õ�Õ§ÙpØ~Ú|Ô�ÑÛFÔ&Ô�ÖmÜ%Ôpé�Ý`ß�É�°¾Õ§Ù{òa¹åe{ � ª�æ:ºUª�Ì@Ì�Ì2ª�Õ�ÚFÑq¹åÇÓ � ª4Öqº&ëµæTÝ@ÓYÙ�×mÜ�Ý\Ø~Þ�Ý�Ø�ß~Ú|ÖmÛ|Ý�Ö�ß~Ô�á�×mÓ�Ø8¼� ½ ¨4Ï1¯W� µ ¨ Ï1¯M�Jâ:¨ Ï1¯�¾�ÚFÜ%Ý ê ×mÙ{ÛIØ~Ô�YÝ`ß�Ôeã�Ô�ß#É�Ø� ¹`Õ§ÙpòQ¹/åT{ � ª�æPº�ª�Ì@Ì�Ì[ª�Õ�Ú|ÑB¹å9Ó � ª4Öqº�º�ë{ÊÇ°p¹�æ­ª � º&ëPå�°p¹�æ­ª � ª@Ì�Ì@Ì2ª4Öqº&çô¤Þ�Ý3Ü�Ý�Ø®Ô�ãQÝ ê ×mÙpØ~Ú|Ô�ÑmÜ�¨ ��� ¯(Ú�Ü®Ü�Ô�ÛFé�Ý@á0×TÜ�ÚFÑ�à§ã�Ô�×�ß�Ø~Þ5Ô�ßSá�ÝYß#�­×�Ñ�à�ÝYå*�%×�Ø�Ø~Ù�ÝYò�Ö�Û|Ú�Ó�Ú|Ø­Ø~Ú|Õ�Ý3ÚFÑ&Ø�ÝYà�ß~Ù{Ø�ÚFÔ�Ñ�ç(ô¤Þ�Ý3Ö=Ô�ÛFä&åÑ�Ô�Õ\Ú�Ù{ÛµÓSÞmÙ�Ô�ÜGß�Ý@Ü�×mÛ±ØSÜ(Ù�ß�Ý�Ó�Ô�Õ\ÖTÙ{ß~Ý`á§Ø~Ô\Ùeß~Ý�ã�ÝYß~ÝYÑTÓ�Ý%Ü�Ô�Û|×�Ø~Ú|Ô�Ñ0Ô�ãkÙeì¯Ô�Ñ&Ø�Ý3í(Ù{ß~Û|Ô\Ü�ÚFÕº×mÛFÙ{Ø�ÚFÔ�Ñ+�©Ú|Ø�Þ � æ�ú�Ü~Ù{Õ�Ö�ÛFÝ`Ü`çIÐÒÑ�ÀÚ|à�×�ß~Ý � Ø�ÞmÝ®ß~Ý`Ü�×�Û±ØSÜÏÙ�ß�Ý­à�Ú|é�ÝYÑ\ã�Ô�ßÏØ�Þ�Ý�ÖTÔ&Ü�Ú|Ø�ÚFÔ�Ñ0Ô{ã�Ø�Þ�Ý�Õ§Ù�Ü~ÜÏÙpØZ«S° � ã�Ô�ßÏØ�Þ�Ý�×�ÑmÓYÝYß�Ø~Ù{ÚFÑ&ØväeÚFÑv(wf1¨ ¬�¯Gá�ÝYß~ÚFé�Ý`áeã�ß~Ô�ÕØ�ÞmÝ�×mÑ�Ú±ã�Ô�ß�Õ á�ÚFÜ�Ø�ß~Ú|æm×�Ø�ÚFÔ�ÑxÔ{ãkØ�Þ�Ý�Õ§Ù�Ü~Ü�( ¨4¬�¯�çQÐÒÑ��ÀÚ|à�×�ß~Ý � Ù3Ú|Ø©Ó`Ù{ÑxæTÝ-Ü�ÝYÝ`Ñ"Ø�ÞTÙpؤØ�Þ�Ý%á�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�Ñ0ã�×mÑmÓ�Ø~Ú|Ô�Ñ[Ô�ãAØ�ÞmÝÔ�×�Ø�Ö�×�Ø­Ú�Ü­Ù�Ö�Ö�ß~ÔNò�ÚFÕ§ÙpØ~Ý`á0Ù�ÓYÓY×�ß~Ù{Ø�Ý`Û|ä§æ�ä0Ù\ÛFÔ��Ô�ßSá�Ý`ß(Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�ÓSÞmÙ�Ô�ܤÝYò�ÖTÙ{ÑmÜ�Ú|Ô�Ñ�ç

kz0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�

Page 7: AIAA-2006-2066-173

−0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.20

0.2

0.4

0.6

0.8

1

position x

prob

abili

ty d

istri

butio

n fu

nctio

n

exactPC (p=4)

> '@?q-�;/)�$�"43*'(�!;[3}7�5/�! %3%"&�!6/-@3&�!02;�0(Fb�0 1 2 3 4 5

10−8

10−6

10−4

10−2

100

polynomial order p

L 1 err

or

meanvariancedistributionLegendre

> 6�?q$�"&"%02"B)�02;�E[$�"&=2$�;/)�$D6EGFIHKJML��KO��R_IatHI]�EG_I[�_nq�]�QRL�STU2VWV�XV�YZJ�E\[RF0YZJM_n`baGLCS�VMHK`R�^L%m�]e]M_5Un[�HZ[Km�LCJ�]MUnE\[�SsU2V�V� 8E¡]�Q�U�HZ[ZEGqr_nJ�S�HZ[Km�L%JW]WUnEt[2]^opbE\V�]MJ�E\`bHI]�EG_I[¢OÐÒÑ+�ÀÚFà�×�ß~Ý � æ�Ø�Þ�Ý�ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�Û=ÓSÞmÙ�Ô�ÜQÝYò�ÖmÙ{ÑmÜ�ÚFÔ�ÑmÜ��©Ú±Ø~Þ[Ó�Û�Ù�Ü~Ü�ÚFÓ`Ù{Û¢£AÝYà�ÝYÑmá�ß~Ý­Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�ÜÏÙ{ÑTá§Ñ�×�Õ�ÝYß~ÚFÓ`Ù{ÛFÛ|ä§Ó�Ô�Õ\å

Ö�×�Ø~Ý`á²Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�Ü3Ù{ß~Ý0ÓYÔ�Õ�ÖmÙ{ß~Ý`á�çxô¤Þ�Ý[ÝYß~ß�Ô�ß3Ó�Ô�Ñ&é�ÝYß~à�Ý`ÑmÓ�Ý\ã�Ô�ß3Ø�Þ�Ýxá�ÚFÜ�Ø�ß~ÚFæ�×�Ø�ÚFÔ�Ñ�ã�×mÑmÓ�Ø~Ú|Ô�Ñ�ëIØ�Þ�Ý[Õ�Ý`Ù{ѽÙ�Ñmá�Ø�ÞmÝépÙ{ß~ÚFÙ�ÑmÓ�Ý%Ô�ã!$M¨ � ª�¬�¯¤Ú�Ü­Ü�Þ�Ô�©ÑxÙ�Ü(ã�×�ÑmÓ�Ø�ÚFÔ�Ñ#Ô{ãÀØ�ÞmÝ-ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{ÛAÔ�ßSá�ÝYß@ç�£�ÝYà�ÝYÑmá�ß�Ý%Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛAÓSÞmÙ�Ô�Ü(ÚFÜ®ÓSÞ�Ô�Ü�ÝYÑ5Ü�ÚFÑmÓYÝØ�ÞmÝYä¯Ù{ß~Ý3Ô�Ö�Ø�ÚFÕ§Ù{Ûkã�Ô�ß�Ø�Þ�Ý\×�Ñ�Ú|ã�Ô�ß~ÕKá�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�Ñ�ç�:�ÑmÛ|äxØ�ÞmÝeÑ�×�Õ�ÝYß~Ú�ÓYÙ{ÛFÛFä5ÓYÔ�Õ�Ö�×�Ø~Ý`á¯Ü�Ý�Ø%Ô{ã·Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�Ü�Ù�ÓSÞ�ÚFÝYé�Ý@ÜÝ�ò�Ö=Ô�Ñ�Ý`Ñ&Ø�Ú�Ù{ÛmÓYÔ�Ñ�é�Ý`ß�à�ÝYÑmÓYÝÏã�Ô�ßQØ�Þ�Ý®Ñ�Ô�Ñ�åÒÜ�Ø~Ù{ÑTá�Ù{ßSáeá�ÚFÜ�Ø�ß~Ú|æm×�Ø�ÚFÔ�ѧÔ�ã¤( f ¨ ¬U¯�çIÐÒÑ8�ÀÚ|à�×�ß�ÝLà%Ø~Þ�Ý­ß~Ý`Ü�×�Û|Ø~ÜIã�Ô�ß�( f ¨4¬�¯QæTÙ�Ü�Ý`áÔ�Ñ5Ø~Þ�Ý3ÛFÔ�à�Ñ�Ô�ß~Õ§Ù{Û�á�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�Ñ5ã�Ô�ß­Õ§Ù�Ü�Ü�( ¨ ¬�¯©Ù�ß�Ý-à�Ú|é�ÝYÑ�ç(ÐÒÑ��ÀÚ|à�×�ß~Ý9à�Ù\Ø�ÞmÝeÙ{ÖmÖ�ß�ÔNò�ÚFÕ§ÙpØ�ÚFÔ�ÑxÔ{ãIØ~Þ�ݺá�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�Ñã�×�ÑmÓ�Ø�ÚFÔ�Ñ�Ô{ã*$²ÚFÜ%à�Ú|é�ÝYÑ¡Ù�Ñmá#Ø~Þ�ݧÝYß~ß~Ô�ß%Ó�Ô�Ñ�é�ÝYß~à�ÝYÑTÓ�Ý3Ú�Ü-Ü�ÞmÔ�©Ñ¯Ø�Ô5æ=Ý�Ý�ò�ÖTÔ�Ñ�ÝYÑ&Ø~ÚFÙ�ÛQÚ|ÑÀ�ÀÚ|à�×�ß�Ýûà�æAç\ô¤Þ�Ý�ß~Ý`Ü�×�Û|Ø~Ü%Ù{ß~ÝÓ�Ô�Õ�ÖmÙ{ß~Ý`á0Ø�Ô"û®ÝYß~Õ\Ú|Ø�ݺÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û�ÓSÞTÙ{Ô�ܤ�©ÞmÚFÓSÞ5Ú�Ü­Ô�Ö�Ø�ÚFÕ�Ù�Ûµã�Ô�ߤØ~Þ�Ý3Ñ�Ô�ß�Õ§Ù�Û�á�ÚFÜ�Ø�ß~ÚFæ�×�Ø�ÚFÔ�ÑAçÏÐbØ�Ó`Ù{Ñ5æTÝeÜ�Ý`ÝYÑxØ�ÞmÙ{Ø­ÚFÜÓ�Ô�Ñ�é�ÝYß~à�Ý`ÑmÓ�Ý@Ü�Ü�Ú|à�Ñ�Ú±õ=ÓYÙ{Ñ&Ø~Û|ä%Ü�ÛFÔ�(ÝYß�Ø~ÞmÙ{ѺØ�Þ�ÝGÑ�×�Õ�ÝYß~ÚFÓ`Ù{ÛFÛ|ä-Ó�Ô�Õ�Ö�×�Ø�Ý@áºÜ�Ý�ØÀÔ{ãmÔ�ß�Ø~Þ�Ô�à�Ô�ÑmÙ�Û{ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛFÜ`çAô¤Þ�Ý@Ü�Ý(ß�Ý@Ü�×mÛ±ØSÜá�Ý`Õ\Ô�ÑmÜ�Ø�ßSÙpØ�Ý�Ø~ÞmÙpغØ�Þ�Ý"Ù�Ö�Ö�ß~Ô�Ù�ÓSÞ�Ô{ã­ÓYÔ�Õ�Ö�×�Ø~Ú|Ñ�à5Ø~Þ�Ý"Ô�Ö�Ø�ÚFÕ§Ù{ÛÏÜ�Ý�غÔ�ã(Ô�ß�Ø~Þ�Ô�à�Ô�ÑmÙ�ÛQÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛFÜ�ã�Ô�ß-Ø�Þ�Ý0Ñ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ßØ�ßSÙ{ÑTÜvã�Ô�ß�Õ§ÙpØ~Ú|Ô�Ñ0Ô�ãAØ�ÞmÝ-×�ÑmÓ�Ý`ß�ØSÙ{ÚFÑ5Ú|Ñ�Öm×�Ø­ÚFÜ®Ù{ÑxÝ[�"Ó�ÚFÝYÑ&Ø®Ù{ÖmÖ�ß�Ô&Ù�ÓSÞ"Ô�ãIá�Ý`Ù{ÛFÚFÑ�à��©Ú±Ø~Þ[Ø~Þ�Ú�ܤØvä�ÖTÝ-Ô�ãkÑ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß~Ú±Ø~Ú|Ý@ÜYç¥+짦#¨��_ôÈõ:÷�÷@ö/õ­ñ_ø�ù*ô�öÐÒÑ[Ø�Þ�Ú�Ü­Ü�Ý`Ó�Ø~Ú|Ô�ÑxÙ\Ñ�Ô�Ñ�ÛFÚ|Ñ�Ý@Ù{ß(Ø�ßSÙ{ÑTÜvã�Ô�ß�Õ§ÙpØ~Ú|Ô�Ñ0ÚFÑ�é�Ô�ÛFé�Ú|ÑmàeÕe×�Û±Ø~Ú|ÖmÛ|Ý-×�ÑTÓ�ÝYß�Ø~Ù�Ú|Ñ[ÖmÙ{ßSÙ{Õ�ÝYØ�ÝYßSÜ(ÚFÜ©ÓYÔ�ÑmÜ�Ú�á�ÝYß~Ý`á[Ú|Ñ[Ø�ÞmÝ3Ó�Ô�Ñ�åØ�ÝYò�Ø�Ô{ã·ÞmÝ`ÙpØ®Ø�ßSÙ{ÑmÜ�ã�ÝYß�Ú|Ñ1Ù§Ø��GÔ�åbá�ÚFÕ�ÝYÑmÜ�ÚFÔ�ÑmÙ�ÛAÓSÞmÙ{ÑmÑ�ÝYÛAâmÔ�-ë=Ü�ÝYÝÇ�kÚFà�×mß�ÝÇç�çl:®ãoØ�Ý`ѯպ×�Û|Ø�ÚFÖ�ÛFÝe×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ#ÖmÙ{ßSÙ{Õ�Ý�Ø~ÝYßSÜÓYÙ�Ñ0æ=Ý-Ó�Ô�Õeæ�ÚFÑ�Ý`á[Ú|Ñ&Ø~Ô\Ù�Ü�Ú|Ñ�à�Û|Ý%Ü�Ú|Õ�ÚFÛFÙ�ß�Ú|Øvä§ÖmÙ{ßSÙ{Õ�ÝYØ�ÝYß(ÚFÑ&é�Ô�ÛFé�Ú|Ñ�àeÑ�Ô�ÑmÛ|ÚFÑ�Ý`Ù�ß(Ø�ßSÙ{ÑmÜ�ã�Ô�ß~Õ§ÙpØ�ÚFÔ�ÑTÜYçM��Ô�ß(Ø~Þ�Ý-Þ�Ý`Ù{ØGØ~ß~Ù�ÑmÜ�åã�ÝYß\ÚFѽØ�ÞmÝ0Ø��(Ô{åÒá�ÚFÕ\Ý`ÑmÜ�Ú|Ô�ÑmÙ{ÛÏÓSÞTÙ{Ñ�Ñ�Ý`ÛÏâTÔ� Ù{ѽ×�ÑTÓ�ÝYß�Ø~Ù�Ú|Ñ�Þ�Ý`Ù{Ø\Ó�Ô�Ñmá�×mÓ�Ø�ÚFé�Ú±Øvä�©_¨ ¬�¯�ëGá�ä�ÑmÙ�Õ\Ú�Ó"é�Ú�Ü�ÓYÔ�Ü�Ú±Øvä54�¨ ¬U¯ºÙ�ÑmáÖ�ß~Ý`Ü~Ü�×mß�Ý�à�ßSÙ�á�ÚFÝYÑ&Ø<ª �ª%« ¨ ¬�¯¤Ó`Ù{ÑxæTÝ3ÓYÔ�Õeæ�Ú|ÑmÝ`á[Ú|Ñ&Ø�Ô"Ù�Ñ[×mÑmÓ�Ý`ß�ØSÙ{ÚFѱ�u¬Ý`ÓYÛ|ÝYØ(Ñ�×�ÕeæTÝ`ß,Ò�~U¨4¬�¯�° ®M¨4¬�¯^­¤Ø2©_¨ ¬�¯�ç·ô¤Þ�Ý-×�ÑmÓYÝYß�åØ~Ù�Ú|Ñ&Øvä0á�ÚFÜ�Ø�ß~Ú|æm×�Ø�ÚFÔ�ÑxÔ{ãSÒ�~Ȩ ¬�¯GÚ�Ü©Ú|Ñ[Ø�ÞTÙpØ®ÓYÙ�Ü�Ý%Ù\Ñ�Ô�ÑmÛ|ÚFÑ�Ý`Ù�ß­Ó�Ô�Õºæ�ÚFÑmÙpØ~Ú|Ô�Ñ0Ô�ãkØ~Þ�Ý%×�ÑmÓ�Ý`ß�ØSÙ{ÚFÑ&Øvä0á�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�ÑmܤÔ{ã�©�¨4¬�¯�ë4�¨ ¬U¯­Ù{ÑT᮪ �ª%« ¨ ¬�¯�ç�ôkÔ"ß�ÝYØ~Ù�Ú|Ñ#ÝYò�Ö=Ô�ÑmÝYÑ&Ø�Ú�Ù{ÛÀÓ�Ô�Ñ�é�ÝYß~à�ÝYÑTÓ�Ý�ë�Ø~Þ�ݺÔ�ß�Ø~Þ�Ô�à�Ô�ÑmÙ�Û�Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�Û�æmÙ�Ü�Ú�Ü­ÞmÙ�Ü©Ø~Ô0æ=ÝeÓ�Ô�ÑmÜ�Ø�ß~×mÓ�Ø~Ý`áÑ�×�Õ�ÝYß~ÚFÓ`Ù{ÛFÛ|ä�ç�ô¤Þ�ÝeÔ�×�Ø~Ö�×�ØSÜ�Ô�ã·Ú|Ñ&Ø~ÝYß~Ý`Ü�Ø�Ù{ß~Ý-Ø~Þ�Ýe×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ�ØväxÚFѯØ�ÞmÝ3Ø�Ý`Õ�ÖTÝ`ß~Ù{Ø�×�ß~Ý3õmÝ`ÛFá¡Ù{Ñmá#Ø�Þ�Ý\×�ÑmÓYÝYß�Ø~Ù{ÚFÑ&Øvä[æTÙ{ßSÜã�Ô�ߤØ~Þ�Ý%Ø�Ý`Õ�ÖTÝ`ß~Ù{Ø�×�ß~Ý%Ô�ÑxØ�Þ�Ý-×mÖ�ÖTÝ`ßL�(Ù�Û|Ûtçô¤Þ�Ý(âmÔ� Ú|ÑeØ�Þ�ݤØ��(Ô{åÒá�ÚFÕ�ÝYÑmÜ�Ú|Ô�ÑmÙ{Û�ÓSÞmÙ�Ñ�Ñ�ÝYÛ�Ú�ÜQÙ�Ü~Ü�×�Õ�Ý`áºØ�Ô%æTÝ­ÙJ�IÔ�ÚFÜ~Ü�Ý`×�ÚFÛ|ÛFÝÏâTÔ���©Þ�ÝYß~Ý(Ø�Þ�Ý©é�ÝYÛFÔ�Ó�Ú|Øvä3Ó�Ô�Õ�Ö=Ô�Ñ�Ý`Ñ&Ø®#ÚFÑ[Ø~Þ�Ý�$�åÒá�ÚFß�Ý@Ó�Ø~Ú|Ô�Ñ[Ú�Ü­Ù�Ü~Ü�×�Õ�Ý`á0Ø�Ô�æTÝ3Ùeã�×�ÑTÓ�Ø�ÚFÔ�Ñ#Ô�ãi¯[Ô�Ñ�ÛFä®M¨�¯P¯�°°{ �� 4 áÖáZ$ ¨3± ñ {5¯ ñ ¯2ª ¯ � y\{�±qª�± | ª ¨ � àȯ

² 0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�

Page 8: AIAA-2006-2066-173

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.40

0.5

1

1.5

2

position x

prob

abili

ty d

ensi

ty fu

nctio

n

exactPC (p=4)

> '@?q-�;/)�$�"43*'(�!;[3}7�5/�! %3%"&�!6/-@3&�!02;�0(Fb�0 1 2 3 4 5

10−8

10−6

10−4

10−2

100

polynomial order p

L 1 err

or

meanvariancedistributionHermite

> 6�?q$�"&"%02"B)�02;�E[$�"&=2$�;/)�$D6EGFIHKJMLe³ZO´�R_IatHI]�EG_I[�_2q*]�QRL,SsU2V�V^X�VWYZJ�Et[RFµYZJW_I`bacLdS¶VWHK`R�^LCm�]�]W_µUn[0HZ[Km�LCJ�]MUnE\[5SsU2V�V· 8Ec]�Q�UµaG_2FI[R_nJ�SsUna!HZ[Km�L%JW]WUnEt[2]^opbE\V�]MJ�E\`bHI]�EG_I[¢O�©Þ�Ý`ß�Ý�±xá�ÝYÑ�Ô�Ø�Ý@Ü·Ø~Þ�Ý%Þ�ÝYÚFà�Þ&ؤÔ{ã�Ø~Þ�Ý�ÓSÞTÙ{Ñ�Ñ�Ý`Û � ±8°^æPÌ æ � Õ÷Ù{ÑTá§Ø�ÞmÝ�Ömß�Ý@Ü�Ü�×�ß~Ý®à�ßSÙ�á�Ú|Ý`Ñ�Ø ª �ª%« ÚFÜ(ÚFÑmá�Ý`ÖTÝ`Ñmá�ÝYÑ&Ø©Ô{ãi$xÙ�Ñmá¯�ç0ô¤Þ�ݧé�ÝYÛFÔ�ÓYÚ±Øvä¡Ó�Ô�Õ�Ö=Ô�Ñ�Ý`Ñ&Ø%ÚFÑ�Ø�ÞmÝe¯�åÒá�ÚFß�Ý@Ó�Ø~Ú|Ô�Ñ¡Ú�Ü3Ù�Ü~Ü�×�Õ�Ý`á1Ø�Ô#æ=Ý��YÝYß~Ômç8�÷ÞmÝ`ÙpØ~Ú|Ñmà#ÝYÛFÝYÕ�ÝYÑ&Ø3Ú�Ü-Ö�ß~Ý`Ü�ÝYÑ&Ø3ÙpØ-Ø�ÞmÝÛFÔ�GÝ`ßz�¤Ù{ÛFÛ�ã�Ô�ß8­ î�¸ $ ¸ ¨3­ î Ó¹­9º%»(¯z�©Þ�Ú�ÓSÞxÕ�Ù�Ú|Ñ&ØSÙ{ÚFÑmÜ­Ù�Ó�Ô�ÑmÜvØSÙ{Ñ&ØL�(Ù�Û|Û�Ø~ÝYÕ�Ö=ÝYßSÙpØ�×mß�Ý·¼�º�»@ëP�©Ú|Ø�Þ½­ î °®­9º%»¤° î� ­­çô¤Þ�ݤÔ{Ø~Þ�ÝYßS�¤Ù{ÛFÛ�ÜÀÙ�ß�ݤÙ�á�ÚFÙ�æmÙpØ~ÚFÓ¤Ù{ÑTáºÚFÑ�âmÔ��Ù�ÑmáeÔ�×�Ø�âmÔ��æ=Ô�×�ÑTá�Ù{ß~ä3Ó�Ô�Ñmá�Ú|Ø�ÚFÔ�ÑmÜQÙ�ß�Ý(Ö�ß~Ý`Ü~Ó�ß~Ú|æ=Ý`áeÙpØ�$±°^æ�Ù�ÑmáT$�°�­­ë�©Ú|Ø�Þ�­^°\æPÌ � Õ5ç%ÐbØ%ÚFÜ%Ù�Ü~Ü�×�Õ�Ý`á5Ø~ÞmÙpØ�Ø�Þ�ÝeØ�Ý`Õ�ÖTÝ`ß~Ù{Ø�×�ß~Ý\á�Ô&Ý@Ü®ÑmÔ{Ø%Ú|Ñ�âT×�ÝYÑmÓYݺØ~Þ�ÝeâmÔ��õmÝYÛ�áµç3ô¤Þ�Ý\Ý ê ×mÙ{Ø�ÚFÔ�ѯÔ�ã·Ø�ÞmÝÞ�Ý@ÙpؤØ�ßSÙ{ÑmÜ�ã�ÝYß©Ú�ܤØ~Þ�ÝYÑ5à�ÚFé�Ý`Ñ[æ�ä ®n¨�¯P¯2¾ ¼¾ $ {0©�¿�¾ ñ@¼¾ $ ñ Ów¾ ñ�¼¾ ¯ ñ�À °^æPª ¨ � ç�¯�©Þ�Ý`ß�ÝJ®n¨�¯P¯(ÚFܤà�Ú|é�ÝYÑxæ�äÕ¨ � àU¯�çIô¤ÞmÝ-ÖmÙ{ßSÙ{Õ�Ý�Ø~ÝYßSÜ�©�¨4¬�¯�ë¢4�¨ ¬�¯�ë ª �ª%« ¨ ¬�¯(ÚFÑ#Ó�Ô�Õºæ�ÚFÑmÙpØ~Ú|Ô�Ñ��©Ú|Ø�ÞxØ~Þ�Ý-æTÔ�×�Ñmá�Ù�ß�ä[Ó�Ô�ÑTá�Ú±Ø~Ú|Ô�Ѽ º�» ¨ ¬�¯¤Ù�ß�Ý�Ù�Ü~Ü�×mÕ\Ý@á§Ø~Ô\æ=Ý%×�ÑmÓYÝYß�Ø~Ù{ÚFÑ5Ù{Ñmá0Ú|Ñmá�ÝYÖ=ÝYÑmá�Ý`Ñ&Ø`ç·ÐÒÑ5ôkÙ�æ�ÛFÝ � Ø�Þ�Ý`Ú|ß©×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ�Ø~Ú|Ý@ܤÙ{ß~Ý�à�Ú|é�ÝYÑ�çQô¤Þ�Ý�Ñ�Ô�Õ�ÚFÑmÙ{ÛépÙ{ÛF×�Ý`ܤÔ�ãkØ~Þ�Ý-×�ÑmÓYÝYß�Ø~Ù�Ú|ÑxÖmÙ{ßSÙ{Õ�ÝYØ�ÝYßSܤÙ{ß~Ý�épÙ�Û|Ú�á[ã�Ô�ß­Ü�Ø~Ù{ÑTá�Ù{ßSá[Ü�Ý`Ù\ÛFÝYé�ÝYÛ�Ó�Ô�Ñmá�Ú|Ø�ÚFÔ�ÑmÜ`çP�U2`baGLTN2O;Á6U2JMUnSsL@]ML%J�VÂEt[u]�QRL�QRL�Ud]�]�JWUn[KV�qrL%J�YZJM_n`baGLCSÃE\[<U�]^ Â_2X�pbEtSTLd[KVWEG_I[RUnaZm�QRUn[Z[RLCa¢Äb_d �Å�pKL@]WLCJ�S�Et[ZE\V�]�E\m�ÆCUna\HRL·Un[KpHZ[Km�LCJ�]MUnE\[2]^oepbEGV^]MJ�E\`bHI]�EG_I[¢O

ÖmÙ{ßSÙ{Õ�Ý�Ø~ÝYß Ü�ä�ÕeæTÔ�Û épÙ{ÛF×�Ý ×mÑ�Ú±Ø á�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�Ñ Ó�Ôpé#ÇNÚFÑ&Ø�ÝYß~épÙ{ÛÓ�Ô�Ñmá�×mÓ�Ø�ÚFÔ�Ñ © � Ì!ç2�Uç#Á � æZh�ñ ÈKÇpü�à�� ÛFÔ�à�Ñ�Ô�ß~Õ§Ù{Û æ­Ì � çé&Ú�Ü~Ó�Ô�Ü�Ú|Øvä 4 � Ìc+��n�1à�Á � æbhQ� ü�àKÇpÕ§Ü ×�ÑmÚ±ã�Ô�ß�Õ æ­Ì � ç

Ö�ß~Ý`Ü~Ü�×�ß�Ý%à�ß~Ù�á�Ú|Ý`Ñ&Ø ª �ª%« å � ç æ ü�àKÇNÕ§ñ�Ü~ñ çdÉ º Ô�ß~á�Ý`ߤÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û yG{ � Ì çP��{9Ì!ç |Þ�Ý`Ù{Ø�ÚFÑ�à�ÝYÛFÝYÕ�ÝYÑ&ؤØ�Ý`Õ�ÖTÝ`ß~Ù{Ø�×�ß~Ý ¼ º%» � + � ç � ç � ×�ÑmÚ±ã�Ô�ß�Õ æ­Ì æUçô¤Þ�Ý\Ó�Ô�Õeæ�ÚFÑ�Ý`á#Ý�=Ý@Ó�Ø�Ô{ãQØ�Þmݺã�Ô�×�ß�×mÑmÓ�Ý`ß�ØSÙ{ÚFѯÖmÙ{ßSÙ{Õ�ÝYØ�ÝYßSÜ©Ô�ãÏôkÙ�æ�Û|Ý � Ú�Ü­ØSÙ{ü�ÝYÑ#ÚFÑ�Ø~Ô[Ù�ÓYÓ�Ô�×�Ñ&Ø�ÚFÑ1Ù0Ü�ÚFÑ�à�ÛFݺÖ=Ô�ÛFä&å

Ñ�Ô�Õ\Ú�Ù{Û©ÓSÞmÙ�Ô�Üeß�×mÑ�çË�®Ü\Õ�ÝYÑ&Ø~Ú|Ô�Ñ�Ý`á�æTÝYã�Ô�ß~Ý0Ø~Þ�ÝxØ�Þ�ß~ÝYÝxÖmÙ{ßSÙ{Õ�ÝYØ�ÝYßSÜu©�¨4¬�¯�ë�4�¨ ¬�¯\Ù{ÑTáʪ �ª%« ¨ ¬U¯eÙ�ß�ÝxÓ�Ô�Õºæ�ÚFÑ�Ý`á�Ú|Ñ Ø�ÞmÝ�u¬Ý`Ó�ÛFÝ�Ø­Ñ�×�ÕeæTÝ`ß�Ò�~Ȩ4¬�¯�ç�ô¤Þ�ݺß~Ý`Ü�×�Û|Ø�ÚFÑ�à§Ø��(Ô"×�ÑmÓYÝYß�Ø~Ù{ÚFÑ#ÖmÙ�ß~Ù�Õ\ÝYØ�Ý`ß~Ü,Ò�~Ȩ4¬�¯®Ù�Ñmáµ¼�º�»/¨ ¬�¯�Ù{ß~Ý3ß~ÝYÖ�ß~Ý`Ü�ÝYÑ&Ø~Ý`áxæ�ä5Ù§Ø��GÔ�åá�ÚFÕ�ÝYÑmÜ�Ú|Ô�ÑmÙ{ÛQÖ=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛQÓSÞmÙ�Ô�Ü�Ý�ò�ÖmÙ�ÑmÜ�ÚFÔ�ÑAç�ø�×mæmÜvØ~Ú±Ø~×�Ø�ÚFÑ�à5Ø~Þ�Ý�ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û·ÓSÞTÙ{Ô�Ü�ÝYò�ÖTÙ{ÑmÜ�Ú|Ô�Ñ1ÚFÑ&Ø�Ôc¨ � ç1¯�Ù{Ñmá¡ÖTÝ`ß�å

iz0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�

Page 9: AIAA-2006-2066-173

heating element The

L

Poisseuille flow

L

L

adiabatic wall

2h

1 he

> '@?q)&�'(;/;/$�<KË0�vD'(<!02;/=L'z�/$�'�3%�!;/=L$�<!$�KL$�;[3 > 62?B5/$s3&$�"%KL�!;/�! %3%�!)_3%$�KL.1$�"&'�3&-/"%$�B$�<!5D6EGFIHKJML·ÌKO;ÍlL�Ud]�]MJMUn[KV^qrLCJ8E\[<U�]^ Â_2X�pbEtSTLd[KVWEG_I[RUnabÁi_IE\V�V�LCHZEta\aGL�m�QRUn[Z[RLda¢Äb_C ®Unac_I[RFsUsQRL�Ud]�E\[RF�LCaGLCSsLd[�]%O

ã�Ô�ß~Õ�Ú|Ñmà�Ù§ö%Ù�Û|Ý`ß�ü�ÚFÑ[Ö�ß~Ô{�vÝ@Ó�Ø~Ú|Ô�Ñ[Ô�Ñ�Ø~ÔºØ~Þ�Ý%Ø��(Ô{åÒá�Ú|Õ�Ý`ÑmÜ�ÚFÔ�ÑTÙ{Û�ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�Û�ÓSÞmÙ{Ô&ÜGæTÙ�Ü�ÚFÜ�¹{· µ º/е�¶ � ß~Ý`Ü�×�Û±ØSܤÚ|Ñ��� �C� â�� î�� �C�´½ ¶ ���W�C� â hµî�� � �

-Ò�~/½ â ¾ ¼B½¾ $ { ¿ ¾ ñ@¼ â¾ $ ñ Ów¾ ñ�¼ â¾ ¯ ñsÀ °^æPª åû°^æPª � ª@Ì�Ì�Ì�ª�Ò¤ª ¨ � �U¯�©Ú|Ø�Þ -Ò�~�½ â ° �¼&· ñ â ¨4Ï1¯�¾ î´µ�¶ � Ò�~ µ ¼&·±½�¨ Ï1¯�· µ ¨ Ï1¯�· â ¨ Ï1¯�¾[Ì ¨ � +1¯ô¤Þ�Ý¡Ô�Ö�Ø~Ú|Õ§Ù{Û�Ü�ÝYØ[Ô�ã-Ø��GÔ�åbá�ÚFÕ�ÝYÑmÜ�ÚFÔ�ÑmÙ�Û­æmÙ�Ü�Ú�Ü"Ô�ß�Ø�ÞmÔ�à�Ô�ÑmÙ{Û©Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�Üð¹{· µ ¨4Ï1¯(º�е�¶ � Ú�Ü0Ô�æ�Ø~Ù�Ú|ÑmÝ`á,æ�ä´Ø�Þ�Ý1Ø�ÝYÑTÜ�Ô�ßÖ�ß~Ô�á�×mÓ�Ø-Ô�ãÏØ�ÞmݧÑ&×mÕ\Ý`ß�Ú�ÓYÙ�Û|ÛFä¯Ó�Ô�Õ�Ö�×�Ø�Ý@á1Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�Ü�ã�Ô�ß�Ò�~Ȩ ¬�¯�Ù�Ñmá�¼ º%» ¨4¬�¯�ç§ô¤Þ�Ý\Ø��(Ô{åÒá�Ú|Õ�Ý`ÑmÜ�ÚFÔ�ÑTÙ{ÛIÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛÓSÞmÙ�Ô�Ü·ÝYò�ÖmÙ{ÑmÜ�ÚFÔ�ѧÚ�Ü·Ø~ß�×�ÑTÓYÙpØ~Ý`á"Ù{Ø_Ö�° � ëU�©Þ�Ú�ÓSÞ"ß�Ý@Ü�×mÛ±ØSÜGÙºÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û=ÓSÞmÙ{Ô&ÜÏÙ�Ö�Ö�ß~ÔNò�ÚFÕ§ÙpØ~Ú|Ô�Ñ\ÚFÑ"Ø~Þ�Ý­õTß~Ü�Ø(Ü�Ú±ò�Ø~ÝYß~Õ�ܨ4�¤°Jç1¯�çM� ê çd¨ � ��¯·ÚFÜ(á�Ú�Ü�ÓYß�ÝYØ�Ú��YÝ`á"×mÜ�ÚFÑ�à\Ù3Ü�Ý@Ó�Ô�Ñmá�åbÔ�ßSá�ÝYßQõTÑ�Ú±Ø~Ý�é�Ô�ÛF×�Õ�Ý®Õ�Ý�Ø~Þ�Ô�á��©Ú|Ø�Þ[ÙeÜ�ÖTÙpØ�Ú�Ù{Û=ÜvØ~ÝYÖmÜ�Ú��YÝ�Îs$±° � æbh=ñpçô¤Þ�Ú�Ü%ß�Ý@Ü�×mÛ±ØSÜ�ã�Ô�ß�Ø�Þ�ݧá�ÝYØ�Ý`ß�Õ�ÚFÑ�ÚFÜ�Ø�Ú�Ó\Ø�Ý`Õ�ÖTÝ`ß~Ù{Ø�×�ß~ݺõTÝYÛ�áµëkÜ�ÝYÝ��ÀÚ|à�×�ß~Ý�ç{æ�ë�ÚFÑ�Ù5á�Ú�Ü�ÓYß�ÝYØ�Ú��`Ù{Ø�ÚFÔ�Ñ1Ý`ß�ß~Ô�ß�Ô{ã � æbhQ�eÚ|Ñ�Ø�ÞmÝ­ ñ åtÑmÔ�ß~ÕxçºÐÒÑ��ÀÚFà�×�ß~Ý���Ù§Ø~Þ�ݧÜvØSÙ{Ñmá�Ù�ß~á¡á�ÝYé�Ú�ÙpØ~Ú|Ô�Ñ1Ô{ãGØ�Þ�ÝeØ�Ý`Õ�ÖTÝ`ß~Ù{Ø�×�ß~Ý\õmÝYÛ�á1ß~Ý`Ü�×�Û|Ø�ÚFÑ�à[ã�ß�Ô�Õ Ø�Þ�Ý�×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ&Ø�ÚFÝ`Ü�ÚFÑÒ�~Ȩ ¬�¯GÙ{ÑTá<¼�º�»/¨ ¬�¯(Ó`Ù{Ñ0æ=Ý%Ü�Ý`ÝYÑ�çIÐÒÑ��kÚFà�×mß�Ý��{æ"Ø~Þ�Ý�Õ\Ý@Ù{ÑxÙ{Ñmá0×�ÑmÓ�Ý`ß�ØSÙ{ÚFÑ&Øvä\æTÙ{ßSÜÏæmÙ�Ü�Ý@á§Ô�Ñ0Ø�Þ�Ý-Ü�Ø~Ù�Ñmá�Ù{ßSá"á�Ý`é�ÚFÙ{Ø�ÚFÔ�ÑÔ{ãQØ~Þ�Ý3Ø~ÝYÕ�Ö=ÝYßSÙpØ�×mß�ÝeÔ�Ñ5Ø�Þmݺ×�Ö�Ö=ÝYß��¤Ù{ÛFÛÀÙ�ß�ݺÜ�Þ�Ô�©Ñ5ã�Ô�ß­Ø�ÞmݺÖ=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛkÓSÞmÙ�Ô�Ü�Ù{Ö�Ö�ß~Ô�Ù�ÓSÞ��©Ú±Ø~Þ8ÖD° � Ù�Ñmá#ì#Ô�Ñ&Ø~Ýí(Ù{ß~ÛFÔ\Ü�Ú|Õe×�Û�ÙpØ�ÚFÔ�ѱ�©Ú|Ø�Þ½Ïm° � æ1æ�ç

ÅnÐÁÆ Ñ5Ê����©Ç®Ê� ÷Î�¡S�ºÇ®ÊÏÇ#��Î�Ç#�q¡IÉmÎÒÈ�Î��_¢�Î�Ç�ÒÏÊ���Ò­Î�Ç8Ó/Ì�Ç®Íd��É�È:¡QÎ�ÇÔÒn¡IÉmÎ�¡�Õ�����¢�­ÑmÔ{Ø�ÞmÝYßIÚ|Õ�Ö=Ô�ß�Ø~Ù�Ñ�ØIØvä�Ö=Ý(Ô�ãmÑ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß~Ú±Ø~Ú|Ý@ÜkÚFÑ�é�Ô�Û|é�ÝÏ×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ\épÙ{ß~ÚFÙ�æ�Û|Ý@ÜAÔ�ãa�©Þ�Ú�ÓSÞeØ�Þ�ݤ×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ&Øvä3á�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�ÑÚ�Ü·ÖmÙ�ß�Ø·Ô�ãTØ~Þ�Ý�Ü�Ô�ÛF×�Ø�ÚFÔ�Ñ�Ô�ãTØ~Þ�Ý­×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ&Øvä ê ×mÙ�Ñ&Ø�Ú|õTÓYÙ{Ø�ÚFÔ�Ñ�Ö�ß~Ô�æ�ÛFÝYÕ¾Ú|Ñ"ÓYÔ�Ñ&Ø�ßSÙ�Ü�Ø��©Ú±Ø~Þ§×�ÑmÓYÝYß�Ø~Ù{ÚFÑ�ÖmÙ{ßSÙ{Õ�ÝYØ�ÝYßSÜn�©Þ�Ú�ÓSÞÞmÙNé�Ý¡Ù ü&ÑmÔ�©Ñ¸×�ÑmÓYÝYß�Ø~Ù{ÚFÑ&Øvä'á�ÚFÜ�Ø�ß~ÚFæ�×�Ø�ÚFÔ�ÑAç ÐÒÑ�Ø~Þ�ÚFÜ5Ü�Ý`Ó�Ø�ÚFÔ�Ñ�Ömß�Ô�æ�Û|Ý`Õ§Ü0ÚFÑ�é�Ô�ÛFé�Ú|Ñmà�Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�Û�ÑmÔ�Ñ�ÛFÚ|ÑmÝ`Ù{ß~Ú|Ø�ÚFÝ`Ü[Ô{ã×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ'épÙ{ß~Ú�Ù{æ�ÛFÝ`Ü"Ù�ß�ݯÓYÔ�ÑmÜ�Ú�á�ÝYß~Ý`áµç ô¤Þ�Ý¡Ù{Ö�Ö�ÛFÚ�ÓYÙpØ~Ú|Ô�Ñ'Ô{ã-Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�Û®ÓSÞmÙ�Ô�Ü�Ø~Ô½ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�Û®Ñ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß~Ú±Ø~Ú|Ý@ܧÚFÜá�Ú�Ü�ÓY×mÜ~Ü�Ý@á1ÚFÑ�à�ÝYÑmÝYßSÙ{ÛQÙ{Ñmá�Ù{ÖmÖ�Û|ÚFÝ`á¡Ø~Ô5ÙxÑ�Ô�Ñ�ÛFÚ|Ñ�Ý@Ù{ß-Ù�á�é�Ý`Ó�Ø�ÚFÔ�Ñ�åÒá�Ú�=×TÜ�ÚFÔ�Ñ�Ö�ß~Ô�æ�ÛFÝYÕ5ç���Ô�ß3Ó�Ô�Õ�ÖmÙ�ß�Ú�Ü�Ô�Ñ1Ù�ÛFÜ�Ô#Ù[ÛFÚFÑ�Ý`Ù�ßÙ�á�é�Ý`Ó�Ø�ÚFÔ�Ñ�åÒá�Ú�=×TÜ�ÚFÔ�ѽÖmß�Ô�æ�Û|Ý`Õ ÚFÜ�ÓYÔ�ÑmÜ�ÚFá�ÝYß~Ý`áµç�ÐÒÑ,Ü�Ý@Ó�Ø�ÚFÔ�Ñ�� ÑmÔ�Ñ�Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û(Ñ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ß�Ú|Ø�ÚFÝ`Ü\ÚFÑ&é�Ô�ÛFé�Ú|Ñ�à1×�ÑmÓ�Ý`ß�ØSÙ{ÚFÑépÙ{ß~ÚFÙ�æ�ÛFÝ`ܤÙ{ß~Ý-á�ÚFÜ~Ó�×TÜ�Ü�Ý`áµç

Ö 0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�

Page 10: AIAA-2006-2066-173

0 0.02 0.04 0.06 0.08 0.1

−0.01

0

0.01

σT

x position

p=2

y po

sitio

n

2

4

6

8

10

12

14

16

18

> '@?B %3&'(;/5'("&5�5/$sE��t'�3%�!02;�B$�<!50 0.02 0.04 0.06 0.08 0.1

280

300

320

340

360

380

flow direction x

tem

pera

ture

Tw

mean

PC (p=2)MC (N=100)

> 62?d-/.�.1$�"qvq'(<!<�3%$�KL.{$�"&'�3&-�"&$D6EGFIHKJML�×KOµØ�[Km�L%JW]MUnE\[2]^o<E\[e]�QRL·]WLdS�YbL%JMUd]�HKJWLlÙbLCa\p;pbHRL·]W_s]�QRL�m�_IST`bE\[RLCpeL@ÚÛL%m�]l_2q6qr_IHKJlHZ[Km�L%JW]WUnEt[<YKU2JMUnSsL@]ML%J�VÂÜ >tÝ ?Þß >àÝ ?Þ�á^âá�ã >tÝ ? Un[Kp<äbå >tÝ ? Oë�ìîë8þMþMò%ó4ý�õ:÷@ó%ïañ ïQù¤þ�ïQò�æqñMïaú�ó%õPò�ýd�MõPïaø8÷@ï�þ�ïQò�æqñMïaú�ó%õPò�ñMïañMò4ó%ñ_ôÈõ:öó4÷@ó4ôÈø�IÔ�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û=ÓSÞmÙ{Ô&ÜÏÚ�ܤÙ{Ö�Ö�ÛFÚ�ÓYÙ{æmÛ|Ý�Ø�Ô\ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�Û=ÑmÔ�Ñ�ÛFÚ|ÑmÝ`Ù{ß~Ú|Ø�ÚFÝ`ÜGÚ|Ñ�é�Ô�Û|é�ÚFÑ�àº×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ0épÙ{ß~ÚFÙ�æ�ÛFÝ`Ü`ë�Þ�Ô�(ÝYé�ÝYß@ë�Ø~Þ�Ý%Ñ�Ô�Ñ�åÛFÚ|Ñ�Ý@Ù{ß~Ú±Ø~Ú|Ý@Ü-ÓYÙ�Ñ¡Ù�Ý`Ó�Ø-Ø�Þ�ݧÓYÔ�×�Ö�ÛFÚFÑ�à5Ô{ãÏØ~Þ�Ý§Ý ê ×mÙ{Ø�ÚFÔ�ÑmÜ�ß~Ý`Ü�×�Û|Ø�ÚFÑ�àxã�ß�Ô�Õ Ø�Þ�ݧÙ�Ö�Ö�ÛFÚFÓ`ÙpØ�ÚFÔ�Ñ�Ô{ã(ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛIÓSÞTÙ{Ô�Ü%Ù�ÑmáØ�ÞmÝ3Ó�Ô�Õ�Ö�×�Ø~ÙpØ~Ú|Ô�ÑmÙ{ÛB�(Ô�ß~ü0ÓYÔ�Õ�ÖmÙ{ß~Ý`á"Ø�Ô"Ù\ÛFÚ|Ñ�Ý@Ù{ß©Ö�ß~Ô�æmÛ|Ý`ÕxçGíGÔ�ÑmÜ�ÚFá�Ý`ߤã�Ô�ß©ÝYò�Ù�Õ�Ö�Û|Ý%Ø~Þ�Ý3Ù{ÖmÖ�Û|Ú�ÓYÙ{Ø�ÚFÔ�ÑxÔ{ãIÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛÓSÞmÙ�Ô�Ü-Ø~Ô¡Ù ê ×TÙ�á�ßSÙpØ~ÚFÓ§ÑmÔ�Ñ�ÛFÚ|ÑmÝ`Ù{ß~Ú|ØväÕ®n¨ ©Sª�¬�¯vñ�ç¡ô¤Þ�ÝxÖ=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛGÓSÞmÙ�Ô�ܺÝ�ò�ÖmÙ{ÑTÜ�ÚFÔ�ѽÙ�Ñmá�Ø�ÞmÝ5ö%Ù�Û|Ý`ß�ü�ÚFѲÖ�ß�Ô1�vÝ`Ó�Ø�ÚFÔ�Ñß~Ý`Ü�×�۱ةÚFÑ ¼ ®n¨ ©Sª�¬�¯ ñ ª�� â ¨4Ï1¯�¾�° �´ ½ ¶ � �´µ�¶ � ®d½�¨4©�¯s® µ ¨ ©�¯�¼���½�¨ Ï1¯M� µ ¨ Ï1¯W� â ¨ Ï1¯�¾2ª åû°^æPª � ª@Ì�Ì�Ì�Ì ¨ � �U¯ô¤Þ�Ý­Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û=ÓSÞmÙ{Ô&Ü·Ù{Ö�Ömß�Ô&Ù�ÓSÞ\ß�Ý@Ü�×�Û|Ø~ÜÏÚ|Ñ"Ùºá�Ô�×�æmÛ|Ý®Ü�×�Õ�Õ§ÙpØ~Ú|Ô�ѧÔpé�ÝYß·Ø�Þ�Ý®Ø�Þ�ß~ÝYÝ©Ø~ÝYß~Õ/ØSÙ{æ�ÛFÝ�¼��½�¨4Ï1¯W� µ ¨ Ï1¯M� â ¨ Ï1¯�¾�ç�IÔ�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛµÓSÞmÙ�Ô�ܤÙ{Ö�ÖmÛ|ÚFÝ`á0Ø�Ô�Ø~Þ�Ý-Û|ÚFÑ�Ý@Ù{ß~Ú��@ÙpØ�ÚFÔ�Ñ;çd¨ ©�¯s®n¨ ©Sª�¬�¯¤Ô�ãkØ~Þ�Ý-ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛµÑ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ß�Ú|Øvä�®M¨4©Sª�¬�¯�ñ-ß~Ý`Ü�×�Û±ØSܤÚ|Ѽ3çd¨ ©�¯s®n¨ ©Sª�¬�¯2ª�� â ¨4Ï1¯�¾�°®çd¨ ©�¯s® â ¨4©�¯[¼�� â ¨ Ï1¯ ñ ¾[ª å�° æ­ª � ª�Ì�Ì@Ì[ª ¨ � �U¯�©Þ�Ú�ÓSÞ#Ó�Ô�ÑTÜ�Ú�ÜvØSܤÔ{ãQÙ§Ü�ÚFÑ�à�Û|Ý�Ø�Ý`ß�Õ á�×�Ý%Ø~Ô�Ø�Þ�Ý-Ô�ß�Ø~Þ�Ô�à�Ô�ÑmÙ�Û|Ú|Øvä§Ö�ß~Ô�Ö=ÝYß�Øvä"Ô{ãÀØ�Þ�Ý-Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�Ü�¹�� µ ¨ Ï1¯2º�ç(ô¤Þ�ÚFÜ©á�ÝYÕ�Ô�Ñ�åÜ�Ø�ßSÙpØ�Ý@ÜIØ�ÞTÙpØ·Ø~Þ�Ý�Ù�Ö�Ö�ÛFÚFÓ`ÙpØ�ÚFÔ�Ñ�Ô�ã�Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛmÓSÞTÙ{Ô�ÜQØ�Ô3Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�Ñ�Ô�Ñ�ÛFÚ|Ñ�Ý@Ù{ß~Ú±Ø~Ú|Ý@ÜQÚ�ÜÏÜ�Ø�ßSÙ{ÚFà�Þ&Ø�ã�Ô�ß��¤Ù{ßSáµëpæ�×�ØÏØ�ÞmÙ{ØÏØ�ÞmÝÑ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß~Ú±Øvä§ß~Ý`Ü�×�Û|Ø~Ü(Ú|Ñ5Ù�ámá�Ú±Ø~Ú|Ô�ÑmÙ{ÛµÜ�×�Õ�Õ§ÙpØ�ÚFÔ�ÑTܤÔ{ãAÚFÑ�Ñ�Ý`ß©Ö�ß�Ô�á�×TÓ�Ø~Ü(Ô�ãkÕº×mÛ±Ø~Ú|Ö�ÛFÝ%ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛFÜ`çQô¤Þ�Ý%Ý[�Ý`Ó�Ø©Ô�ãAØ�ÞmÝ`Ü�ÝÙ�ámá�Ú±Ø~Ú|Ô�ÑmÙ{Û�Ü�×�Õ�Õ§ÙpØ~Ú|Ô�ÑmܤÚFÜ®á�ÚFÜ~Ó�×TÜ�Ü�Ý`á[Ú|ÑxØ~Þ�Ý-Ñ�Ý�ò�Ø­Ü�Ý`Ó�Ø~Ú|Ô�Ñ�ç� ì íðïQñ_ò%ó4ñMôUõPö8õbè6éQôÈý{÷@ó%ïañ6�MèMó�êZü_ø�ó4ïQñ� Ô�ÑmÝ�åÒá�Ú|Õ�Ý`ÑmÜ�ÚFÔ�ÑTÙ{Û©Ñ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß"Ù�á�é�Ý`Ó�Ø�ÚFÔ�Ñ�åÒá�Ú�=×TÜ�ÚFÔ�Ñ´Ö�ß~Ô�æ�ÛFÝYÕ ÚFÜ"ÓYÔ�ÑmÜ�Ú�á�ÝYß~Ý`á�Ø�Ô½Ü�Ø�×má�ä½Ø�ÞmݯÝ[�Ý`Ó�Ø0Ô{ã%ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛÑ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß~Ú±Ø~Ú|Ý@ÜAÚFÑ�é�Ô�ÛFé�Ú|Ñmà­Ø�ÞmÝG×�ÑTÓ�ÝYß�Ø~Ù�Ú|ÑeépÙ{ß~ÚFÙ�æ�Û|ÝGÔ�ѺØ�Þ�ݤÙ{Ö�ÖmÛ|Ú�ÓYÙ{Ø�ÚFÔ�ѺÔ�ãmÖ=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�Û&ÓSÞTÙ{Ô�Ü`çAô¤Þ�Ý©á�Ý`à�ß~ÝYÝGÔ{ãTÓYÔ�×�Ö�ÛFÚFÑ�àÔ{ãIØ~Þ�ݺÜ�ä�Ü�Ø�Ý`Õ Ù�ÑmáxØ�Þ�ݺß�Ý`ÛFÙ{Ø�ÚFé�Ý3Ó�Ô�Õ�Ö�×�Ø~Ù{Ø�ÚFÔ�ÑmÙ�ÛAÓ�Ô&ÜvØSÜ­Ù{ß~Ý-Ó�Ô�Õ\ÖTÙ{ß~Ý`á[Ø�Ô§Ø~Þ�Ô�Ü�Ý%ã�Ô�ß­Ø�Þ�Ý3ÛFÚFÑ�Ý`Ù�ß®Ù�á�é�Ý`Ó�Ø�ÚFÔ�Ñ�åÒá�Ú�=×mÜ�ÚFÔ�ÑÖ�ß~Ô�æ�ÛFÝYÕ5çQô¤Þ�Ý3Ü�Ø�Ý@Ù�á�ä"ÑmÔ�Ñ�ÛFÚ|ÑmÝ`Ù{ß­Ù�á�é�Ý@Ó�Ø~Ú|Ô�Ñ�åbá�Úx�×mÜ�Ú|Ô�Ñ0Ý ê ×mÙ{Ø�ÚFÔ�Ñ[ã�Ô�ߤØ~Þ�Ý-é�Ý`Û|Ô�Ó�Ú|Øvä8®n¨�$_ª�¬�¯(Ú�Ü­Ó�Ô�ÑmÜ�ÚFá�Ý`ß�Ý@á®�¾ ®¾ $ {�4�¨4¬�¯2¾ ñ ®¾ $ ñ ° æ­ª $ � y æPª ��| ª ¨ � æU¯

� h¤0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�

Page 11: AIAA-2006-2066-173

�©Ú|Ø�Þ Ù�Ñ�×�ÑTÓ�ÝYß�Ø~Ù�Ú|Ñ�é&Ú�Ü~Ó�Ô�Ü�Ú|Øvä54�¨4¬�¯\Ü�×mæ:�vÝ`Ó�ØeØ~Ô�Ù1Û|Ô�à�Ñ�Ô�ß�Õ§Ù{Û(á�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�ÑË�©Ú|Ø�Þ�Õ�Ý@Ù{Ñ 4ä° � Ù{Ñmá½ÓYÔ�Ý[�"Ó�ÚFÝYÑ&Ø�Ô{ãépÙ{ß~ÚFÙ{Ø�ÚFÔ�Ñ´Ó�ÔpéZëΰmæ­Ì � ç,ø�ÚFÑmÓYÝxÑmÔ�ÓYÛFÙ�Ü�Ü�Ú�ÓYÙ{Û(Ô�ß�Ø�ÞmÔ�à�Ô�ÑmÙ{Û(Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛFÜ�Ô�ã®Ø~Þ�Ýð�®Ü�ü�ÝYä Ü�ÓSÞ�Ý`Õ�ÝxÓYÔ�ß~ß�Ý@Ü�Ö=Ô�Ñmá½Ø�Ô�Ø�ÞmÝÛFÔ�à�ÑmÔ�ß~Õ�Ù�Û(á�ÚFÜ�Ø�ß~Ú|æm×�Ø�ÚFÔ�Ñ�ëGØ�ÞmÝ#Ô�Ö�Ø~Ú|Õ§Ù{Û©Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û­ÓSÞmÙ�Ô�Ü\Ý�ò�ÖmÙ�ÑmÜ�Ú|Ô�Ñ�æmÙ�Ü�Ú�Ü\ÚFÜ"ÓYÔ�Õ�Ö�×�Ø~Ý`á Ñ�×�Õ�ÝYß~ÚFÓ`Ù{ÛFÛ|ä�Ø~Ô²ß�ÝYØ~Ù{ÚFÑÝ�ò�Ö=Ô�Ñ�Ý`Ñ&Ø�Ú�Ù{Û®ÓYÔ�Ñ�é�Ý`ß�à�ÝYÑmÓYÝ�ç'ô¤Þ�ݯæ=Ô�×�ÑTá�Ù{ß~ä�Ó�Ô�Ñmá�Ú|Ø�ÚFÔ�ÑmÜ0Ù{ß~ݯÓSÞ�Ô�Ü�ÝYÑ Ø�Ô²æ=Ý�®n¨4æPª�¬�¯�° � Ù{ÑmáÑ®n¨ � ª�¬�¯�°gæ�ç ô¤ÞmÝÜ�Ô�ÛF×�Ø�ÚFÔ�ѲÔ�ã(Ø~Þ�Ý[á�Ý�Ø�Ý`ß�Õ�ÚFÑ�Ú�ÜvØ~ÚFÓ§épÙ�ß�Ú�Ù{Ñ&Ø3Ô{ã�¨ � æU¯%ÚFÜeÜ�ÞmÔ�©Ñ�ÚFÑË�ÀÚFà�×�ß~Ý,+Næ�ã�Ô�ßeÜ�ÝYé�Ý`ß~Ù�Û·épÙ{ÛF×�Ý`Ü3Ô{ã©Ø�ÞmÝ"é�ÚFÜ~Ó�Ô&Ü�Ú|Øvä�4^°¹�æ­��æ­Ì � ç:��æ­Ì çP� � � � ��}²º&ç�ø�×mæmÜvØ~Ú±Ø~×�Ø�ÚFÔ�Ñ,Ô�ã®Ø�ÞmÝxÖ=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û©ÓSÞmÙ�Ô�ÜeÝ�ò�ÖmÙ{ÑTÜ�ÚFÔ�ÑmÜ\Ô�ãl4�¨ ¬U¯eÙ�Ñmá�®M¨�$�ª�¬U¯ºÚFÑp¨ � æU¯\Ù{Ñmá Ùö%Ù{ÛFÝYß~ü�Ú|ÑxÖ�ß~Ô{�vÝ@Ó�Ø�ÚFÔ�Ñ[Ô�Ñ&Ø�Ô�Ý@Ù�ÓSÞxÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{ÛµæTÙ�Ü�ÚFÜ�¹2� µ ¨4Ï1¯(º �µ�¶ � ß~Ý`Ü�×�Û±ØSܤÚ|Ñ

�´ ½ ¶ � �´µ�¶ � ¼� ½ � µ �Jâ{¾�® µ ¾ ®d½¾ $ { ��� �C� â�� î�� �C�´½ ¶ ���W�C� â hµî�� � �-4 ½ â ¾ ñ[®d½¾ $ ñ ° æ­ª å�° æ­ª � ª�Ì�Ì@Ì[ª4Ö_ª ¨ �:� ¯

�©Ú|Ø�Þ -4q½ â ° î´µ�¶ � ¼���½�� µ � â ¾^4 µ Ì ¨ �1� ¯ÐÒÑD�ÀÚFà�×�ß~ÝT�{æ5Ø~Þ�ݺէÙ{Ø�ß~Ú±ò5æ�ÛFÔ�ÓSü#Ü�Ø�ß~×mÓ�Ø~×�ß~Ý3Ô{ãIØ�ÞmÝeÓ�Ô�×�Ö�ÛFÝ`á¯Ü�Ý�Ø�Ô{ãQÝ ê ×mÙ{Ø�ÚFÔ�ÑmÜǨ �P� ¯¤ß~Ý`Ü�×�Û|Ø�ÚFÑ�à§ã�ß~Ô�Õ Ø�Þ�Ý\Ù{Ö�ÖmÛ|Ú�ÓYÙ{Ø�ÚFÔ�ÑÔ{ãÀØ�ÞmÝ-ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û�ÓSÞTÙ{Ô�Ü©Ù�Ö�Ö�ß~Ô�Ù�ÓSÞ[ÓYÙ�Ñ[æ=ݺÜ�ÝYÝ`Ñ�ç·ô¤Þ�Ý3á�Ô�×�æ�ÛFÝ3Ü�×�Õ�Õ§ÙpØ~Ú|Ô�Ñ[Ôpé�ÝYߤØ~Þ�Ý%Ø�Þ�ß~ÝYÝ%Ø~ÝYß~Õ Ø~Ù�æ�ÛFÝ8¼�� ½ � µ �Jâ1¾ã�Ô�ß�É(ª&Ê�ª�å+°Iæ­ª � ª�Ì�Ì@Ì[ª4Ö#ÚFÑ#Ø~Þ�ݺÙ�á�é�Ý`Ó�Ø�ÚFÔ�Ñ5Ø~ÝYß~Õß®�ìCíì « ß~Ý`Ü�×�Û|Ø~Ü©ÚFÑ1Ù\ã�×�ÛFÛkæmÛ|Ô�ÓSüxÕ�Ù{Ø�ß~Ú±òµç©ô¤Þ�ݺá�Úx�×mÜ�Ú|Ô�Ñ5Ø�ÝYß~Õî4�¨ ¬�¯Iì íì « ß~Ý`Ü�×�Û±ØSÜ�Ô�Ñ�Û|ä%ÚFѺÙ{ÑeÙ�á�á�Ú|Ø�ÚFÔ�ÑTÙ{Û{Ø~ß�Ú�á�Ú�Ù{à�Ô�ÑmÙ�Û�æ�ÛFÔ�ÓSü�Õ§ÙpØ~ß�Ú|òµëpÜ�ÚFÑmÓ�Ý�¼���½�� µ � â ¾�ã�Ô�ß�Én°¸Õ§ÙpòQ¹/åi{ � ª�æPº�ª@Ì�Ì�Ì2ª�Õ�Ú|ÑB¹åBÓ � ª4Öqº&ëåÕ°]æ­ª � ª�Ì�Ì@Ì[ª4Ö²Ù�Ñmá±Êð° æ­ª � ÚFÜ-Ù[Ø~ß�Ú�á�Ú�Ù{à�Ô�ÑmÙ{ÛÀØSÙ{æ�ÛFÝ�ç§ô¤Þ�ݧÓ�Ô�×�Ö�ÛFÝ`á�Ü�ÝYØ-Ô{ãGÝ ê ×mÙpØ~Ú|Ô�ÑmÜ%ÚFÜ-Ü�Ô�ÛFé�Ý@á¯Ú|Ø�Ý`ß~Ù{Ø�ÚFé�ÝYÛFä#×mÜ�Ú|ÑmàÙ"æmÛ|Ô�ÓSü#ö%Ù{×mÜ~Üvåvø�Ý`ÚFá�Ý`ÛkÚ|Ø�Ý`ß~Ù{Ø�ÚFÔ�Ñð�©Ú|Ø�Þ�Ù{Ñ1Ù�ÓYÓ�×mß~Ù�Ó�ä"Ø~Þ�ß~Ý`Ü�Þ�Ô�Û�á#Ô{ã�ï�° � æbhQ�pçJ�­ÖmÖ�ß�ÔNò�ÚFÕ§ÙpØ�Ýeè­Ý��¤Ø~Ô�ѯÛFÚFÑ�Ý`Ù�ß�Ú��`Ù{Ø�ÚFÔ�ÑÚ�Ü%×mÜ�Ý@á#ã�Ô�ß%Ø�Þ�Ý�Ñ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß�Ø�Ý`ß�Õ§Ü�ÝYÕ�Ö�ÛFÔpä�Ú|Ñ�àxÔ�ÑmÝ\æmÛ|Ô�ÓSü1ö%Ù{×TÜ�Ü�åÒø�ÝYÚ�á�ÝYÛIÚ|Ø�ÝYßSÙpØ~Ú|Ô�Ñ1Ö=ÝYß3è®Ý��¤Ø�Ô�Ñ¡ÛFÚFÑ�Ý`Ù�ß�Ú��`Ù{Ø�ÚFÔ�Ñ�çºô¤ÞmÝß~Ý�ã�ÝYß~ÝYÑTÓ�Ý%Ü�Ô�Û|×�Ø~Ú|Ô�Ñ"Ú�Ü©Ùeì#Ô�Ñ&Ø�Ý3í(Ù{ß~Û|Ô\Ü�ÚFÕº×�Û�ÙpØ~Ú|Ô�Ñ�çS�IÔ�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛTÓSÞTÙ{Ô�ÜGß~Ý`Ü�×�Û±ØSÜGã�Ô�ß(Ø�Þ�Ý-á�Ú�Ü�Ø�ß~Ú|æ�×�Ø�ÚFÔ�Ñ"ã�×mÑmÓ�Ø~Ú|Ô�Ñ[Ô�ãAØ�ÞmÝé�Ý`Û|Ô�ÓYÚ±Øvä�Ù{ß~Ý©à�ÚFé�Ý`ѧÚ|Ñ+�ÀÚFà�×�ß~Ý���Ù�çÀÐÒÑ+�ÀÚ|à�×�ß�Ý8��æ§Ø�ÞmÝ®Ý�ò�Ö=Ô�Ñ�Ý`Ñ&Ø�Ú�Ù{ÛTÓYÔ�Ñ�é�Ý`ß�à�ÝYÑmÓYÝ(ã�Ô�ß·Ø~Þ�Ý®Õ�Ý@Ù{Ñ�ë�Ø�Þ�Ý�épÙ{ß~ÚFÙ�ÑmÓ�Ý­Ù�Ñmá�Ø�ÞmÝá�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�Ñxã�×�ÑmÓ�Ø~Ú|Ô�Ñ#ÓYÙ{Ñxæ=Ý3Ü�Ý`ÝYÑ�ç

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

µ=0→←µ=0.25

←µ=0.5

←µ=1µ=2→µ=∞→

sensor location x=0.5→

position x

tem

pera

ture

T

> '@?d<!�!;/$�'("0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

µ=0→

←µ=0.25

←µ=0.5←µ=1µ=2→

µ=∞→

sensor location x=0.5→

position x

velo

city

u

> 62?B;/02;/<!�!;�$�'("D6EGFIHKJML�ðROÃñ�L@]ML%J�SuE\[ZE\V�]�EGm<V�_IatHI]�EG_I[KVT_nq8]�QRL�atEt[RL�U2JeUn[Kp�[R_I[Za\Et[RL�U2J<U2pKÆ2L%m�]�Ec_R[IX�pbEcÚ�HKVWEG_I[�qr_nJeV^L%Æ2L%JMUna*ÆCUna\HRLCVu_2q8]�QRLÆZEGVWm�_nVWEc]^o�OôkÔ"ÜvØ~×má�äxØ�Þ�ÝeÝ[�Ý`Ó�Ø®Ô{ãQØ~Þ�Ý3Ñ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß�Ù�á�é�Ý@Ó�Ø~Ú|Ô�Ñ[Ø~ÝYß~Õ Ô�Ñ5Ø�Þ�Ýeß�Ý@Ü�×mÛ±Ø~Ú|Ñ�à[Ó�Ô�×mÖ�Û|Ý@á#Ü�ä�Ü�Ø�Ý`Õ Ù�Ñmá#Ó�Ô�Õ\Öm×�Ø~Ù{Ø�ÚFÔ�ÑmÙ�Û

Ó�Ô&ÜvØSܤÔ{ãIØ�ÞmÝ-ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{ÛkÓSÞmÙ{Ô&Ü©Ù{Ö�Ö�ß~Ô�Ù�ÓSÞ�ë&Ø�Þ�ÝeÙ{Ö�Ö�ÛFÚ�ÓYÙpØ~Ú|Ô�Ñ[Ô�ãÀÖ=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛAÓSÞTÙ{Ô�Ü(Ø~Ô§Û|ÚFÑ�Ý@Ù{ß­Ù�á�é�Ý`Ó�Ø�ÚFÔ�Ñ�åÒá�Ú�=×TÜ�ÚFÔ�ÑxÚFÜ�2� 0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�

Page 12: AIAA-2006-2066-173

O

O

D

D

D DA

A

A

A

A

D D

DD

D

D

D

D

D

> '@?q<!�!;/$�'("�'(5�E[$�)s3&�!02;%A}5/� T1-/ &�!02;

D

D

D DA

A

A

A

A

D D

DD

D

D

D

DA A A A

A

A

A

A

A

A

A

A

A

AD

A

A

A A A

A

> 62?Q;/02;/<!�!;�$�'("�'(5�E[$�)s3%�!02;CA}5/� T1-/ %�!02;D6EGFIHKJMLsòKO�ó�Ud]MJ�Ecô;`bac_Km�õµV�]MJ�HKm�]�HKJMLT_2q�]�QRL�m�_RHKYbacLCp�V^oKV�]MLCS'JML%VWHZac]�Et[RFeq�JM_ISö]�QRLuYb_IaGoK[R_RS�EGUna÷m�QRU�_nV�U2YZYZJW_nU2m�Q; 8Ec]�Qø8ù = m@_I[KVWE\V�]�E\[RFs_2q6U2pKÆ2LCm�]�EG_I[½ú3û�ü�Un[KpvpbEcÚ�HKVMEc_I[�ú�ñ�üÂ`baG_Rm�õ<SsUd]MJ�E\m�L%V@OÙ{Û�Ü�Ô�ÓYÔ�ÑmÜ�Ú�á�ÝYß~Ý`á ç!¾ ®¾ $ {54�¨ ¬U¯2¾ ñ[®¾ $ ñ °ÎæPª $ � y æ­ª �@| ª ¨ �{� ¯�©Þ�Ý`ß�ÝvçÀ° � ÚFÜ3Ø�Þ�Ý[Ù�á�é�Ý`Ó�Ø~Ú|Ô�Ñ�é�ÝYÛFÔ�Ó�Ú|Øvä�çðÿ�Ý�Ø�Ý`ß�Õ�ÚFÑ�Ú�ÜvØ~ÚFÓ"Ü�Ô�ÛF×�Ø~Ú|Ô�ÑmÜ3Ô�ã9¨ �{� ¯-Ù{ß~Ý"Ü�ÞmÔ�©Ñ�ÚFÑË�ÀÚFà�×�ß~Ý,+pÙxã�Ô�ßeÜ�ÝYé�Ý`ß~Ù�ÛépÙ{ÛF×�Ý`ܤÔ�ãkØ~Þ�Ý-é�ÚFÜ~Ó�Ô&Ü�Ú|Øvä�çn�­ÖmÖ�Û|Ú�ÓYÙ{Ø�ÚFÔ�Ñ5Ô{ãÀØ�ÞmÝ-ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û�ÓSÞTÙ{Ô�Ü(ß~Ý`Ü�×�Û|Ø~ܤÚFÑ

çd¼�� ñ â ¾ ¾ ® ½¾ $ { ��� �d� â@� î�� �C�´½ ¶ �����C� â h�î�� � �-4q½ â÷¾ ñ�® ½¾ $ ñ °ÎæPª å�° æPª � ª@Ì�Ì�Ì�ª4Ö�ª ¨ � àȯ

�©Ú|Ø�Þý-4q½ â à�ÚFé�Ý`Ñ�æ�ä^¨ �1� ¯�ç´ô¤ÞmÝ#Õ�Ù{Ø�ß~Ú±ò½æmÛ|Ô�ÓSü�Ü�Ø�ß~×mÓ�Ø~×�ß~Ý[ß~Ý`Ü�×�Û|Ø�ÚFÑ�à�ã�ß�Ô�Õ ¨ � àU¯eÓYÙ{Ñ´æ=ݯÜ�Ý`ÝYÑ´Ú|Ѳ�ÀÚ|à�×�ß�ݵ��Ùmç´ô¤ÞmÝÛFÚ|Ñ�Ý@Ù{ß3Ù�á�é�Ý@Ó�Ø�ÚFÔ�Ñ¡Ø�Ý`ß�Õþç ì%íì « ß�Ý`Õ�Ù�Ú|ÑTÜ-Ù#Ü�ÚFÑ�à�ÛFÝ\Ø�Ý`ß�Õ Ú|Ñ�é�Ô�Û|é�ÚFÑ�àÀ¼���ñ â ¾%Ù{ãoØ�ÝYß%Ø~Þ�Ý0Ù�Ö�Ö�ÛFÚFÓ`ÙpØ�ÚFÔ�Ñ�Ô{ã(ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛÏÓSÞmÙ{Ô&Üá�×�Ý�Ø~ÔeØ~Þ�Ý%Ô�ß�Ø�Þ�Ô�à�Ô�ÑmÙ{ÛFÚ±Øvä\Ö�ß~Ô�Ö=ÝYß�Øvä§Ô{ãkØ�Þ�Ý%æmÙ�Ü�Ú�Ü(ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û�Ü�¹�� µ º �µ�¶ � ç·ô¤Þ�Ý`ß�ÝYã�Ô�ß~Ý®Ø~Þ�Ý-Ù�á�é�Ý`Ó�Ø~Ú|Ô�Ñ"Ø~ÝYß~Õ ÞmÙ�Ü(Ô�ÑmÛ|äÙ5Ó�Ô�Ñ�Ø~ß�ÚFæ�×�Ø~Ú|Ô�Ñ1Ô�ѯØ�Þmݧæ�Û|Ô�ÓSü¯á�Ú�Ù{à�Ô�ÑmÙ�ÛÀã�Ô�ß�Ø�Þ�Ý�ÛFÚ|ÑmÝ`Ù{ß3Ó`Ù�Ü�Ý�ë�Ü�ÝYÝ��ÀÚ|à�×�ß�Ý � æ�Ù�çeô¤Þ�ݧá�Úx�×mÜ�Ú|Ô�Ñ1Ø~ÝYß~ÕKß~Ý`Ü�×�Û±ØSÜ%Ú|Ñ¡Ø�ÞmÝÜ~Ù{Õ�ݺØ~ß�Ú�á�Ú�Ù{à�Ô�ÑmÙ�Ûkæ�ÛFÔ�ÓSü¯ÜvØ~ß�×TÓ�Ø�×mß�ݧÙ�Ü�ã�Ô�ß�Ø~Þ�Ý�Ñ�Ô�ÑmÛ|ÚFÑ�Ý`Ù�ß%Ý ê ×mÙpØ~Ú|Ô�Ñ�ç��mÔ�ß�Ø�Þ�Ú�Ü-Ü�Ú|Õ�Ö�ÛFÝ�Ý�ò�Ù{Õ�Ö�ÛFÝeÚ|غÓYÙ�Ñ1æ=ݧÜ�Ý`ÝYÑ1æ�äÓ�Ô�Õ�ÖmÙ{ß~Ú|Ñmà�¨ �:� ¯­Ù�Ñmác¨ � àU¯(Ø~ÞmÙpØ�ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{ÛkÑ�Ô�ÑmÛ|ÚFÑ�Ý`Ù�ß�Ú|Ø�ÚFÝ`Ü®ß�Ý@Ü�×m۱خÚFÑ1Ù�ѯÚ|ÑTÓ�ß~Ý`Ù�Ü�Ý`áxÑ�×�ÕeæTÝ`ß�Ô�ãQÜ�×�Õ�Õ§ÙpØ~Ú|Ô�ÑmÜ®Ù{ãoØ�ÝYßØ�ÞmÝ0Ù{Ö�ÖmÛ|Ú�ÓYÙ{Ø�ÚFÔ�Ñ�Ô{ã¤ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛÏÓSÞmÙ{Ô&ÜYç[ô¤Þ�Ý`Ü�Ý"Ü�×mÕ\Õ§Ù{Ø�ÚFÔ�ÑmÜ-ß~Ý`Ü�×�Û|غÚFÑ�Ù1ÜvØ~ß�Ô�Ñ�à�Ý`ß%ÓYÔ�×�ÖmÛ|Ý@á�æ�Û|Ô�ÓSü¡Õ�Ù{Ø�ß~Ú±ò�á�×�ݧØ~ÔØ�ÞmÝxÙ�á�á�Ú|Ø�ÚFÔ�ÑmÙ�ÛÏæ�ÛFÔ�ÓSü�ܺÚ|Ñ�Ø�Þ�Ý[æ�ÛFÔ�ÓSü�Õ§ÙpØ~ß�Ú|ò½ÙpãoØ~ÝYßeá�ÚFÜ~Ó�ß~Ý�Ø~Ú��@ÙpØ�ÚFÔ�ÑAë·Ü�Ý`Ý+�ÀÚ|à�×�ß�Ý � æmç1ô¤Þ�Ú�ÜeÚ�ÜeÝYé�Ý`ѲÕ�Ô�ß~Ý[Ó�ÛFÝ`Ù�ß3ã�Ô�ßÕ�Ô�ß~Ý\ÓYÔ�Õ�Ö�ÛFÝ�ò¯Ñ�Ô�Ñ�ÛFÚ|Ñ�Ý@Ù{ß�Ö�ß~Ô�æ�ÛFÝYÕ§Ü�Ú|Ñ�é�Ô�ÛFé&ÚFÑ�à[Ö�ß�Ô�á�×TÓ�Ø~Ü�Ô�ãGÕ�Ô�ß~Ý3Ø~ÞmÙ{Ñ¡Ø��GÔ[×�ÑmÓYÝYß�Ø~Ù{ÚFÑ�éNÙ�ß�Ú�Ù{æmÛ|Ý@Ü�Ù�Ü®ã�Ô�ß%Ý�ò�Ù{Õ�Ö�ÛFÝÝYÑTÓ�Ô�×mÑ�Ø~ÝYß~Ý`á[Ú|Ñxâm×mÚFá5á�ä�ÑmÙ�Õ\Ú�ÓYÜ`ç îs�ô¤Þ�ÝeÝYß~ß�Ô�ß�Ó�Ô�Ñ�é�ÝYß~à�Ý`ÑmÓ�Ý3Ù�Ü­ã�×mÑmÓ�Ø~Ú|Ô�Ñ1Ô{ã(Ó�Ô�Õ�Ö�×�Ø~ÙpØ~Ú|Ô�ÑmÙ{ÛM�(Ô�ß~ü[Ú�Ü%Ó�Ô�Õ�ÖmÙ�ß�Ý@á5Ú|Ñ��ÀÚ|à�×�ß�Ý � æ"ã�Ô�ß�æTÔ�Ø�Þ1ÛFÚFÑ�Ý`Ù�ß�Ù�Ñmá

Ñ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß0Ù�á�é�Ý@Ó�Ø�ÚFÔ�Ñ�åbá�Ú��×mÜ�ÚFÔ�ÑAç^� æ�ÛFÔ�ÓSü,ö%Ù�×mÜ�Ü�åvø�ÝYÚ�á�ÝYÛ©Ú|Ø�Ý`ß~Ù{Ø�ÚFÔ�Ñ´Ø�Þmß�Ý@Ü�Þ�Ô�ÛFá´Ô�ã�ïnÿ�� Ä ° � æZh�ó5ÚFÜ0×mÜ�Ý@á,Ù{Ñmá´ã�Ô�ßØ�ÞmݧÑ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ß-Ö�ß�Ô�æ�ÛFÝYÕ Ø~Þ�Ý"è­Ý@�¤Ø�Ô�Ñ�ÛFÚ|ÑmÝ`Ù{ß~Ú��`ÙpØ~Ú|Ô�ѯØ~Þ�ß�Ý@Ü�ÞmÔ�Û�á�ï����GÚ�Ü-Ô�Ñ�Ý�Ô�ßSá�ÝYß%Ô{ã¤Õ§Ù{à�Ñ�Ú±Ø~×má�Ý"Ü�Õ§Ù{ÛFÛ|Ý`ß`ç��÷à�Ô�Ô�áÕ�Ý`Ù�Ü�×�ß~Ý(ã�Ô�ßQØ~Þ�Ý­Ó�Ô�Õ�Ö�×�Ø~Ù{Ø�ÚFÔ�ÑmÙ�ÛP�(Ô�ß~ü3ÚFÜQØ�Þ�Ý©Ý ê ×mÚ|épÙ{ÛFÝYÑ&Ø·Ñ�×�ÕeæTÝ`ß·Ô{ãµá�Ý�Ø~ÝYß~Õ�Ú|Ñ�Ú�Ü�Ø�Ú�Ó­Ü�Ô�Û|é�Ý`Ü`çAÐÒÑ��ÀÚ|à�×�ß~Ý � æ�Ù®Ø�ÞmÝ­ÝYß~ß�Ô�ßÓ�Ô�Ñ�é�ÝYß~à�Ý`ÑmÓ�Ý­Ù�ÜQã�×�ÑTÓ�Ø�ÚFÔ�Ñ[Ô{ãµØ~Þ�Ý�ÓYÔ�Õ�Ö�×�ØSÙpØ~Ú|Ô�ÑmÙ{Ûa�GÔ�ß�üeã�Ô�ßÏØ�Þ�Ý�Û|ÚFÑ�Ý@Ù{ßGÙ�á�é�Ý@Ó�Ø~Ú|Ô�Ñ�åbá�Úx�×mÜ�Ú|Ô�Ñ§Ý ê ×mÙpØ~Ú|Ô�ÑmÜGÚFÜGà�ÚFé�Ý`Ñ�çÀô¤ÞmÝÓ�Ô�Õ�Ö�×�Ø~Ù{Ø�ÚFÔ�ÑmÙ�ÛS�GÔ�ß�ü5ã�Ô�ßLÖc°`æ[Ú�Ü-Ý ê ×mÚ|épÙ{ÛFÝYÑ&Ø%Ø�Ô5Ô�Ñ�Ý"á�Ý�Ø~ÝYß~Õ�Ú|Ñ�Ú�Ü�Ø�Ú�Ó\Ü�Ô�ÛFé�Ý�ç§ô¤Þ�Ý"Ó�Ô�Õ\Öm×�Ø~Ù{Ø�ÚFÔ�ÑmÙ�ÛM�(Ô�ß~ü¯ÚFÑmÓ�ß~Ý`Ù�Ü�Ý@ÜãMÙ�Ü�Ø�Ý`ߧØ�ÞTÙ{Ñ´Ø�Þ�Ý1ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û®Ô�ßSá�ÝYߧá�×�ݯØ�Ô�Ø�ÞmݯÚ|ÑTÓ�ß~Ý`Ù�Ü�ÚFÑ�à�Ñ&×mÕºæ=ÝYß"Ô�ã%ÓYÔ�×�Ö�ÛFÝ`á´Ý ê ×TÙpØ�ÚFÔ�ÑTÜ"Ù{Ñmá Ø~Þ�ݯÚFÑmÓ�ß~Ý`Ù�Ü�ÚFÑ�àÑ�×�Õºæ=ÝYß(Ô{ã�æmÛ|Ô�ÓSü"ö%Ù{×mÜ~Ü�åÒø�Ý`ÚFá�ÝYÛ=Ú±Ø~ÝYßSÙpØ�ÚFÔ�ÑTÜYçÀÐÒÑ��ÀÚFà�×�ß~Ý � æ�æ§Ø�ÞmÝ�ÓYÔ�Õ�Ö�×�ØSÙpØ�ÚFÔ�ÑTÙ{Û­�(Ô�ß~üeã�Ô�ßGØ�Þ�Ý�Ñ�Ô�ÑmÛ|ÚFÑ�Ý`Ù�ßGÙ�á�é�Ý@Ó�Ø�ÚFÔ�Ñ�åá�Ú�=×TÜ�ÚFÔ�Ñ#Ö�ß~Ô�æmÛ|Ý`Õ Ú�Ü­à�Ú|é�ÝYÑ�ç¤��Ô�ß©Ø�Þ�ݺÑ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ß­Ö�ß~Ô�æ�ÛFÝYÕ Ø�Þ�ÝºÝ ê ×�ÚFéNÙ�Û|Ý`Ñ&Ø®Ñ�×�Õºæ=ÝYß�Ô{ãQá�Ý�Ø~ÝYß~Õ\ÚFÑ�Ú�ÜvØ~ÚFÓeÜ�Ô�Û|é�Ý`Ü©Ó`Ù{Ñ#æ=ÝÚFÑ�Ø~ÝYß~Ö�ß~Ý�Ø�Ý@á%Ú|ѺØ��GÔ#�(ÙNä�Ü`ç��ÀÚ|ßSÜvØ@ëpÙ®á�ÝYØ�ÝYß~Õ�Ú|ÑmÚFÜ�Ø�Ú�ÓÏÜ�Ô�ÛFé�ÝÏÓ`Ù{Ñ-æ=Ý(ÓYÔ�ÑmÜ�Ú�á�ÝYß~Ý`á-Ù�Ü�Ô�ÑmÝ®f loj�d~�{q­á�Ý�Ø~ÝYß~Õ�Ú|Ñ�Ú�Ü�Ø�Ú�ÓÏÜ�Ô�ÛFé�Ý�ë��©Þ�Ú�ÓSÞ

� j¤0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�

Page 13: AIAA-2006-2066-173

Ú�Ü®Ý ê ×mÚ|épÙ{ÛFÝYÑ&Ø�Ø�Ô0Ô�Ñ�Ýeè®Ý��¤Ø~Ô�Ñ1Ú|Ø�Ý`ß~Ù{Ø�ÚFÔ�Ñ#ÚFÑ¡Ü�Ô�ÛFé�Ú|Ñ�à"Ø�Þ�Ý\Ñ�Ô�ÑmÛ|ÚFÑ�Ý`Ù�ß�á�ÝYØ�Ý`ß�Õ�ÚFÑ�ÚFÜ�Ø�Ú�Ó3Ömß�Ô�æ�Û|Ý`Õ5ç�ÐÒÑ#Ø~ÞmÙpØ-ÓYÙ�Ü�Ý�¨Má�Ù�Ü�Þ�Ý@áÛFÚ|Ñ�Ý@ÜIÚ|Ñ��ÀÚ|à�×�ß~Ý � æ{æa¯�Ø�Þ�ݤÝ`ß�ß~Ô�ßIÓ�Ô�Ñ&é�ÝYß~à�Ý`ÑmÓ�Ý·Ú�ÜÀã�Ô�ßÀØ~Þ�ݤÑ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ßÀÖmß�Ô�æ�Û|Ý`ÕÁÜ�ÛFÔ�(ÝYßkØ�ÞmÙ�Ѻã�Ô�ßkØ~Þ�ݤÛ|ÚFÑ�Ý@Ù{ßIÖ�ß~Ô�æ�ÛFÝYÕ5ë�á�×mÝØ�ÞmÝ%Ø�Þ�ݺÙ�á�á�Ú|Ø�ÚFÔ�ÑmÙ�Û�è­Ý@�¤Ø�Ô�Ñ5Ú|Ø�Ý`ß~Ù{Ø�ÚFÔ�ÑmÜ`çÏø�Ý`ÓYÔ�Ñmá�Û|ä�ë�Ø�Þ�ÝeÓ�Ô�Õ�Ö�×�Ø~ÙpØ~Ú|Ô�ÑmÙ{ÛB�(Ô�ß~ü�ã�Ô�ß�Ù5j=upjTf loj=dS�{qeá�ÝYØ�ÝYß~Õ�Ú|ÑmÚFÜ�Ø�Ú�Ó-Ü�Ô�Û|é�ÝÓYÙ�Ñ[æ=Ý-Ó�Ô�ÑmÜ�Ú�á�Ý`ß�Ý@á0Ù�ÜGØ~Þ�Ý-á�Ý�Ø~ÝYß~Õ\ÚFÑ�Ú�ÜvØ~ÚFÓ-Ù�Õ�Ô�×�Ñ&ؤÔ{ãÀÓYÔ�Õ�Ö�×�ØSÙpØ~Ú|Ô�ÑmÙ{Ûd�(Ô�ß~ü=çÀÐÒÑxØ�ÞmÙ{Ø­ÓYÙ�Ü�Ý�¨oã�×�ÛFÛ�ÛFÚ|Ñ�Ý%ÚFѱ�ÀÚFà�×�ß~Ý � æ�æa¯Ø�ÞmÝ­Ó�Ô�Õ�Ö�×�Ø~ÙpØ~Ú|Ô�ÑmÙ{Û­�GÔ�ß�ü3Ú�Ü·Ù�ãMÙ�Ó�Ø�Ô�ßQã�Ô�×�ß·Ü�Õ§Ù{ÛFÛ|Ý`ß`ë�Ü�ÚFÑmÓ�ݤã�Ô�ßÏÜ�Ô�ÛFé&ÚFÑ�à�Ø�ÞmÝ©Ñ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ßÏá�Ý�Ø~ÝYß~Õ�Ú|Ñ�Ú�Ü�Ø�Ú�Ó¤Ö�ß�Ô�æ�ÛFÝYÕr¨xÖ±° æU¯ã�Ô�×mß®è­Ý@�¤Ø�Ô�Ñ5Ú|Ø�Ý`ß~Ù{Ø�ÚFÔ�ÑmÜ­Ù�ß�Ý%ß~Ý ê ×�ÚFß�Ý@áµçÏô¤Þ�ÝeÜ�Ý@Ó�Ô�ÑTá0Õ�Ý`Ù�Ü�×mß�Ý-Ô�ãkØ~Þ�ݺÓ�Ô�Õ�Ö�×�Ø~Ù{Ø�ÚFÔ�ÑmÙ�ÛB�GÔ�ß�ü0Ú|Ñ5Ø~ÝYß~Õ�Ü­Ô{ãIÝ ê ×�ÚFéNÙ�Û|Ý`Ñ&ØÑ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß0á�ÝYØ�Ý`ß�Õ�ÚFÑ�ÚFÜ�Ø�Ú�Ó¯Ü�Ô�ÛFé�Ý@ܧÕ�Ú|à�Þ�Ø[æTÝ#Ø~Þ�ݯæ=Ý`Ü�Ø[ß~ÝYÖ�ß~Ý`Ü�ÝYÑ&ØSÙpØ�ÚFÔ�Ñ´Ô{ã%Ø~Þ�Ý1Ó�Ô�Õ�Ö�×�Ø~Ù{Ø�ÚFÔ�ÑmÙ�Û,�(Ô�ß~ü=ë©Ü�Ú|ÑTÓ�ݯÔ�Ñ�Ý#ÚFÜÚFÑ�Ø~ÝYß~Ý`Ü�Ø�Ý@áeÚFÑ\Ø�Þ�Ý©ßSÙpØ~Ú|Ô%Ô�ã=ÚFÑmÓ�ß~Ý`Ù�Ü�ݤÔ�ã�ÓYÔ�Õ�Ö�×�ØSÙpØ�ÚFÔ�ÑTÙ{ÛP�(Ô�ß~ü%ã�Ô�ß·ß~ÝYÖmÛFÙ�Ó�ÚFÑ�à-Ù�á�Ý�Ø�Ý`ß�Õ�ÚFÑ�Ú�ÜvØ~ÚFÓ­Ü�Ô�ÛFé�Ý(æ�äeÙ�Ñ\×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ�Øväê ×mÙ{Ñ&Ø~Ú±õTÓ`ÙpØ~Ú|Ô�Ñ�çeÐÒÑ¡Ø�ÞmÝ\Û�ÙpØ�Ø�Ý`ß-Õ\Ý@Ù�Ü�×�ß~Ý�ë�Ü�Ô�ÛFé&ÚFÑ�à[Ø�Þ�ݧÑmÔ�Ñ�ÛFÚ|ÑmÝ`Ù{ß%Ö�ß~Ô�æmÛ|Ý`ÕKÚFÜ%ß~ÝYÛ�ÙpØ~Ú|é�ÝYÛFä#ÓSÞ�Ý`Ù�ÖTÝ`ß�Ø~ÞmÙ{ѲÜ�Ô�ÛFé�Ú|Ñ�à[Ø�ÞmÝÛFÚ|Ñ�Ý@Ù{ߤÖ�ß~Ô�æ�ÛFÝYÕ5ç·ô¤Þ�Ú�ܤÚ�Ü©ÓYÙ{×TÜ�Ý@á0æ�ä§Ø~Þ�Ý-á�Ý`ÓYß�Ý@Ù�Ü�Ý�Ô�ãAØ�ÞmÝ-Ñ&×mÕºæ=ÝYß©Ô{ãIè­Ý@�¤Ø�Ô�ÑxÚ|Ø�ÝYßSÙpØ~Ú|Ô�ÑmÜGØ�ÞmÙ{Ø­ÚFܤß~Ý ê ×�ÚFß~Ý`á§Ø~Ô�ß�Ý@Ù�ÓSÞï ��� �©Ú|Ø�Þ5ÚFÑmÓ�ß~Ý`Ù�Ü�ÚFÑ�à\æ�Û|Ô�ÓSü[ö%Ù�×mÜ~Üvåvø�ÝYÚ�á�Ý`ÛµÚ±Ø~ÝYßSÙpØ�ÚFÔ�Ñ5Ü�Ø�ÝYÖTÜYç

0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.660

5

10

15

20

25

30p=4

velocity u

prob

abili

ty d

ensi

ty

exactPC

> '@?B.�"&026'(6/�!<!� 3}7#5/�! 43&"%�!6/-�3%�!02;�F�-/;/)s3%�!02;0 1 2 3 4 5

10−6

10−4

10−2

100

polynomial order pL 1 e

rror

meanvariancedistribution

> 6�?q$�"&"%02"B)�02;�E[$�"&=2$�;/)�$D6EGFIHKJML��KO,Ái_IaGoK[R_ISuEGUnabm�QRU�_nV*U2YZYbatEcLCpu]W_e[R_I[ZatE\[RL%U2J�U2pKÆ2L%m�]�Ec_R[IX�pbEcÚ�HKVWEG_I[¢O

ô¤Þ�Ú�Ü(Ø�Ý@ÜvØ­Ö�ß~Ô�æmÛ|Ý`Õ¼á�ÝYÕ�Ô�ÑmÜvØ~ß~Ù{Ø�Ý@ÜÏØ~ÞmÙpØ®Ù�Ö�Ö�ÛFÚFÓ`ÙpØ~Ú|Ô�Ñ[Ô�ãÀÖ=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛµÓSÞmÙ{Ô&ÜGØ�Ô§Ö�ß~Ô�æ�ÛFÝYէܤ�©Ú|Ø�Þ5Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛµÑ�Ô�Ñ�åÛFÚ|Ñ�Ý@Ù{ß~Ú±Ø~Ú|Ý@Ü3Ô�ã­×�ÑmÓ�Ý`ß�ØSÙ{ÚFѲépÙ{ß~ÚFÙ�æ�Û|Ý@Ü3ÓYÙ�ѽß~Ý`Ü�×�Û±ØeÚ|Ñ�Ù1ÜvØ~ß�Ô�Ñ�à�Ý`ߺÓ�Ô�×�Ö�ÛFÝ`á½Ü�Ý�غÔ�ã­Ý ê ×mÙ{Ø�ÚFÔ�ÑmÜeÓ�Ô�Õ�ÖmÙ�ß�Ý@á¡Ø�Ô¯Ø~Þ�Ý0ÛFÚFÑ�Ý`Ù�ßÖ�ß~Ô�æ�ÛFÝYÕ5ëTæ�×�Ø®Ø~ÞmÙpØ�Ø�Þ�ݺß~Ù{Ø�ÚFÔ§Ô{ã·ÚFÑmÓ�ß~Ý`Ù�Ü�Ý-Ô�ãQÓYÔ�Õ�Ö�×�ØSÙpØ~Ú|Ô�ÑmÙ{ÛAÓYÔ�Ü�Ø~Ü®Ô{ãIØ�Þ�ÝeÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛAÓSÞmÙ{Ô&Ü­Ù{ÖmÖ�ß�Ô&Ù�ÓSÞ5Ó�Ô�Õ\ÖTÙ{ß~Ý`áØ�ÔºØ�Þ�Ý%á�ÝYØ�ÝYß~Õ�Ú|ÑmÚFÜ�Ø�Ú�Ó­Ö�ß~Ô�æmÛ|Ý`Õ Ú�Ü(Ñ�Ô{Ø(Ñ�Ý`ÓYÝ`Ü~Ü�Ù�ß�ÚFÛFäeÞ�ÚFà�ÞmÝYß·ã�Ô�ßÏØ~Þ�Ý�Ñ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ßÏÖ�ß~Ô�æ�ÛFÝYÕ Ø~ÞmÙ{ѧã�Ô�ßÏØ~Þ�Ý�Û|ÚFÑ�Ý`Ù�ßGÖmß�Ô�æ�Û|Ý`Õ5ç

ÐÁÆ è ÊÏÇ,�-ÊZ�s�­Ç­Ê� ÷Î�¡n�\Ç­ÊGÇ#��Î�Ç,�q¡IÉmÎvÈ�Î���¢�Î�Ç�ÒÏÊZ��Ò­Î�Ç�Ó/Ì®Ç�Íd��É�È:¡QÎ�ÇÔÒM¡IÉmÎ�¡�Õ����_¢�IÔ�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛQÓSÞmÙ�Ô�Ü-ÓYÙ�Ñ�á�Ú|ß~Ý`Ó�Ø�ÛFä1æ=Ý0Ù{Ö�ÖmÛ|ÚFÝ`á¡Ø�Ô#Ý ê ×mÙpØ~Ú|Ô�ÑmÜJ�©Ú±Ø~Þ²ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�Û·Ñ�Ô�ÑmÛ|ÚFÑ�Ý`Ù�ß�Ú|Ø�ÚFÝ`Ü`ç8�­ÖmÖ�Û|ä�ÚFÑ�à¯Ö=Ô�ÛFä&åÑ�Ô�Õ\Ú�Ù{Û%ÓSÞmÙ�Ô�ܧØ~Ô�Ö�ß~Ô�æmÛ|Ý`Õ�Ü[ÚFÑ&é�Ô�ÛFé�Ú|Ñ�à²ÑmÔ�Ñ�Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�Ñ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ß�Ú|Ø�ÚFÝ`Ü[ÓYÙ{Ñ�æTÝ�Õ�Ô�ß~Ý1á�Ú��§ÓY×�Û|Ø`ç÷ÐÒѸØ~Þ�ÚFÜxÜ�Ý`Ó�Ø~Ú|Ô�ÑÑ�Ô�Ñ�ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛAÑ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß~Ú±Ø~Ú|Ý@Ü©Ú|Ñ�é�Ô�ÛFé&ÚFÑ�à�Ø~Þ�ݺ×�ÑTÓ�ÝYß�Ø~Ù�Ú|Ñ#épÙ�ß�Ú�Ù{æ�ÛFÝ`Ü®Ù{ß~ݺÜvØ~×má�ÚFÝ`á¯Ù�Ñmá5Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛkÓSÞmÙ�Ô�Ü©Ú�Ü®Ù�Ö�Ö�ÛFÚ|Ý@áØ�Ô§Ù�Ñ�Ô�Ñ�ÛFÚ|Ñ�Ý@Ù{ߤÖ=ÝYÑmá�×mÛ|×�Õ Ö�ß~Ô�æ�ÛFÝYÕ5çë�ìîë8þMþMò%ó4ý�õ:÷@ó%ïañ ïQù¤þ�ïQò�æqñMïaú�ó%õPò�ýd�MõPïaø8÷@ïÀñ_ïQñ_þ�ïaò�æqñ_ïQú�ó%õPò�ñMïañMò4ó%ñ_ôÈõ:öó ÷�ó4ôÈø�©Ý@Ó�ÝYÑ&Ø~Û|ä�ëÀÜ�ÝYé�ÝYßSÙ{ÛQÕ�Ý�Ø~Þ�Ô�á�Ü3ÞmÙNé�ݧæTÝ`ÝYѲÖ�ß~Ý`Ü�ÝYÑ&Ø~Ý`á�ã�Ô�ߺÝYépÙ{ÛF×mÙpØ~Ú|Ñmà#Ñ�Ô�Ñ�Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛQã�×mÑmÓ�Ø~Ú|Ô�ÑmÜ3Ô�ã¤épÙ�ß�Ú�Ù{æ�ÛFÝ`Ü%ß~ÝYÖmß�ÝYåÜ�ÝYÑ&Ø�Ý@á§æ�ä§ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û�ÓSÞmÙ�Ô�Ü·ÝYò�ÖmÙ{ÑmÜ�ÚFÔ�ÑmÜ@�^î"ôkÙNä�ÛFÔ�ßÏÜ�ÝYß~ÚFÝ`ÜGÝ�ò�ÖmÙ{ÑTÜ�ÚFÔ�Ñ"Ô�ãµØ�Þ�Ý�Ñ�Ô�Ñ�ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�Ûmã�×�ÑTÓ�Ø�ÚFÔ�ÑAë�Ù�Ñ0ÚFÑ&Ø�ÝYà�ß~Ù{åØ�ÚFÔ�ѧÕ�ÝYØ�Þ�Ô�á§Ù{ÑTá§Ù3Ü�Ù�Õ�Ö�Û|ÚFÑ�à�åtæmÙ�Ü�Ý@áeÜ�Ö=Ý`Ó�Ø~ß~Ù�ÛmÖ�ß~Ô{�vÝ@Ó�Ø~Ú|Ô�Ñ\Õ�ÝYØ�Þ�Ô�áµçIÐÒÑ�Ø�ÞmÚFÜGÜ�Ý@Ó�Ø�ÚFÔ�Ñ"Ù�Ñ�Ô{Ø~Þ�ÝYßGÙ{Ö�Ö�ß~Ô�Ù�ÓSÞeÚ�ÜÏá�ÚFÜ~Ó�×mÜ~Ü�Ý`áã�Ô�ß®ÝYépÙ{ÛF×mÙpØ~Ú|Ñmà§Ñ�Ô�Ñ�Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�ÑmÔ�Ñ�ÛFÚ|ÑmÝ`Ù{ß~Ú|Ø�ÚFÝ`ܤæ�ä[Ñ�×�Õ�Ý`ß�Ú�ÓYÙ�Û|ÛFä"ÚFÑ�Ø~ÝYà�ß~Ù{Ø�ÚFÑ�à�Ø�Þ�ݺÑ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ߤØ�ÝYß~Õ§�©Þ�ÚFÓSÞ¯Ó`Ù{Ñ�ÑmÔ{Ø­æ=Ý

��ê 0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�

Page 14: AIAA-2006-2066-173

0 10 20 30 40 50 60 7010

−6

10−4

10−2

100

work [# number of deterministic solves]

L 1 err

or

meanvariancedistribution

> '@?B<!�!;/$�'("�'(5�E[$�)s3&�!02;CAy5/� T1-/ &�!02;0 10 20 30 40 50 60 70

10−6

10−4

10−2

100

work [# (non)linear deterministic solves]

L 1 err

or

mean [# lin.]variancedistributionmean [# nl.]variancedistribution

> 6�?q;/02;�<!�!;/$�'("M'(5�E[$�)s3%�!02;CA}5/� T1-/ %�!02;D6EGFIHKJMLuN�KO��_IS�YbHI]WUd]�Ec_R[RUnab �_nJ�õeqr_nJ�]�QRLTYb_RacoZ[R_ISuEcUnaÛm�QRU�_nV�U2YZYbatEcLCp<]M_va\Et[RL�U2J8Un[Kpµ[R_R[Za\Et[RL�U2J�U2pKÆ2L%m�]�Ec_R[IX�pbEcÚ�HKVWEG_I[Et[u]WLCJ�SsVÂ_2q¤]�QRL·L�KHZEcÆCUnacLd[�]�[RHZS�`#L%J�_2q!pKL@]WLCJ�S�Et[ZE\V�]�E\m8V�_IacÆnL%V�OÝYépÙ�Û|×mÙ{Ø�Ý@á á�ÚFß~Ý`Ó�Ø~Û|ä�ç,íGÔ�ÑmÜ�Ú�á�Ý`ßeã�Ô�ß�Ý�ò�Ù{Õ�Ö�ÛFÝxÙ�Ñ Ý ê ×mÙpØ~Ú|Ô�Ñ�ÚFÑ&é�Ô�ÛFé�Ú|Ñ�à¡Ø�ÞmÝ5Ü ê ×mÙ{ß~Ý0ß~Ô�Ô{Ø�Ô{ã%Ù{Ñ´×�ÑmÓ�Ý`ß�ØSÙ{ÚFÑ�épÙ{ß~Ú�Ù{æ�ÛFÝ� ®M¨4©Sª�¬�¯YçºÐÒÑ¡Ø�ÞmÙ{Ø3ÓYÙ�Ü�ÝeØ�Þ�ݧÓ�Ô�Õºæ�ÚFÑmÙ{Ø�ÚFÔ�Ñ1Ô�ãÏØ~Þ�Ý�ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{ÛQÓSÞmÙ�Ô�Ü�Ý�ò�ÖmÙ�ÑmÜ�ÚFÔ�Ñ�Ù{Ñmá¯Ø�Þ�Ý0ö%Ù�Û|Ý`ß�ü�ÚFѯÖ�ß~Ô{�vÝ@Ó�Ø�ÚFÔ�Ñ¡ÚFÜÑ�Ô�Ø%á�ÚFß~Ý`Ó�Ø~Û|ä¯Ù�Ö�Ö�ÛFÚFÓ`Ù{æ�ÛFÝ3Ø~Ô0Ø~Þ�ÝeØ�ÝYß~Õ ÚFѯØ�ÞmÝ\Ý ê ×mÙpØ~Ú|Ô�ѯÚ|Ñ�é�Ô�ÛFé&ÚFÑ�à � ®n¨ ©Sª�¬U¯�çºø�×mæmÜvØ~Ú±Ø~×�Ø�ÚFÑ�àxØ�ÞmÝeÖ=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛQÓSÞmÙ{Ô&ÜÝ�ò�ÖmÙ�ÑmÜ�Ú|Ô�Ñ[Ô�ã_®n¨ ©Sª�¬U¯¤Ù{ÑmáxÖ=ÝYß�ã�Ô�ß~Õ\ÚFÑ�à§Ù§ö%Ù{ÛFÝYß~ü�Ú|ÑxÖ�ß~Ô{�vÝ`Ó�Ø�ÚFÔ�Ñ[ß�Ý@Ü�×�Û|Ø~Ü(ã�Ô�ß � ®n¨ ©Sª�¬�¯¤ÚFÑ

¼�� ®M¨4©Sª�¬�¯�ª�� â ¨4Ï1¯�¾�° á�� �´ ½ ¶ � ®Q½�¨ ©�¯W��½�¨ Ï1¯ � ª�� â ¨4Ï1¯ ã ª å�°Îæ­ª � ª�Ì�Ì@Ì2Ì ¨ � ç�¯ÐÒѽà�ÝYÑ�Ý`ß~Ù�ÛðëÀØ�ÞmÝ[Ù{ÖmÖ�ß�Ô&Ù�ÓSÞ�Ô{ã­ÚFÑ&Ø�ß~Ô�á�×mÓ�ÚFÑ�à1Ù1Ñ�Ý�� ×mÑmÓ�Ý`ß�ØSÙ{ÚFѲéNÙ�ß�Ú�Ù{æmÛ|Ý� d¨ ©Sª�¬�¯�° � ®n¨ ©Sª�¬�¯eÙ�ܺá�Ý`Ü~Ó�ß~Ú|æ=Ý`á²ÚFÑ�Ü�Ý`Ó�åØ�ÚFÔ�Ñ0Ð�Ð�Ð�ç � Ú|Ñ0Ó`Ù�Ü�Ý©Ô{ãAÙºÑ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ßQØ�ßSÙ{ÑmÜ�ã�Ô�ß~Õ§ÙpØ~Ú|Ô�Ñ\Ô�ãAÙ{Ñ"×mÑmÓ�Ý`ß�ØSÙ{ÚFѧÖmÙ{ßSÙ{Õ�Ý�Ø~ÝYßSÜ·á�Ô�Ý`ÜÏÑ�Ô{ؤÙ{Ö�ÖmÛ|ä�ÚFѧØ�Þ�Ú�Ü(ÓYÙ�Ü�Ý�ë&Ü�ÚFÑmÓYÝØ�ÞmÝ0×�ÑmÓYÝYß�Ø~Ù�Ú|ѽépÙ�ß�Ú�Ù{æ�ÛFÝ�®n¨ ©Sª�«�¯º×mÜ�×mÙ{ÛFÛFä�Ù�Ö�ÖTÝ@Ù{ßSÜ3ÚFѽÕ�Ô�ß~ݧØ�ÞTÙ{ѽÔ�Ñ�ݧØ�Ý`ß�ÕýÔ�ã©Ø�Þ�Ý[Ý ê ×mÙpØ~Ú|Ô�Ñ�çD� ÜvØ~ß~Ù�Ú|à�Þ�Ø�ã�Ô�ß��(Ù�ß~áÙ{ÖmÖ�ß�Ô&Ù�ÓSÞ0Ø�Ô"Ù{ÖmÖ�Û|ä0ÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û�ÓSÞTÙ{Ô�Ü(Ø~Ô"Ù{ÑxÝ ê ×mÙ{Ø�ÚFÔ�ÑxÚ|Ñ�é�Ô�Û|é�ÚFÑ�à§Ù\Ñ�Ô�Ñ�Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛµÑ�Ô�Ñ�ÛFÚ|Ñ�Ý@Ù{ß~Ú±Øvä0Ô{ãQÙ{Ñx×�ÑmÓ�Ý`ß�ØSÙ{ÚFÑépÙ{ß~ÚFÙ�æ�ÛFÝeÚ�Ü%Ø�Ô5ÚFÑ&Ø�ÝYà�ß~Ù{Ø�ÝeØ�Þ�Ý0ö%Ù�Û|Ý`ß�ü�ÚFÑ1Ö�ß~Ô{�vÝ@Ó�Ø~Ú|Ô�Ñ¡Ô�ãÏØ�ÞmݧÑ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ß�Ø~ÝYß~ÕKÑ�×�Õ�ÝYß~ÚFÓ`Ù{ÛFÛ|ä1×mÜ�ÚFÑ�à¯ö%Ù{×TÜ�Ü ê ×mÙ�á�ß~Ù{Ø�×�ß~ÝÚFÑ�Ø~ÝYà�ß~Ù{Ø�ÚFÔ�Ñ�çIÐÒÑ5ÓYÙ�Ü�Ý%Ô{ãQÙ�ÖbÉ º Ô�ßSá�ÝYß©Ù�Ö�Ö�ß~ÔNò�ÚFÕ§ÙpØ~Ú|Ô�Ñ0Ô{ãn®n¨ ©Sª�¬U¯GØ�Þ�Ú�Ü©ß~Ý`Ü�×�Û±ØSÜ(ã�Ô�ß�¨ � ç1¯GÚFÑ

¼�� ®n¨ ©Sª�¬U¯2ª�� â ¨ Ï1¯�¾Z°^ Ä�

� �´ ½ ¶ � ®d½�¨ ©�¯M��½�¨4Ï1¯��� � â ¨4Ï1¯sÆ�¨4Ï1¯vá:ÏÇ°�´� ¶ î � �´ ½ ¶ � ®d½�¨4©�¯W��½�¨ Ï � ¯l� � â ¨ Ï � ¯�Æ � ª ¨ � �U¯

ã�Ô�ß#å�°^æPª � ª�Ì@Ì�Ì2ª%ÖAëP�©Þ�Ý`ß�ÝJÏ � Ù{Ñmá�Æ � Ù{ß~Ý�Ø�Þ�Ý�Ï ê ×mÙ�á�ß~Ù{Ø�×�ß~Ý�Ö=Ô�ÚFÑ&Ø~Ü©Ù�Ñmá±�GÝ`Ú|à�Þ�ØSÜYë�ß~Ý`Ü�ÖTÝ@Ó�Ø~Ú|é�ÝYÛFä�çô¤Þ�Ú�Ü©Ù{Ö�Ömß�Ô&Ù�ÓSÞ[Ó`Ù{ÑxæTÝ3Ù�Ö�Ö�ÛFÚ|Ý@á"Ø~Ô�Ø�Þ�Ý%Ø~ÝYß~Õ ÚFÑ5Ù�Ñ[Ý ê ×mÙpØ~Ú|Ô�ÑxÓYÔ�Ñ&Ø~Ù�Ú|Ñ�ÚFÑ�à§Ù\Ñ�Ô�ÑmÖTÔ�Û|ä�Ñ�Ô�Õ\Ú�Ù{Û�Ñ�Ô�Ñ�ÛFÚ|Ñ�Ý@Ù{ß~Ú±Øvä0Ô{ãÀÙ�Ñ×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ¡éNÙ�ß�Ú�Ù{æmÛ|Ý\Ô�Ñ�ÛFä�ç�ôkÔ0Ø~Þ�Ý�Ô{Ø�ÞmÝYß�Ø�Ý`ß�Õ§Ü%Ô{ã¤Ù{Ñ¡Ý ê ×mÙpØ~Ú|Ô�Ñ1Ø~Þ�Ý�Ñ�Ô�ß~Õ§Ù{ÛIÖ=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛQÓSÞmÙ{Ô&Ü�Ù�Ö�Ö�ß~Ô�Ù�ÓSÞ¡ÓYÙ{Ñ�æ=ÝÙ{ÖmÖ�Û|ÚFÝ`á�ç

� = 0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�

Page 15: AIAA-2006-2066-173

� ì����Mô�ñMïañMò4ó%ñ_ôÈõ:ö�þ�ôÈñ!è_üMò4üMú þMöï#"nò4ôÈúô¤Þ�Ý¡Ñ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ß0Ö=ÝYÑmá�×mÛ|×�Õ Ömß�Ô�æ�Û|Ý`Õ ÚFÜ[Ó�Ô�ÑTÜ�Ú�á�ÝYß~Ý`á'Ù�Ü[Ù�Ñ'Ý�ò�Ù{Õ�Ö�ÛFÝ1Ô�ãºÙ²Ö�ß~Ô�æ�ÛFÝYÕ ÚFÑ�é�Ô�ÛFé�Ú|Ñmà½Ù½Ñ�Ô�Ñ�ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛÑ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß~Ú±Øvä§Ô�ãIÙ�Ñ[×�ÑTÓ�ÝYß�Ø~Ù�Ú|Ñ5épÙ{ß~Ú�Ù{æ�ÛFÝ�ë�Ü�ÝYÝ9�ÀÚ|à�×�ß�Ý ��� çQô¤Þ�Ý-Õ�Ô{Ø~Ú|Ô�Ñ[Ô�ãkØ~Þ�Ý-ÖTÝ`Ñmá�×�ÛF×�Õ ÚFܤà�Ôpé�Ý`ß�Ñ�Ý@á"æ�ä

á�ñ@$á:« ñ Ó�� �GÜ�ÚFÑ�$±° æ­ª «*)Ñæ­ª ¨ � +1¯�©Þ�Ý`ß�ݤØ~Þ�Ý�Ù�Ñ�à�ÛFÝ�$0á�ÝYõmÑ�Ý`Ü·Ø�Þ�Ý�ÖTÔ&Ü�Ú|Ø�ÚFÔ�ѧÔ�ã�Ø�Þ�Ý­Ö=ÝYÑTá�×�ÛF×�Õ5ë � ÚFÜ·Ø�Þ�Ý�à�ßSÙNé&Ú|Ø~Ù{Ø�ÚFÔ�ÑmÙ�Û�Ù�ÓYÓYÝYÛFÝYßSÙpØ~Ú|Ô�Ñ\Ù�Ñmá � Ú�ÜQØ�ÞmÝ­Û|Ý`Ñ�à{Ø~ÞÔ{ã(Ø�ÞmݧÖTÝ`Ñmá�×�ÛF×�Õ5ç"ÐÒÑ�Ú|Ø�Ú�Ù{ÛFÛ|ä1Ø�Þ�ݧÖ=ÝYÑTá�×�ÛF×�Õ Ú�Ü-ÙpØ3ß~Ý`Ü�Ø3ÙpØ�$Ë° � ç"ô¤Þ�Ý�Û|Ý`Ñ�à{Ø~Þ � ¨ ¬�¯�Ú�Ü3Ù�Ü�Ü�×�Õ�Ý`á1Ø�Ô5æ=ݧ×�ÑmÓ�Ý`ß�ØSÙ{ÚFÑ�©Þ�Ý`ß�ÝeØ�Þ�Ý�×�ÑTÓ�ÝYß�Ø~Ù�Ú|Ñ&Øvä¯Ó`Ù{Ñ1æ=ݧá�Ý`Ü~Ó�ß~ÚFæTÝ@á1æ�ä¯Ù[×�Ñ�Ú|ã�Ô�ß~Õ á�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�ÑÕ�©Ú|Ø�Þ54��Z° � Ù�Ñmá1ÓYÔpé��Z°rÌ � çeô¤Þ�ݧÜ�Ý`Ó�Ô�Ñmá

Ø�Ý`ß�Õ�Ô{ã©Ø~Þ�Ý0Ý ê ×mÙpØ~Ú|Ô�ѽÓ�Ô�Ñ&ØSÙ{ÚFÑmÜ3Ù¯ÑmÔ�Ñ�ÛFÚ|ÑmÝ`Ù{ß3Ø�ÝYß~Õ Ü�ÚFÑ�$Aç�ô¤Þ�Ý"Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛGÓSÞmÙ�Ô�Ü-ÝYò�ÖmÙ{ÑmÜ�ÚFÔ�ѽÙ{ÑTá�Ø�Þ�Ý5ö%Ù{ÛFÝYß~ü&ÚFÑÖ�ß~Ô{�vÝ@Ó�Ø�ÚFÔ�Ñ5ÓYÙ�Ñ�Ñ�Ô�Ø©á�Ú|ß~Ý`Ó�Ø�ÛFä0æ=ݺÙ{Ö�ÖmÛ|ÚFÝ`á[Ø�Ô\Ø~Þ�ÚFÜ­Ø�Ý`ß�Õ5ç��®Ö�Ö�ÛFä&ÚFÑ�à�Ø~Þ�Ý-ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛAÓSÞmÙ{Ô&ܤÙ{Ö�Ö�ß~Ô�Ù�ÓSÞ"Ø~Ô�¨ � +1¯(ß�Ý@Ü�×mÛ±ØSÜÚFÑ ¼�� ñ â ¾ á�ñ%$ âá:« ñ Ó � ¼ � hµî Ü�ÚFÑ�$�ª�� â ¾�° æ­ª å�° æ­ª � ª�Ì�Ì@Ì Ö�Ì ¨ � �U¯ÐbØ(Ó`Ù{Ñ"æ=Ý%Ü�Ý`ÝYÑ�Ø�ÞTÙpØ(Ø�Þ�Ý�Ñ�Ô�ß~Õ§Ù{ÛmÖ=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�ÓSÞmÙ{Ô&Ü·Ù{ÖmÖ�ß�Ô&Ù�ÓSÞ§ÓYÙ�Ñ"æ=Ý�Ù�Ö�Ö�ÛFÚ|Ý@á�Ø�ÔºØ�Þ�Ý®õmß~Ü�ØÏØ~ÝYß~Õ5çn�®Ñ0Ù{ÖmÖ�ß�Ô&Ù�ÓSÞ�ã�Ô�ßá�Ý@Ù{ÛFÚ|Ñ�à��©Ú|Ø�Þ\Ø�Þ�Ý©Ü�Ý`ÓYÔ�ÑmáºØ�Ý`ß�ÕÁÚ�ÜkØ�Ô%ÚFÑ�Ø~ÝYà�ß~Ù{Ø�Ý(ڱطÑ�×�Õ�ÝYß~Ú�ÓYÙ{ÛFÛFä3ÙpãoØ�Ý`ßQÜ�×�æTÜvØ~Ú±Ø~×�Ø�ÚFÑ�à%Ø�ÞmÝ(Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�ÓSÞmÙ�Ô�ÜÀÝ�ò�ÖmÙ�ÑmÜ�ÚFÔ�ÑAë�©Þ�Ú�ÓSÞ5ß�Ý@Ü�×�Û|Ø~ܤÚFÑ

¼ � h�î Ü�ÚFÑ�$�ª�� â ¾�° �´� ¶ î î´µ�¶ � � fµ � µ ¨ Ï � ¯�Ü�ÚFÑ � �´ ½ ¶ � $Q½�¨4«�¯W��½�¨ Ï � ¯ � â ¨4Ï � ¯sÆ � ª å�°Îæ­ª � ª�Ì�Ì@Ì2ª ¨ � �U¯

�©Þ�Ý`ß�Ý � f{¨4¬�¯�° � hµî¨ ¬�¯3ã�Ô�ßû�©Þ�Ú�ÓSÞ²Ø�Þ�Ý[Ô�Ö�Ø~Ú|Õ§Ù�Û(Ü�Ý�Ø\Ô{ã©Ô�ß�Ø�Þ�Ô�à�Ô�ÑmÙ{Û·Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�Ü3Ú�ÜeÓYÔ�Õ�Ö�×�Ø~Ý`á²Ñ�×�Õ�Ý`ß�Ú�ÓYÙ�Û|ÛFä�ç1ô¤ÞmÝÑ�×�Õ�ÝYß~ÚFÓ`Ù{Û(Ú|Ñ&Ø~ÝYà�ßSÙ{Û(ÚFÑI¨ � ��¯ºÓ�Ô�Ñ�ØSÙ{ÚFÑmܺØ�Þ�Ý5Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�Û(ÓSÞTÙ{Ô�Ü\Ó�Ô�Ý��§ÓYÚ|Ý`Ñ&Ø~Ü�$ ½ ¨ «�¯ºã�Ô�ßûÉû°mæPª � ª�Ì@Ì�Ì2ª4ÖÑ�©Þ�Ú�ÓSÞ Ù�ß�ÝxÙã�×�ÑmÓ�Ø�ÚFÔ�Ñ¡Ô{ã·Ø~Ú|Õ�Ý�ç9��Ü�Ù[Ó�Ô�ÑmÜ�Ý ê ×�ÝYÑmÓYÝ-Ø�ÞmÝeÑ�×�Õ�ÝYß~Ú�ÓYÙ{ÛÀÚFÑ�Ø~ÝYà�ß~Ù�ÛAÞmÙ�Ü­Ø~Ô[æ=Ý\ÓYÔ�Õ�Ö�×�Ø~Ý`á1ÝYé�Ý`ß�ä[Ø�ÚFÕ�Ý\Ü�Ø�Ý`Ö1Ô�ß�Ü�Ø~Ù{à�ݺÚFÑÓYÙ�Ü�ÝGÔ{ã�Þ�ÚFà�ÞmÝYßÀÔ�ßSá�ÝYßAØ�ÚFÕ\ݤÚFÑ&Ø�ÝYà�ß~Ù{Ø�ÚFÔ�Ñ�çkô¤Þ�ÚFÜÀÓYÙ�Ѻæ=Ý(Ù�Ü�Ý`ß�ÚFÔ�×TÜAá�ßSÙ/�©æmÙ�ÓSü�ÚFÑ-Ø�Ý`ß�Õ§ÜÀÔ{ã�Ø~Þ�Ý(ÓYÔ�Õ�Ö�×�ØSÙpØ~Ú|Ô�ÑmÙ{Û&Ý[�"Ó�ÚFÝYÑTÓ�ä�çô¤Þ�ÝeÑ&×mÕ\Ý`ß�Ú�ÓYÙ�ÛµØ�ÚFÕ�ݺÚFÑ&Ø�ÝYà�ß~Ù{Ø�ÚFÔ�Ñ#Ô�ãÀØ~Þ�ݺ×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ�Øvä ê ×mÙ{Ñ&Ø~Ú±õTÓ`ÙpØ~Ú|Ô�Ñ#Ö�ß�Ô�æ�ÛFÝYÕ ¨ � �U¯©Ù{ÑTá�¨ � ��¯¤Ú�Ü­Ö=ÝYß�ã�Ô�ß~Õ�Ý`á5×mÜ�Ú|ÑmàÙºã�Ô�×�ß�Ø�Þ"Ô�ß~á�ÝYß(Ý�ò�Ö�ÛFÚFÓYÚ±ØL�©×�Ñ�à�Ý�ås�%×�Ø�ØSÙºÜ~ÓSÞ�ÝYÕ�Ý�çIô¤ÞmÝ�ß~Ý`Ü�×�Û|ØGã�Ô�ßÏØ~Þ�Ý-á�ÚFÜ�Ø�ß~ÚFæ�×�Ø�ÚFÔ�Ñ0ã�×�ÑmÓ�Ø�ÚFÔ�Ñ[Ù{Øz«�° � ÓYÙ�Ñ0æ=Ý�Ü�ÝYÝ`Ñ0ÚFÑ�ÀÚ|à�×�ß~Ý �/� Ù�çAÐÒÑû�ÀÚFà�×�ß~Ý �/� æ-Ø~Þ�ݤÝ[�Ý`Ó�ØIÔ�ãmØ~Þ�ÝGÑ�×�ÕeæTÝ`ßQÔ{ã=ö%Ù�×mÜ~Ü ê ×mÙ�á�ßSÙpØ�×mß�ÝGÖTÔ�Ú|Ñ&Ø~Ü�ϼÚ|ÑeØ�Þ�ݤÑ�×�Õ�ÝYß~Ú�ÓYÙ{Û&ÚFÑ�Ø~ÝYà�ß~Ù{Ø�ÚFÔ�ÑÔ{ã(Ø�ÞmݧÑ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ß%Ø�ÝYß~Õ Ú�Ü3Ü�Ø�×má�Ú|Ý@áµç"ÐÒÑ�Ø�Þ�Ú�Ü3Ó`Ù�Ü�Ýeã�Ô�ßTÏ ° � æ[Ø�ÞmݧÝYß~ß�Ô�ß�ÚFÑ�Ø�Þ�Ý"á�ÚFÜ�Ø�ß~Ú|æm×�Ø�ÚFÔ�Ñ�ã�×�ÑmÓ�Ø�ÚFÔ�ѽÓYÔ�Ñ�é�Ý`ß�à�Ý`ÜÝ�ò�Ö=Ô�Ñ�Ý`Ñ&Ø�Ú�Ù{ÛFÛ|ä5Ø~Ô5Ù�Ñ¡Ý`ß�ß~Ô�ß�Ô{ã � æbhµî��3ã�Ô�ßLÖÀ° � æmçÇÿ�Ý`ÓYß�Ý@Ù�Ü�Ú|Ñ�à0Ø�Þ�ݧÑ�×�ÕeæTÝ`ß-Ô{ã©ö%Ù{×mÜ~Ü ê ×TÙ�á�ßSÙpØ~×�ß�ÝeÖTÔ�Ú|Ñ&ØSÜ�ã�Ô�ß%Ø�ÞmÝÑ�×�Õ�ÝYß~ÚFÓ`Ù{ÛQÚFÑ&Ø�ÝYà�ß~Ù{Ø�ÚFÔ�Ñ�Ô{ã(Ø�Þ�ݧÑmÔ�Ñ�ÛFÚ|ÑmÝ`Ù{ß%Ø~ÝYß~ÕKØ~Ô�Ï ° � çxÜvØSÙ{ß�Ø~Ü%Ø~Ôxß~Ý`á�×mÓ�Ý�Ø~Þ�Ý"Ó�Ô�Ñ&é�ÝYß~à�Ý`ÑmÓ�Ýeã�ß�Ô�ÕrÖ�°�+�ç+��Ô�ßÝYé�ÝYÑxÛ|Ô�(ÝYߤÑ�×�ÕeæTÝ`ß©Ô{ã ê ×TÙ�á�ßSÙpØ~×�ß�Ý�Ö=Ô�ÚFÑ�ØSÜ�Ïm° � æeÙ�Ñmá,Ïm°Ãç�ë�Ø~Þ�Ý3Ù{Ö�Ömß�ÔNò�ÚFÕ�Ù{Ø�ÚFÔ�Ñ[Ô{ãAØ~Þ�Ý3á�Ú�ÜvØ~ß�ÚFæ�×�Ø~Ú|Ô�Ñxã�×�ÑmÓ�Ø~Ú|Ô�Ñæ=Ý`Ó�Ô�Õ�Ý`Ü­ÛFÝ`Ü~Ü­Ù�ÓYÓ�×mß~Ù{Ø�Ý9�©Ú|Ø�Þ¯ÚFÑmÓ�ß~Ý`Ù�Ü�ÚFÑ�à�Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛAÔ�ß~á�Ý`ß®Ü�Ø~Ù{ß�Ø�ÚFÑ�à�ã�ß~Ô�Õ]Ö�°z�"Ù�Ñmá�Ö�°ÃàTëmß~Ý`Ü�ÖTÝ@Ó�Ø~Ú|é�ÝYÛFä�ç¤ÐbØ�ÓYÙ�Ñæ=Ý[Ó�Ô�ÑmÓ�ÛF×má�Ý@á�Ø�ÞTÙpØeÚ|Ø�ÚFÜeÚ|Õ�Ö=Ô�ß�Ø~Ù�Ñ�غØ�Ô1ÚFÑ&Ø�Ý`à�ßSÙpØ�Ý"Ø�Þ�Ý[Ñ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ß3Ø�Ý`ß�Õ§ÜÇ�©Ú±Ø~Þ�Ü�×:�"Ó�ÚFÝYÑ&Ø\Ù�ÓYÓ�×mß~Ù�Ó�ä¡Ø�Ô1ÙNé�Ô�Ú�á�Ø�ÞmÙ{ØØ�ÞmÝ0ÜvØ~Ô�ÓSÞmÙ�Ü�Ø�Ú�Ó�ÝYß~ß�Ô�ß-ÚFÜeá�Ô�Õ�ÚFÑmÙpØ~Ý`á�æ�ä1Ø~Þ�Ý"ÚFÑ&Ø�ÝYà�ß~Ù{Ø�ÚFÔ�Ñ�ÝYß~ß~Ô�ß%Ô{ã¤Ø�ÞmÝ"Ñ�Ô�ÑmÛ|ÚFÑ�Ý`Ù�ß-Ø�Ý`ß�Õ§Ü`ë_�©ÞmÚFÓSÞ½Ó`Ù{Ñ�ÝYé�ÝYѲÛFÝ`Ù�á¡Ø~Ôá�ÚFé�Ý`ß�à�ÝYÑmÓYÝ�Ô�ãkØ~Þ�Ý3Ü�Ô�Û|×�Ø�ÚFÔ�Ñ[ã�ß�Ô�Õ¼Ù�Ó�ÝYß�Ø~Ù�Ú|ÑxÖ=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�Û�ÓSÞmÙ{Ô&Ü(Ô�ßSá�ÝYß@çÐÒѲôkÙ�æ�ÛFÝ � Ø�Þ�Ý0ÓYÔ�Õ�Ö�×�ØSÙpØ~Ú|Ô�ÑmÙ{ÛIØ~Ú|Õ�ݧã�Ô�ߺÜ�Ô�Û|é�ÚFÑ�àxØ�Þ�Ý0×�ÑmÓ�Ý`ß�ØSÙ{ÚFÑ&Øvä ê ×mÙ{Ñ&Ø~Ú±õTÓ`ÙpØ~Ú|Ô�Ñ¡Ö�ß~Ô�æmÛ|Ý`ÕÔ¨ � ��¯%Ù�ÑmᲨ � �U¯�ÚFÜ

à�ÚFé�Ý`Ñ¡Ù�Ü�ã�×�ÑmÓ�Ø~Ú|Ô�Ñ�Ô{ãÏØ~Þ�ݧÑ�×�Õºæ=ÝYß-Ô�ã¤ö%Ù�×mÜ�Ü ê ×mÙ�á�ßSÙpØ�×mß�ÝeÖTÔ�Ú|Ñ&Ø~ÜTÏ ã�Ô�ß�ÚFÑ&Ø�Ý`à�ßSÙpØ�ÚFÑ�à[Ø�Þ�ݧÑmÔ�Ñ�ÛFÚ|ÑmÝ`Ù{ß�Ø~ÝYß~Õxç�ÐÒÑ�Ø�ÞmÝÜ�Ý`Ó�Ô�ÑmáxÓ�Ô�ÛF×�Õ�Ñ[Ø�Þ�ÝJ�¤Ù{ÛFÛ�Ó�ÛFÔ�ÓSü�Ø~Ú|Õ�Ý%ã�Ô�ß­Ü�Ô�Û|é�ÚFÑ�àºØ~Þ�Ý%ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�Û�ÓSÞmÙ{Ô&ÜÏÖ�ß~Ô�æmÛ|Ý`Õ ÚFܤà�Ú|é�ÝYÑ0ã�Ô�ß­Ù{ÑxÚFÕ\ÖmÛ|Ý`Õ\Ý`Ñ&Ø~ÙpØ~Ú|Ô�ÑÚFÑ[ì��Qô�£_���'Ô�Ñ[Ù � ç à&ö%û,���IÝYÑ&Ø�ÚF×�Õ`à��(í�ç�ô¤Þ�Ý%Ú|ÑTÓ�ß~Ý`Ù�Ü�Ý­ÚFÑxÓYÔ�Õ�Ö�×�ØSÙpØ~Ú|Ô�ÑmÙ{ÛTØ�ÚFÕ�Ý��©Ú|Ø�Þ[ß�Ý@Ü�Ö=Ý`Ó�ØGØ~ÔºØ~Þ�Ý%Û|ÚFÑ�Ý@Ù{ß~Ú��`Ý`áÖ=ÝYÑmá�×mÛ|×�Õ Ömß�Ô�æ�Û|Ý`Õg¨3Ïg° � ¯¤Ú�Ü­à�ÚFé�Ý`Ñ#Ú|ѯØ�Þ�Ý-Ø~Þ�ÚFß~á¯ÓYÔ�ÛF×�Õ�Ñ�ç#�mß�Ô�Õ¼Ø~Þ�Ý3ÝYß~ß~Ô�ß®ÓYÔ�Ñ�é�Ý`ß�à�ÝYÑmÓYÝ�ÚFÑD�ÀÚFà�×�ß~Ý ��� æ5Ú±Ø��(Ù�ÜÓ�Ô�ÑmÓ�ÛF×má�Ý@á5Ø�ÞTÙpØ®ã�Ô�ß�Ö�° � æ§Ù"Ñ�×�Õºæ=ÝYß�Ô{ã ê ×mÙ�á�ß~Ù{Ø�×�ß~Ý-Ö=Ô�ÚFÑ�ØSÜ®Ô�ãÂÏg° � æ§Ú�Ü®Ñ�Ý`Ý`á�Ý@á5Ø�Ô[ÙNé�Ô�Ú�áxØ�ÞTÙpØ®Ø~Þ�ݺÑ�×�Õ�Ý`ß�Ú�ÓYÙ�ÛÚFÑ�Ø~ÝYà�ß~Ù{Ø�ÚFÔ�ѧÝYß~ß~Ô�ß·ÚFÑ�âm×�Ý`ÑmÓ�Ý@ÜQØ~Þ�Ý�Ý`ß�ß~Ô�ßGÓ�Ô�Ñ�é�ÝYß~à�ÝYÑTÓ�Ý�çAÐÒѧØ�ÞmÙ{ؤÓ`Ù�Ü�Ý©Ø�ÞmÝ�Ó�Ô�Õ�Ö�×�Ø~Ù{Ø�ÚFÔ�ÑmÙ�Û�Ø~Ú|Õ�Ý®ÚFÑmÓYß�Ý@Ù�Ü�Ý`Ü��©Ú|Ø�Þ � ç:Ìc+��Ó�Ô�Õ�ÖmÙ{ß~Ý`á"Ø�Ô\Ø�Þ�Ý%ÛFÚ|ÑmÝ`Ù{ß~Ú��YÝ`á0ÖTÝ`Ñmá�×�ÛF×�Õ Ö�ß�Ô�æ�ÛFÝYÕ5çQÐbØ­ÓYÙ{Ñ[æTÝ3ÓYÔ�ÑmÓYÛ|×Tá�Ý`á0Ø�ÞmÙ{ؤØ�Þ�Ý-Ü�×:�"Ó�ÚFÝYÑ&Ø�ÛFä0Ù�Ó`Ó�×�ßSÙpØ~Ý�Ñ�×�Õ�Ý`ß�Ú�ÓYÙ�ÛÚFÑ�Ø~ÝYà�ß~Ù{Ø�ÚFÔ�Ñ,Ô{ãºÙ²ÑmÔ�Ñ�ÛFÚ|ÑmÝ`Ù{ß0Ø�Ý`ß�Õ ß~Ý`Ü�×�Û±ØSÜ"ÚFÑ�Ù½Ü�Ú|à�Ñ�Ú|õTÓYÙ�Ñ�Ø[Ú|ÑmÓYß�Ý@Ù�Ü�Ý#Ú|Ñ Ó�Ô�Õ�Ö�×�Ø~ÙpØ~Ú|Ô�ÑmÙ{Û®Ø�ÚFÕ�Ý�ÓYÔ�Õ�ÖmÙ{ß~Ý`á Ø~Ô½Ø�ÞmÝÛFÚ|Ñ�Ý@Ù{ß~Ú��`Ý`á[Ö�ß~Ô�æ�ÛFÝYÕ5ç

� f¤0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�

Page 16: AIAA-2006-2066-173

m

lx

g

> '@?q.{$�;/5/-/<!-�K

t=0t=1

> 62?ÛB/E[$�5/$s3&$�"%KL�!;/�! %3%�!)n"&$�'(<!����'�3%�!02;/ D6EcFRHKJWL�N2NnO,P�QRL�[R_I[ZatE\[RL%U2J�YbLd[KpbHZa\HZSýYZJM_n`baGLCS

−1 −0.98 −0.96 −0.94 −0.920

0.2

0.4

0.6

0.8

1

angle x

prob

abili

ty d

istri

butio

n fu

nctio

n

exactPC

> '@?q5/�! 43&"%�!6/-�3%�!02;#F�-/;/)s3%�!02;0 2 4 6 8 10

10−15

10−10

10−5

100

polynomial order p

L 1 err

or

N=20

N=15

N=10

N=5

distribution

> 62?B�!;CË-�$�;�)�$��D6EGFIHKJMLeNCjRO�Ái_IaGoK[R_RS�EGUna�m�QRU�_nV�JML%VMHZa¡]�V�qr_nJ�]�QRLTpbE\V�]�J�E\`bHI]�EG_I[vq�HZ[Km�]�EG_I[vUn[Kp;]�QRL�L�Ú¢LCm�]l_2q!]�QRL�[RHZSTLCJ�EGm�Una÷Et[2]WL�FIJWUd]�Ec_R[_I[e]�QRL�m�_I[nÆ2L%JMF2Ld[Km�L·qr_nJ�]�QRL�[R_R[Za\Et[RL�U2J�Y#LC[KpbHZatHZSýYZJM_n`bacLdS

� k¤0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�

Page 17: AIAA-2006-2066-173

P�U2`baGLujRO���LCaGUd]�EGÆ2L�m�_IS�YbHI]WUd]�Ec_I[RUna÷]�EtSsLTqr_nJ�]�QRLv[IHZSsL%J�E\m�Una6Et[2]WL�FIJWUd]�Ec_R[;_2q9]�QRLe[R_R[Za\Et[RL�U2J�]ML%J�SîE\[½]�QRLeYbLd[KpbHZa\HZSYZJM_n`baGLCSÊqr_nJ ø8ù � h O Ï �¤Ù{ÛFÛ�Ó�ÛFÔ�ÓSü"Ø�ÚFÕ\Ý<y Ü | �÷ÚFÑ&Ø�ÝYà�ß~Ù{Ø�ÚFÔ�Ñ� �mç � æ æç �mç �K+ � æ�ç �� æ +�ç àR� � ��ç à� ç +�ç �K+ � ��ç!ç� æ �mç ç�ç � ç�çc+иÅmÆ �¡ÊÏÇ�ÍB��Ì�¢&Î�ÊÏÇ#¢

ÐÒѽØ~Þ�Ú�ÜeÖTÙ{Ö=ÝYßeØ�Þ�Ý#Ù{ÖmÖ�Û|Ú�ÓYÙ{Ø�ÚFÔ�ѽÔ�ã®Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�Û(ÓSÞTÙ{Ô�ܺØ�Ô�×�ÑmÓYÝYß�Ø~Ù�Ú|Ñ&Øvä ê ×mÙ{Ñ&Ø�Ú|õTÓ`ÙpØ�ÚFÔ�Ñ�Ú|Ñ Ö�ß�Ô�æ�ÛFÝYÕ§Ü\Ú|Ñ�é�Ô�Û|é�ÚFÑ�àÑ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß~Ú±Ø~Ú|Ý@ܺÞmÙ�ܺæ=ÝYÝ`Ñ á�Ú�Ü~Ó�×mÜ~Ü�Ý@áµç�ô¤Þ�Ý[Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û(ÓSÞmÙ{Ô&ÜeÙ�Ö�Ö�ß~Ô�Ù�ÓSÞTñSïtîvó�Ú�Ü\Ù{Ñ�ÝYò�Ø�ÝYÑTÜ�ÚFé�Ý`Û|ä�×mÜ�Ý`á½×mÑmÓ�Ý`ß�ØSÙ{ÚFÑ�Øväê ×mÙ{Ñ&Ø~Ú±õTÓ`ÙpØ~Ú|Ô�ÑxÕ�ÝYØ�Þ�Ô�áµç®û®Ô�GÝ`é�Ý`ß`ë�Ú|Ø®Ú�Ü,�©Ú�á�ÝYÛFä#Ù�ÓSü�Ñ�Ô�©ÛFÝ`á�à�Ý`á0Ø�ÞmÙ{Ø®Ñ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß~Ú±Ø~Ú|Ý@Ü®ÓYÙ�ѯÓYÙ�×mÜ�Ý3á�Úx�"Ó�×mÛ±Ø~Ú|Ý@Ü­ã�Ô�ß�Ù{Ö�åÖ�ÛFä�Ú|Ñ�àxÖ=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�ÛÀÓSÞTÙ{Ô�Ü`ç î�ïtîs�,ô¤Þ�ß~ÝYÝeØvä�ÖTÝ@Ü�Ô�ãÏÑ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß~Ú±Ø~Ú|Ý@Ü®Ø~ÞmÙpØ-ÓYÙ�ѯã�ß~Ý ê ×�Ý`Ñ�Ø~Û|ä#æ=Ý\ÝYÑmÓYÔ�×�Ñ&Ø�Ý`ß�Ý@á¯ÚFÑ1Ö�ßSÙ�Ó�Ø�Ú�ÓYÙ�ÛÖ�ß~Ô�æ�ÛFÝYÕ§Ü�ÞmÙNé�ݺæ=ÝYÝYѲÓYÔ�ÑmÜ�ÚFá�ÝYß~Ý`áB�û¨vls¯%è­Ô�ÑmÛ|ÚFÑ�Ý`Ù�ß�Ø�ßSÙ{ÑmÜ�ã�Ô�ß~Õ§ÙpØ�ÚFÔ�ÑTÜ®Ô�ãÏ×�ÑmÓYÝYß�Ø~Ù{ÚFÑ�Ú|Ñ�Öm×�Ø-ÖmÙ{ßSÙ{Õ�Ý�Ø~ÝYßSÜYëd�©Þ�Ú�ÓSÞ�Ô{ãoØ�Ý`Ñß~Ý`Ü�×�۱ؤÚFÑ0Ñ�Ô�Ñ�åbÜ�Ø~Ù�Ñmá�Ù�ß~á§á�Ú�Ü�Ø�ß~Ú|æ�×�Ø�ÚFÔ�ÑmÜ`çS�Qò�ÖTÔ�Ñ�ÝYÑ&Ø~ÚFÙ�Û=ÓYÔ�Ñ�é�Ý`ß�à�ÝYÑmÓYÝ®ÓYÙ�Ñ"æTÝ�ß~Ý�Ø~Ù�Ú|ÑmÝ`á§ã�Ô�ßÏØ~Þ�Ú�ÜÏØvä�Ö=Ý�Ô�ãAÑ�Ô�Ñ�Û|ÚFÑ�Ý@Ù{ß~Ú±Ø~Ú|Ý@Üæ�äeÓYÔ�Õ�Ö�×�Ø~Ú|Ñ�à%Ñ�×�Õ�ÝYß~Ú�ÓYÙ{ÛFÛFäºÙ{Ñ�Ô�Ö�Ø�ÚFÕ�Ù�ÛmÜ�Ý�Ø·Ô{ã=Ô�ß�Ø�Þ�Ô�à�Ô�ÑmÙ{Û�æmÙ�Ü�Ú�ÜIÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�ÛFÜÀã�Ô�ßQØ�Þ�Ý©ÑmÔ�Ñ�ÛFÚ|ÑmÝ`Ù{ßIØ~ß~Ù�ÑmÜ�ã�Ô�ß~Õ�Ù{Ø�ÚFÔ�ÑeÔ{ãØ�ÞmÝ�×�ÑTÓ�ÝYß�Ø~Ù�Ú|Ñ[Ú|ÑmÖ�×�Ø©×mÜ�Ú|Ñ�à\Ù{Ñ[Ô�ß�Ø�Þ�Ô�à�Ô�ÑTÙ{ÛFÚ��@ÙpØ�ÚFÔ�ѧÙ�Û|à�Ô�ß~Ú±Ø~Þ�Õ��d¨�lol*¯(Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û=Ñ�Ô�ÑmÛ|ÚFÑ�Ý`Ù�ß�Ú|Ø�ÚFÝ`ÜGÚFÑ&é�Ô�ÛFé�Ú|Ñ�àº×�ÑmÓ�Ý`ß�ØSÙ{ÚFÑépÙ{ß~ÚFÙ�æ�ÛFÝ`Ü`ëU�©ÞmÚFÓSÞxÔ�ÓYÓ�×mß(ÚFÑxÕ�Ù�Ñ�ä§Ñ�Ô�Ñ�ÛFÚFÑ�Ý`Ù�ß(Ömß�Ô�æ�Û|Ý`Õ§ÜYçS�IÔ�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û=Ñ�Ô�ÑmÛ|ÚFÑ�Ý`Ù�ß�Ú|Ø�ÚFÝ`ܤÓ`Ù{Ñ0ß~Ý`Ü�×�Û|Ø~Ü(ÚFÑxÙ�Ü�Ø�ß~Ô�Ñ�à�ÝYߤÓ�Ô�×�åÖ�ÛFÝ`á[Ü�ä�Ü�Ø�ÝYÕ5ë�æ�×�Ø(Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�Ñ�Ô�ÑmÛ|ÚFÑ�Ý`Ù�ß�Ú|Ø�ÚFÝ`Ü(á�Ô\Ñ�Ô{ؤÑ�Ý@Ó�Ý`Ü~Ü~Ù{ß~Ú|ÛFäeÚFÑmÓYß�Ý@Ù�Ü�Ý­Ø�ÞmÝ�ß~ÝYÛ�ÙpØ~Ú|é�Ý�Ù{Õ�Ô�×�Ñ&Ø(Ô�ãkÓ�Ô�Õ\Öm×�Ø~Ù{Ø�ÚFÔ�ÑmÙ�Û�(Ô�ß~üeÔ�ã�Ø�Þ�Ý�Ö=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�Û=ÓSÞTÙ{Ô�ÜGÙ{Ö�Ömß�Ô&Ù�ÓSÞ"Ó�Ô�Õ�ÖmÙ{ß~Ý`á\Ø~Ô\ÙºÛ|ÚFÑ�Ý`Ù�ßGÖmß�Ô�æ�Û|Ý`Õ��d¨�lolol*¯(ÑmÔ�Ñ�Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{ÛmÑmÔ�Ñ�ÛFÚ|ÑmÝ`Ù{ß~Ú|Ø�ÚFÝ`ÜGÚ|Ñ�åé�Ô�Û|é�ÚFÑ�à�×�ÑmÓ�Ý`ß�ØSÙ{ÚFÑeépÙ�ß�Ú�Ù{æ�ÛFÝ`ÜkØ~Ô��©ÞmÚFÓSÞeØ�Þ�Ý©Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�ÓSÞmÙ�Ô�ÜIÙ{Ö�Ömß�Ô&Ù�ÓSÞeÓ`Ù{Ñ�ÑmÔ{ØQá�Ú|ß~Ý`Ó�Ø�ÛFä3æ=Ý­Ù{Ö�Ö�ÛFÚFÝ`áµçn�­Ñ§Ù{ÖmÖ�ß�Ô&Ù�ÓSÞØ�Ô0Ø�ß~Ý`Ù{Ø®Ý ê ×mÙ{Ø�ÚFÔ�ÑmÜ#�©Ú±Ø~Þ1Ñ�Ô�Ñ�ÖTÔ�Û|ä�Ñ�Ô�Õ�ÚFÙ�Û�ã�×�ÑmÓ�Ø�ÚFÔ�ÑmÜ�Ú�Ü­Ø~Ô"Ñ�×�Õ�ÝYß~ÚFÓ`Ù{ÛFÛ|äxÚ|Ñ&Ø~ÝYà�ßSÙpØ~Ý-Ø�ÞmݺÚFÑ�Ñ�ÝYß�Ö�ß~Ô�á�×mÓ�ØSÜ­Ô�ãQØ�Þ�Ý\Ñ�Ô�Ñ�åÖ=Ô�ÛFä&ÑmÔ�Õ�ÚFÙ�Û�Ø~ÝYß~Õ�ÜZ�©Ú|Ø�Þ§Ø~Þ�Ý®æmÙ�Ü�Ú�Ü·Ö=Ô�ÛFä�Ñ�Ô�Õ�Ú�Ù{Û�Ü·ß~Ý`Ü�×�Û|Ø�ÚFÑ�à-ã�ß~Ô�Õ Ø�Þ�Ý%ö%Ù{ÛFÝYß~ü�Ú|ѧÖ�ß~Ô{�vÝ@Ó�Ø~Ú|Ô�ѧ×mÜ�ÚFÑ�àeö%Ù�×mÜ~Ü ê ×mÙ�á�ß~Ù{Ø�×�ß~ÝÚFÑ�Ø~ÝYà�ß~Ù{Ø�ÚFÔ�Ñ�çIè­×�Õ�Ý`ß�Ú�ÓYÙ�Û=ß~Ý`Ü�×�Û|Ø~ÜGÞmÙNé�Ý�æTÝ`ÝYÑ[Ö�ß�Ý@Ü�Ý`Ñ&Ø�Ý`á"ã�Ô�ߤٺէÙ�Ü�Ü�åbÜ�Ö�ß~Ú|ÑmàºÜ�ä�Ü�Ø�Ý`Õxë&Ø~Þ�Ý%Þ�Ý`Ù{ØGØ~ß~Ù�ÑmÜvã�Ý`ßGÖmß�Ô�æ�Û|Ý`Õ÷Ú|ÑxÙØ��(Ô{åÒá�Ú|Õ�Ý`ÑmÜ�ÚFÔ�ÑTÙ{Û�ÓSÞmÙ{Ñ�ÑmÝYÛ�âmÔ�-ë�ÛFÚ|Ñ�Ý@Ù{ß­Ù�Ñmá"Ñ�Ô�Ñ�ÛFÚ|Ñ�Ý@Ù{ß©Ù�á�é�Ý`Ó�Ø�ÚFÔ�Ñ�åÒá�Ú�=×TÜ�ÚFÔ�Ñ[Ý ê ×mÙpØ~Ú|Ô�ÑmܤÙ{ÑmáxÙ\Ñ�Ô�ÑmÛ|ÚFÑ�Ý`Ù�ߤÖTÝ`Ñmá�×�ÛF×�ÕÖ�ß~Ô�æ�ÛFÝYÕ5çQô¤Þ�Ý-ß~Ý`Ü�×�Û|Ø~ܤÞmÙNé�Ý�æ=ÝYÝ`Ñ5é�ÝYß~Ú|õmÝ`á0æ&äxÓ�Ô�Õ\ÖTÙ{ß~ÚFÜ�Ô�Ñ��©Ú±Ø~Þxß~Ý`Ü�×�Û|Ø~ܤÔ�ãÀì#Ô�Ñ�Ø~Ýeí(Ù{ß~Û|Ô�Ü�Ú|Õe×�ÛFÙ{Ø�ÚFÔ�ÑmÜ`ç

�Ñ��é��AÉP�AÇ®Íd�_¢å bMV /V@YM$�6/-/ & %)*�/$�"&$2C"!�V G�V�G_' u KJC X V X V X�#$�6'�7[C{HZV NûV%$n;��!0/C"&SV lZV�lM�'(;/$�KJCHZV X V%'È$nN9')( * 3%"&$2C�G_-/KL$�"%�!)�'(<)&�'(<!<!$�;/=2$� ��!;¤3%�/$n-� &$M0(F.{02< 7�;�02KL�t'(<U)*�'(02 �"&$�./"%$� &$�;�3&'�3%�!02;/ _F�02"q %3%0@)*�/'( %3%�!)�.�"&0�)�$� % &$� �C1?�|}RnN� /V{?�)��yV,+d02KL./-�3�V!C.-a02<Èj2k > j2h2h = ?sC�G_0/V/j�C./.UV/k Ö iMA ² ��Ö V/ &SV lZV{lM�'(;/$�KJC X V?@.'(;/02 �C�?�3&0�)*�'( 43&�!)�B;�� 3&$�$�<!$�KL$�;[3% 10a'Z %.1$�)s3%"*'(<:'(./.�"&0['()*�UC�?�./"&�!;/=2$�"�A2-Q$�"%<t'(=/C/G�$�v43a02"�5{C ��Ö�Ö�� V6 &SV lZVÈlM�/'(;/$�KJC7&SV NûV�$n"&-�=2$�"�C�G_-/KL$�"&�!)�'(<: %02<!-�3&�!02;�0(FQ &.{$�)�3%"*'(<­ %3%0�)*�'( %3%�!)�B/;/� 3&$Z$�<!$�KL$�;[3S %7@ %3%$�KL �C7+d02KL./-�3�V1NJ$s3&�/0�5/ �RM.�./<yVNJ$�)*�UVOa;/=/V!C.-a02< � j Ö > ��Ö�Ö k@?*C�.�.UV{j2i Ö A ê h ê V8 &SV lZV/lM�/'(;/$�KJC@|};�=2"&$�5/�!$�;�3% BF�02"Q'�=2$�;/$�"&'(<�./-�"&.{02 &$� %3%0@)&�'( %3%�!)!B;/� 3%$M$�<!$�KL$�;�3QF�02"%Kz-/<t'�3%�!02;UC+d02KL.UV�NJ$s3&��V@R_././<yV@NJ$�)*�UV@OQ;/=2"%=/V!C

-a02< � k2i > ��Ö�Ö�Ö ?sC1./.UV ��Ö A ê = V9 &SV1lM�'(;/$�KJC1?�3%0�)*�'( %3%�!)9B;/� 3%$Z$�<!$�KL$�;�3% �v�� 3%�JKz-�< 3&�!./<!$M"*'(;/5/02K�;/02;CA4ln'(-� & &�t'(;9.�"&02.{$�"%3%�!$� �C� /VOQ;�=2"&=/V{NJ$�)*�UV!C,-a02<yV � j2f > ��Ö�Ö�Ö ?sCG_0/V � C./.UV/j2kMA = h�V: &SV­lM�'(;/$�KJC; V<&q$�5CA=!�02"& &$2C X "%02.'(=['�3%�!02;+0(F_-/;/)�$�"%3&'(�!;[3}7��!;�)�02KL./<!$seû.���7@ &�!)�'(<B %7@ %3%$�KL z-/ %�!;/=�'J %3%0�)*�'( %3%�!)lB/;/� 3&$#$�<!$�KL$�;[3% '(././"%0['()*�UC X ��7@ &�!)�'¤YZC-Q02<yV ��ê2ê > ��Ö�Ö�Ö ?*C ��ê ² A � =�= V> ?1V?!�02 &5/$�"�C@&SV AÕVQAû'(< 3%$�"& �C@&SV X $�"&$A�2CnR^;/02;CA}�!;[3&"%-/ &� E[$�.{02< 7@;/02KL�t'(<M)*�/'(02 JKL$s3&�/0�5±F�02",-/;/)�$�"%3&'(�!;�3}7ð.�"&02.'(=['�3%�!02;ð�!;4+d>­Y &�!KZ-/<t'�3%�!02;/ �C@RM|}R_RiA j2h2h2kMA i Ö�� C =�= 3&�JR_|}RMR�RM$�"%02 &.'()�$Z?@)��!$�;/)�$� MNJ$�$�3%�!;/=L'(;�5�Oae@�/�!6/� 3�C�&q$�;/0/C{G_$sE('(5'�C� ['(;UV Ö A � j�C/j2h2h2k�VB NûV ['("&5'C5{C�+MV A=!�V/?�-UC{lZV O�V"$S'("%;/�t'(5'C5��! �C/?@.{$�)s3%"*'(<�.102< 7@;/02KL�t'(<1)*�'(02 q &02<!-@3&�!02;/ q0(FÈ3%�/$S 43&0�)*�'( 43&�!)�'(5�E[$�)s3%�!02;�$AD@-/'�3&�!02;UC, V/?@)��yV

+d02KL./-�3�V!C.-a02< � ² > j2h2h2j@?sC1./.�V ê���Ö A ê2ê i�VE lZV�'È�!;UC�+nV AF!�V1?@-�C�lZV O_V�$S'("%;/�t'(5'C5��! �C�?�3%0@)&�'( %3%�!)Z %02< E[$�"& _F�02"q3&�/$�OQ-�<!$�"�$AD@-/'�3&�!02;/ �C = ê "&5JR_|}RMR�R_$�"&02 &./'()�$z?@)��!$�;/)�$� SNJ$�$�3%�!;/='(;/5�Oae@�/�!6/� 3�C�&�$�;/0/CG_$sE2'(5'�Cj2h2h2f�VåHG 'QV�N9'�3%�/$�<!�!;UC@NûV 3zVI!�-/ & &'(�!;/�yC[�SV RZV%J1'(;/=/C�?�3%0@)*�/'( %3%�!)S'(././"%0['()*�/$� d3&0�-/;�)�$�"43*'(�!;[3}7KD�-'(;�3%� B)�'�3%�!02;��!;L+d>­Y± %�!Kz-/<t'�3%�!02;/ �C%MON.P�Q1RTS

UWVYX[Z�\]Z ^%_ R Ua`Wb P�c*C.-Q02<yV ê i�C./.UV{j2h Ö A}j ê k�C{j2h2h2f�V� ² 0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�

Page 18: AIAA-2006-2066-173

å�å RZV +MV��P'�7�<!02"�Cd'QV 'aV�lM"&$�$�;UC X V RZV�G�$�v�K,'(;UC�NûV NûV X -�3�520/C2?@02KL$a'(5�E2'(;/)�$�5�)�02;�)�$�.�3% P�!;S5/�! &)�"%$�3%$a'($�"%0@5@7�;'(KL�!)­ &$�;/ %� 3&� E@� 3}7S'(;/'(< 7� %�! �CRM|}R_R X '(.{$�"�j2h2h � A j2f2j Ö C{�!;�R_|}RMR � f�3&��+B02KL.�-�3*'�3%�!02;'(<È>P<!-/�!5�Y_7@;'(KL�!)� e+B02;@F�$�"&$�;/)�$2CR_;'(�/$��!KJC�+dRZC" 2-/;/$�j2h2h � Vå / #Oð�:-/"%=2$�02;�C­YZV X $�<!<!$s3&�!$�"�C� /VÈbd02"%=2=['2'("&5UCÈRM././<!�!)�'�3%�!02;/ �0(FB)�02;�3%�!;�-/02-/ � &$�;/ &� 3%� E@� 3}7�$)D�-'�3%�!02;/ �3&0�Ë/0�v� Sv_� 3%�93&$�KL.{$�"*'�3%-/"&$�A_5/$�A.{$�;/5/$�;�3M./"&02.{$�"%3%�!$� �C�MON.Pgf�hOQ X[`Oi R X[j cakTQ1R�l<m X R `n\ C-Q02<yV =�= C./.UV/k �2� A k2j = C1j2h2h ê Vå 6 &SV AÕV:AÇ'(< 3%$�"% �C;'QVn!_-�7@ &$2C;o_;/)�$�"%3&'(�!;�3}7+'(;/'(< 7� %�! zF�02"ÂË-/�!5�KL$�)*�'(;��!)� Zv_� 3%�8'(././<!�!)�'�3%�!02;/ �CQ�:$�)*�/;/�!)�'(<�"&$�.{02"%3¤|H+dRn?@Op&q$�.{02"43j2h2h2jMA � 0a|H+dRS?@Oq0G_Rn?@Rr':'(;/=2<!$s7s&�$� &$�'("%)*�t+d$�;�3%$�"�C�!_'(KL.�3&02;�C.-PRZC/j2h2h2j�Vå 8 &SV AÕV1AÇ'(< 3&$�"& �C:�:0�vB'("&5/ S 43&0�)*�'( 43&�!)�Ë-/�!5�KL$�)*�'(;��!)� SE@�t'�.{02< 7@;/02KL�t'(<P)*�'(02 9A_|};[E@� 3&$�5UC:R_|}RMR X '(.{$�"�j2h2h ê A h = ��ê CPR_|}RMR = � 43RM$�"%02 &.'()�$Z?@)��!$�;/)�$� _NJ$�$s3&�!;/=,'(;�5�Oae@�/�!6/� 3�C,&q$�;/0/C{G�$�E('(5'�C1j2h2h ê Vå 9du V:AÇ'(;UCdlZV O_V<$S'("%;/�t'(5'C5��! �CQbd$s7[02;/5�A±�!$�;/$�"3AyRM �52$s7û$�e@.'(;/ %�!02;/ 10v!_'(;/5/<!�!;/=�'("%6/� 3&"&'("%7 X YM>P �C; /Va?@)��yV;+d02KL./-�3�V!C X -/6/<!�! %�/$�502;/<!�!;/$[0aYM$�)�$�Kz6{$�"Mj2h2h2f�Vå : G�V@A±�!$�;/$�"�C/�q�/$��/02KL02=2$�;/$�02-/ �)*�'(02 �C/RMKL$�"�V. /VN9'�3%�UV!C"-a02<yVk2h > ��Ö2ê i@?sC{./.�V{i Ö ² A Ö2ê k�Vå > /V RZV ?1V�A±� 343&$sE[$�$�;UC,!�V�bd� u <yC2NJ0@5/$�<!�!;/=�'("%6/� 3&"&'("%7Z-/;/)�$�"%3&'(�!;�3%�!$� d-/ %�!;/=ZlM"*'(K�A4?@)&�/KL�!5�3P.{02< 7�;�02KL�t'(<)*�'(02 �C�R_|}RMRiA}j2h2h2kMA i Ö k�C =�= 3%�RM|}R_RÕR_$�"&02 &./'()�$�?�)��!$�;/)�$� nNJ$�$s3%�!;/=L'(;/5JO­e����!6/� 3�C"&�$�;/0/CG_$sE2'(5'�C, �'(;�V Ö A � j�C{j2h2h2k�Vå B YZV u �!-UC2lZV O�V)$S'("&;/�t'(5/'C5@�! �C��q�/$PA±�!$�;/$�"�A}RM w52$�7n.102< 7@;/02KL�t'(<2)*�/'(02 UF�02"U %3%0@)*�/'( %3%�!)Q5/� T1$�"&$�;[3&�t'(<[$AD@-/'�3&�!02;/ �C�?@|}RMNx /V�?@)��yVd+d02KL./-�3�V!C-a02<yVj = > j2h2h2j@?sC�G_0/V/j�C/./.�V{k ��Ö A k =�= Vå E YZV u �!-�C/YZV%'È-�)�02"�C�+MV A=!�V@?@-UC{lZV O�V$S'("&;/�t'(5'C5��! �C/?�3%0@)&�'( %3%�!)SKL0@5�$�<!�!;�=�0(FRË0�v¤A} 43&"%-/)s3&-/"%$n� 3%$�"&'�3&�!02;� B-/ &�!;/=Z=2$�;/$�"*'(<!����$�5�.{02< 7@;/0�AKL�t'(<1)*�'(02 �C, V�>P<!-/�!5�Oa;/=2"&=/V!C-Q02< � j = > j2h2h2j@?*C1./.UVf � A}f Ö V/ G YZV u �!-UCQlZV O_V<$S'("%;/�t'(5'C5��! �CaNJ0@5�$�<!�!;�=9-/;/)�$�"%3&'(�!;�3}7��!;û %3%$�'(5@7� %3&'�3%$�5/� T1-/ %�!02;�./"%026/<!$�KL �E@�t'J=2$�;/$�"*'(<!����$�5�.102< 7@;/02KL�t'(<d)&�'(02 �C+d02KL./-�3�VNJ$s3%�/0@5� _R_././<yVNJ$�)*��VOQ;/=2"%=/V!C.-a02< ��Ö�� > j2h2h2j@?sC1./.UV = Ö j ² A = Ö = i�V

/ å YZV u �!-UCUlZV O_V�$S'("&;��t'(5'C5@�! �C�NJ0@5/$�<!�!;/=,-/;�)�$�"43*'(�!;[3}7��!;·Ë0�vÕ &�!KZ-/<t'�3%�!02;/ _E@�t'L=2$�;�$�"&'(<!����$�5�.{02< 7@;/02KL�t'(<:)*�'(02 �Cy /V7+d02KL.UV X �[7@ �V!C-a02<yV � i ² > j2h2h ê ?sC1./.UV ��ê ² A � k ² V

/T/ YZV u �!-UC:lZV O_V�$S'("%;/�t'(5'C5��! �C:R�;/$sv� 43&0�)*�'( 43&�!)#'(././"%0['()*�û3%0�3%"*'(;/ %�!$�;[3��/$�'�3�)�02;�5/-/)s3&�!02;�KL0�5/$�<!�!;/=#v_� 3%��-/;/)�$�"%3&'(�!;�3}7[C­|};�3�Vz /V!_$�'�3_N9'( & ��:"*'(;�V!C-a02< = k > j2h2h ê ?sC{./.�V = k2i � A = k Ö2ê V

/Y6 YZV u �!-UC" V"!_$� %3%�'�E[$�;�C�!_�!=2�CA}02"%5/$�"q)�02<!<!0�)�'�3&�!02;�KL$�3%�/0�5/ BF�02"q5/� T1$�"&$�;[3&�t'(<U$)D�-'�3%�!02;/ _v�� 3%��"&'(;/5/02KÑ�!;�./-�3% �C{?�|}RnN� /V{?�)��yV,+d02K�A./-�3�V!C"-a02<Èj ² > j2h2h2f@?sC�G�0/V ê C/.�.UV �2�2� iMA �2��ê�Ö V

� i¤0(F � iRMKL$�"%�!)�'(;,|};/ 43&� 3%-�3%$Z0(FPR_$�"&02;'(-�3%�!)� n'(;/5�RM 43&"%02;'(-�3%�!)�