Mollified Collocation Method with Localhp-Refinement for Steady Thermoelastic Problems Graduate Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Aerospace Engineering from Bandung Institute of Technology By Syahrir Ginanjar NIM: 23623005 (Aerospace Engineering Graduate Program) INSTITUT TEKNOLOGI BANDUNG FEBRUARY 2025 ABSTRACT Mollified Collocation Method with Localhp-Refinement for Steady Thermoelastic Problems By Syahrir Ginanjar NIM: 23623005 (Aerospace Engineering Graduate Program) This study introduces a framework for solving thermo-mechanical problems by using Mollified Collocation Method (MCM), that is, method that employs Mollified Basis Functions in the Point Collocation Method (PCM). By directly evaluating Partial Differential Equations (PDEs) in their strong form, offering a more straightforward and computationally efficient alternative to weak form approach. The strong form evaluation enables precise enforcement of boundary and governing equations, making the method particularly interesting for steep gradient problem solutions. A key feature of MCM is the flexibility of its basis functions, which possess arbitrary smoothness and polynomial degrees. This characteristic allows for seamless implementation of local refinement strategies, includingp- (locally increasing polynomial degree),h-refinement (locally reducing element size) andhp-refinement (combining both strategies). These refinements allow for control over accuracy and resolution of the solution, focusing computational effort in regions with complex physical phenomena, such as high-temperature gradients. The proposed framework demonstrates significant potential for addressing thermoelasticity problem with locally steep gradients by combining mathematical rigor with practical efficiency. This study numerically investigates the effects of local refinements on solving high-gradient thermo and thermoelasticity problems, aiming to balance solution accuracy and computational cost. Results indicate that local refinement strategies significantly enhance convergence rates and reduce errors, as measured by relative error discretisation in terms ofL 2 -norm andH 1 -seminorm error. Keywords: Localhp-refinement, Mollified basis functions, Point collocation method, Thermoelasticity. i ABSTRAK Metode KolokasiMollifieddenganhp-Penyempurnaan Lokal untuk Permasalahan Termoelastis Tunak Oleh Syahrir Ginanjar NIM: 23623005 (Program Studi Magister Teknik Dirgantara) Studi ini memperkenalkan kerangka kerja untuk menyelesaikan masalah termo- mekanis dengan Metode Kolokasi dengan Fungsi BasisMollified. Metode ini menggunakan Fungsi BasisMollifieddalam Metode Kolokasi Titik, di mana Persamaan Diferensial Parsial (PDE) dievaluasi secara langsung dalam bentuk kuat. Pendekatan ini menawarkan alternatif yang lebih sederhana dan efisien secara komputasi dibandingkan pendekatan bentuk lemah. Evaluasi dalam bentuk kuat memungkinkan penerapan yang presisi terhadap persamaan batas dan umum, sehingga metode ini cocok untuk menangani masalah dengan gradien tajam. Keunggulan dari metode ini adalah fleksibilitas fungsi batas, yang memiliki kehalusan dan derajat polynomial yang dapat diatur secara bebas. Karakteristik ini memungkinkan implementasi strategi penyempurnaan lokal, antara lainp-penyempurnaan (meningkatkan derajat polinomial secara lokal),h-penyempurnaan (mengurangi ukuran elemen secara lokal), dan hp-penyempurnaan (menggabungkan kedua strategi tersebut). Strategi penyempurnaan lokal ini memungkinkan pengendalian akurasi dan resolusi solusi, sehingga komputasi dapat difokuskan pada area dengan fenomena fisik kompleks, seperti gradien suhu tinggi. Kerangka kerja ini menunjukkan potensi yang signifikan dalam menyelesaikan masalah termoelastisitas yang memiliki gradien tinggi di daerah tertentu dengan menggabungkan ketelitian matematis dan efisiensi praktis, yang bertujuan untuk menginvestigasi secara numerik pengaruh strategi penyempurnaan lokal dalam menyelesaikan masalah termo yang memiliki gradien tinggi dan termo-mekanis yang bertujuan untuk menyeimbangkan akurasi solusi dan beban komputasi. Hasil menunjukkan bahwa strategi penyempurnaan lokal meningkatkan laju konvergensi dan mengurangi kesalahan, yang diukur melalui kesalahan diskretisasi relatif dalam normaL 2 dan seminormaH 1 . Kata kunci:hp-penyempurnaan lokal, Fungsi basisMollified, Metode kolokasi titik, Termoelastis ii Mollified Collocation Method with Localhp-Refinement for Steady Thermoelastic Problems Approval Page By Syahrir Ginanjar NIM: 23623005 (Aerospace Engineering Graduate Program) Institut Teknologi Bandung Approved Advisors On 11 February 2025 First Advisor Second Advisor Prof. Ir. Lavi Rizki Zuhal, Ph.D. Pramudita Satria Palar, S.T., M.T., Ph.D NIP. 197301162006041001 NIP. 198806222012121001 Third Advisor University of Glasgow Dr. Eky Febrianto THESIS USAGE GUIDELINES This Thesis is registered and available in the library of Institut Teknologi Bandung (ITB) and is open for public with copyright on the author and by following the rules of intellectual property rights (HaKI) in ITB. Literature references are allowed to be recorded but citations are only allowed with permission from the author. The citation to this Tesis in Indonesian can be written as follows: Ginanjar, Syahrir. (2025): Metode Kolokasi Mollified denganhp- Penyempurnaan Lokal untuk Permasalahan Termoelastis Tunak, Tesis Program Magister, Institut Teknologi Bandung and in English can be written as follows: Ginanjar, Syahrir. (2025):Mollified Collocation Method with Localhp- Refinement for Steady Thermoelastic Problems, Master’s Thesis, Institut Teknologi Bandung To copy or to publish a part or the entirety of this Thesis is only allowed with the permission from Director of Graduate School, ITB. iv To EveryoneStrugglingtoAchieveTheir Dreams ”Verily, with every difficulty there is relief” - Al-Quran, As-Sharh:5-6 To those who stood by my side, unwavering in the storm- Through trials, doubt, and burdens worn, In every moment, on every stepping stone. This journey is not mine alone. It isours. With mywarmestheart,Thank You. ”Percayalah, sepanjang kita punya mimpi, punya rencana, walaupun kecil tapi masuk akal, tidak boleh sekalipun rasa sedih, rasa tak berguna itu datang mengganggu pikiran.” - TERE LIYE Learnfrom Yesterday,Livefor Today,Hopefor Tomorrow. The important thing is to never stop asking questions. - ALBERT EINSTEIN ACKNOWLEDGEMENT All praise, gratitude, and thanks be to God Almighty for His mercy, who has blessed me throughout my life and guided me to always choose kindness. I would like to express my endless thanks to my parents, Nanang Kamal and R Ida Sri Widianingsih, for their unconditional support and love throughout my life, especially in achieving my dreams and shaping me into who I am. Hopefully, my hard work on this project can bring you happiness and joy to our future life. I am also deeply grateful to my brothers, Syahrul Shiddiq and Andhiki Supono, as well as my sister, Syahrillita Oktarina, for their tremendous support, especially for their logistical assistance in fulfilling my physiological needs. Special acknowledgement goes to Prof. Ir. Lavi Rizki Zuhal, Ph.D., Pramudita Satria Palar, S.T., M.T., Ph.D., and Dr. Eky Febrianto, who have placed immense trust in me, allowed me to explore ideas, inspired me to not only improve my research skills in computational mechanics field but also to understand the meaning of life and the importance of life planning. Working under their supervision has been both inspiring and challenging throughout this Graduate Thesis research project. I would also like to express my gratitude to my excellent discussion partners, Dewangga Alfarisy and I Made Wiragunarsa, who have helped me gain a deeper understanding of physical phenomena, computational algorithms, and insights from their experiences in computational mechanics research. Lastly, to everyone I have met and interacted with, I extend my heartfelt appreciation for your presence in my life. I believe that nothing in this world happens by coincidence; rather, every encounter has a purpose in helping me find deeper meaning in life. Thank you for bringing that meaning whether I have realized it consciously or unconsciously or will come to understand it in the future. I hope that every small act of kindness you have shared will return to you in even greater abundance.