What comes out of your instrument?

In order to understand quickly these principles, let’s take a simple tube. This one will resonate on specific frequencies called natural harmonics.
Musicians are used to hear natural harmonics as following: (C) C G C E G Bb C …
Except that blowing into the instrument closes one side of the tube. It is now resonating on one over two natural harmonic. The harmonics that stay are: (C) G E Bb D …
This is where we can see the magical effect of the mouthpiece and the bell.
The bell tends to heighten the lower harmonics, when the mouthpiece will lower the higher harmonics. Thanks to these effects, the harmonics are getting closer to the natural harmonics.
The study "From cylinder to trumpet" shows this path.

Therefore we are not playing with natural harmonics but with something close which depends a lot on the mouthpiece, the lead pipe and the bell.
Indeed, some manufacturer are using modifications of the lead pipe to correct the pitch of the instrument.

 Mouthpiece

Then, we understand the importance of the mouthpiece which is more than the place where sound is created, it can also change the pitch of the instrument.
According to Wiener Klangstil (Institute of Music Acoustics), changing just the mouthpiece shank can shift some harmonics up to 30 cents!
 

Materials and thickness

B.B Ninob wrote a study presented on Joël Eymard website which shows that thickness or materials can have a strong influence on the pitch.
The material can also disperse more or less the energy, which can explain that we feel less sound volume with plastic mouthpiece than metallic ones.

Acoustic simulation software:

• TUTT (Windows / Gratuit) - de B.B Ninob, available on his page (with lots of interesting studies) : http://la.trompette.free.fr/Ninob/Ninob.php
• WIAT (Python / Domaine public) - by Antoine Lefebvre at McGill University : http://www.music.mcgill.ca/caml/wiat/

Simulation softwares as COMSOL MultiPhysics can also be used. However, these have to be paid and physical and acoustical knowledge is required. Also, results can be more precised but are taking a lot more time to calculate.

References :

Brass instrument (lip reed) acoustics: an introduction - http://newt.phys.unsw.edu.au/jw/brassacoustics.html
Producing a harmonic sequence of notes with a trumpet -http://hyperphysics.phy-astr.gsu.edu/hbase/music/brassa.html
Examination of the Influence of Different Mouthpiece Forms on the Resonance Behavior of Trumpets - https://iwk.mdw.ac.at/?page_id=86&sprache=2
Étude de l'influence des différentes formes d'embouchures sur les résonances d'une trompette. - http://la.trompette.free.fr/Vienne/influence_embouchure.htm
La justesse d'une trompette. - http://translate.google.com/translate?hl=en&sl=fr&u=http://la.trompette.free.fr/justesse.htm
Propagation du son dans un tuyau à parois souples - http://translate.google.com/translate?hl=en&sl=fr&u=http://la.trompette.free.fr/Ninob/Tube_non_rigide.html